The fact that most fundamental laws of physics, notably those of electrodynamics and quan- tum mechanics, have been formulated in mathematical language as ...
Nonlinear physics (solitons, chaos, discrete breathers) N. Theodorakopoulos Konstanz, June 2006
Contents
Foreword
vi
1 Background: Hamiltonian mechanics 1.1 Lagrangian formulation of dynamics . . . . . . . . . . . . . . 1.2 Hamiltonian dynamics . . . . . . . . . . . . . . . . . . . . . . 1.2.1 Canonical momenta . . . . . . . . . . . . . . . . . . . 1.2.2 Poisson brackets . . . . . . . . . . . . . . . . . . . . . 1.2.3 Equations of motion . . . . . . . . . . . . . . . . . . . 1.2.4 Canonical transformations . . . . . . . . . . . . . . . . 1.2.5 Point transformations . . . . . . . . . . . . . . . . . . 1.3 Hamilton-Jacobi theory . . . . . . . . . . . . . . . . . . . . . 1.3.1 Hamilton-Jacobi equation . . . . . . . . . . . . . . . . 1.3.2 Relationship to action . . . . . . . . . . . . . . . . . . 1.3.3 Conservative systems . . . . . . . . . . . . . . . . . . . 1.3.4 Separation of variables . . . . . . . . . . . . . . . . . . 1.3.5 Periodic motion. Action-angle variables . . . . . . . . 1.3.6 Complete integrability . . . . . . . . . . . . . . . . . . 1.4 Symmetries and conservation laws . . . . . . . . . . . . . . . 1.4.1 Homogeneity of time . . . . . . . . . . . . . . . . . . . 1.4.2 Homogeneity of space . . . . . . . . . . . . . . . . . . 1.4.3 Galilei invariance . . . . . . . . . . . . . . . . . . . . . 1.4.4 Isotropy of space (rotational symmetry of Lagrangian) 1.5 Continuum field theories . . . . . . . . . . . . . . . . . . . . . 1.5.1 Lagrangian field theories in 1+1 dimensions . . . . . . 1.5.2 Symmetries and conservation laws . . . . . . . . . . . 1.6 Perturbations of integrable systems . . . . . . . . . . . . . . .
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1 1 1 1 2 2 2 3 3 3 4 4 5 5 6 6 7 7 7 7 8 8 8 9
2 Background: Statistical mechanics 2.1 Scope . . . . . . . . . . . . . . . 2.2 Formulation . . . . . . . . . . . . 2.2.1 Phase space . . . . . . . . 2.2.2 Liouville’s theorem . . . . 2.2.3 Averaging over time . . . 2.2.4 Ensemble averaging . . . 2.2.5 Equivalence of ensembles 2.2.6 Ergodicity . . . . . . . . .
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FPU paradox The harmonic crystal: dynamics . . . . . . . . . . . . . . . . . . . . . . . . . The harmonic crystal: thermodynamics . . . . . . . . . . . . . . . . . . . . . The FPU numerical experiment . . . . . . . . . . . . . . . . . . . . . . . . . .
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3 The 3.1 3.2 3.3
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Contents 4 The Korteweg - de Vries equation 4.1 Shallow water waves . . . . . . . . . . . . . . . . . . . 4.1.1 Background: hydrodynamics . . . . . . . . . . 4.1.2 Statement of the problem; boundary conditions 4.1.3 Satisfying the bottom boundary condition . . . 4.1.4 Euler equation at top boundary . . . . . . . . . 4.1.5 A solitary wave . . . . . . . . . . . . . . . . . . 4.1.6 Is the solitary wave a physical solution? . . . . 4.2 KdV as a limiting case of anharmonic lattice dynamics 4.3 KdV as a field theory . . . . . . . . . . . . . . . . . . 4.3.1 KdV Lagrangian . . . . . . . . . . . . . . . . . 4.3.2 Symmetries and conserved quantities . . . . . . 4.3.3 KdV as a Hamiltonian field theory . . . . . . .
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5 Solving KdV by inverse scattering 5.1 Isospectral property . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Lax pairs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Inverse scattering transform: the idea . . . . . . . . . . . . . . . . . . . . . . 5.4 The inverse scattering transform . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.1 The direct problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.2 Time evolution of scattering data . . . . . . . . . . . . . . . . . . . . . 5.4.3 Reconstructing the potential from scattering data (inverse scattering problem) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.4 IST summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5 Application of the IST: reflectionless potentials . . . . . . . . . . . . . . . . . 5.5.1 A single bound state . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5.2 Multiple bound states . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.6 Integrals of motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.6.1 Lemma: a useful representation of a(k) . . . . . . . . . . . . . . . . . 5.6.2 Asymptotic expansions of a(k) . . . . . . . . . . . . . . . . . . . . . . 5.6.3 IST as a canonical transformation to action-angle variables . . . . . .
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6 Solitons in anharmonic lattice dynamics: 6.1 The model . . . . . . . . . . . . . . . 6.2 The dual lattice . . . . . . . . . . . . 6.2.1 A pulse soliton . . . . . . . . 6.3 Complete integrability . . . . . . . . 6.4 Thermodynamics . . . . . . . . . . .
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the Toda . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
lattice . . . . . . . . . . . . . . . . . . . .
7 Chaos in low dimensional systems 7.1 Visualization of simple dynamical systems . . . . . 7.1.1 Two dimensional phase space . . . . . . . . 7.1.2 4-dimensional phase space . . . . . . . . . . 7.1.3 3-dimensional phase space; nonautonomous of freedom . . . . . . . . . . . . . . . . . . . 7.2 Small denominators revisited: KAM theorem . . . 7.3 Chaos in area preserving maps . . . . . . . . . . . 7.3.1 Twist maps . . . . . . . . . . . . . . . . . . 7.3.2 Local stability properties . . . . . . . . . . 7.3.3 Poincar´e-Birkhoff theorem . . . . . . . . . . 7.3.4 Chaos diagnostics . . . . . . . . . . . . . . 7.3.5 The standard map . . . . . . . . . . . . . . 7.3.6 The Arnold cat map . . . . . . . . . . . . . 7.3.7 The baker map; Bernoulli shifts . . . . . . .
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Contents
7.4
7.3.8 The circle map. Frequency locking . . . . . . . . . . . . . . . . . . . . 66 Topology of chaos: stable and unstable manifolds, homoclinic points . . . . . 67
8 Solitons in scalar field theories 8.1 Definitions and notation . . . . . . . . . . . . 8.1.1 Lagrangian, continuum field equations 8.2 Static localized solutions (general KG class) . 8.2.1 General properties . . . . . . . . . . . 8.2.2 Specific potentials . . . . . . . . . . . 8.2.3 Intrinsic Properties of kinks . . . . . . 8.2.4 Linear stability of kinks . . . . . . . . 8.3 Special properties of the SG field . . . . . . . 8.3.1 The Sine-Gordon breather . . . . . . . 8.3.2 Complete Integrability . . . . . . . . .
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9 Atoms on substrates: the Frenkel-Kontorova model 9.1 The Commensurate-Incommensurate transition . . . . . . . . . . . . 9.1.1 The continuum approximation . . . . . . . . . . . . . . . . . 9.1.2 The special case ² = 0: kinks and antikinks . . . . . . . . . . 9.1.3 The general case ² > 0: the soliton lattice . . . . . . . . . . . 9.2 Breaking of analyticity . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2.1 FK ground state as minimizing periodic orbit of the standard 9.2.2 Small amplitude motion . . . . . . . . . . . . . . . . . . . . . 9.2.3 Free end boundary conditions . . . . . . . . . . . . . . . . . . 9.3 Metastable states: spatial chaos as a model of glassy structure . . .
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10 Solitons in magnetic chains 10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2 Classical spin dynamics . . . . . . . . . . . . . . . . . . . . . . 10.2.1 Spin Poisson brackets . . . . . . . . . . . . . . . . . . . 10.2.2 An alternative representation . . . . . . . . . . . . . . . 10.3 Solitons in ferromagnetic chains . . . . . . . . . . . . . . . . . . 10.3.1 The continuum approximation . . . . . . . . . . . . . . 10.3.2 The classical, isotropic, ferromagnetic chain . . . . . . . 10.3.3 The easy-plane ferromagnetic chain in an external field 10.4 Solitons in antiferromagnets . . . . . . . . . . . . . . . . . . . . 10.4.1 Continuum dynamics . . . . . . . . . . . . . . . . . . . . 10.4.2 The isotropic antiferromagnetic chain . . . . . . . . . . 10.4.3 Easy axis anisotropy . . . . . . . . . . . . . . . . . . . . 10.4.4 Easy plane anisotropy . . . . . . . . . . . . . . . . . . . 10.4.5 Easy plane anisotropy and symmetry-breaking field . . .
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11 Solitons in conducting polymers 11.1 Peierls instability . . . . . . . . . . . . . . . . . 11.1.1 Electrons decoupled from the lattice . . 11.1.2 Electron-phonon coupling; dimerization 11.2 Solitons and polarons in (CH)x . . . . . . . . . 11.2.1 A continuum approximation . . . . . . . 11.2.2 Dimerization . . . . . . . . . . . . . . . 11.2.3 The soliton . . . . . . . . . . . . . . . . 11.2.4 The polaron . . . . . . . . . . . . . . . .
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Contents 12 Solitons in nonlinear optics 12.1 Background: Interaction of light with matter, Maxwell-Bloch equations . . . 12.1.1 Semiclassical theoretical framework and notation . . . . . . . . . . . . 12.1.2 Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.2 Propagation at resonance. Self-induced transparency . . . . . . . . . . . . . . 12.2.1 Slow modulation of the optical wave . . . . . . . . . . . . . . . . . . . 12.2.2 Further simplifications: Self-induced transparency . . . . . . . . . . . 12.3 Self-focusing off-resonance. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.3.1 Off-resonance limit of the MB equations . . . . . . . . . . . . . . . . . 12.3.2 Nonlinear terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.3.3 Space-time dependence of the modulation: the nonlinear Schr¨odinger equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.3.4 Soliton solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
122 122 122 123 123 123 125 126 126 127 128 129
13 Solitons in Bose-Einstein Condensates 132 13.1 The Gross-Pitaevskii equation . . . . . . . . . . . . . . . . . . . . . . . . . . . 132 13.2 Propagating solutions. Dark solitons . . . . . . . . . . . . . . . . . . . . . . . 132 14 Unbinding the double helix 14.1 A nonlinear lattice dynamics approach . . . . . . . . . 14.1.1 Mesoscopic modeling of DNA . . . . . . . . . . 14.1.2 Thermodynamics . . . . . . . . . . . . . . . . . 14.2 Nonlinear structures (domain walls) and DNA melting 14.2.1 Local equilibria . . . . . . . . . . . . . . . . . . 14.2.2 Thermodynamics of domain walls . . . . . . . .
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15 Pulse propagation in nerve cells: the Hodgkin-Huxley model 15.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.2 The Hodgkin-Huxley model . . . . . . . . . . . . . . . . . . . . . . . 15.2.1 The axon membrane as an array of electrical circuit elements 15.2.2 Ion transport via distinct ionic channels . . . . . . . . . . . . 15.2.3 Voltage clamping . . . . . . . . . . . . . . . . . . . . . . . . . 15.2.4 Ionic channels controlled by gates . . . . . . . . . . . . . . . . 15.2.5 Membrane activation is a threshold phenomenon . . . . . . . 15.2.6 A qualitative picture of ion transport during nerve activation 15.2.7 Pulse propagation . . . . . . . . . . . . . . . . . . . . . . . .
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16 Localization and transport of energy in proteins: The Davydov soliton 16.1 Background. Model Hamiltonian . . . . . . . . . . . . . . . . . . . . . . 16.1.1 Energy storage in C=O stretching modes. Excitonic Hamiltonian 16.1.2 Coupling to lattice vibrations. Analogy to polaron . . . . . . . . 16.2 Born-Oppenheimer dynamics . . . . . . . . . . . . . . . . . . . . . . . . 16.2.1 Quantum (excitonic) dynamics . . . . . . . . . . . . . . . . . . . 16.2.2 Lattice motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.2.3 Coupled exciton-phonon dynamics . . . . . . . . . . . . . . . . . 16.3 The Davydov soliton . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.3.1 The heavy ion limit. Static Solitons . . . . . . . . . . . . . . . . 16.3.2 Moving solitons . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Contents 17 Nonlinear localization in translationally invariant systems: 17.1 The Sievers-Takeno conjecture . . . . . . . . . . . . . 17.2 Numerical evidence of localization . . . . . . . . . . . 17.2.1 Diagnostics of energy localization . . . . . . . . 17.2.2 Internal dynamics . . . . . . . . . . . . . . . . 17.3 Towards exact discrete breathers . . . . . . . . . . . .
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breathers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
A Impurities, disorder and localization A.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.1.1 Electrons . . . . . . . . . . . . . . . . . . . . . . . . . . . A.1.2 Phonons . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.2 A single impurity . . . . . . . . . . . . . . . . . . . . . . . . . . . A.2.1 An exact result . . . . . . . . . . . . . . . . . . . . . . . . A.2.2 Numerical results . . . . . . . . . . . . . . . . . . . . . . . A.3 Disorder . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.3.1 Electrons in disordered one-dimensional media . . . . . . A.3.2 Vibrational spectra of one-dimensional disordered lattices Bibliography
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Foreword The fact that most fundamental laws of physics, notably those of electrodynamics and quantum mechanics, have been formulated in mathematical language as linear partial differential equations has resulted historically in a preferred mode of thought within the physics community - a “linear” theoretical bias. The Fourier decomposition - an admittedly powerful procedure of describing an arbitrary function in terms of sines and cosines, but nonetheless a mathematical tool - has been firmly embedded in the conceptual framework of theoretical physics. Photons, phonons, magnons are prime examples of how successive generations of physicists have learned to describe properties of light, lattice vibrations, or the dynamics of magnetic crystals, respectively, during the last 100 years. This conceptual bias notwithstanding, engineers or physicists facing specific problems in classical mechanics, hydrodynamics or quantum mechanics were never shy of making particular approximations which led to nonlinear ordinary, or partial differential equations. Therefore, by the 1960’s, significant expertise had been accumulated in the field of nonlinear differential and/or integral equations; in addition, major breakthroughs had occurred on some fundamental issues related to chaos in classical mechanics (Poincar´e, Birkhoff, KAM theorems). Due to the underlying linear bias however, this substantial progress took unusually long to find its way to the core of physical theory. This changed rapidly with the advent of electronic computation and the new possibilities of numerical visualization which accompanied it. Computer simulations became instrumental in catalyzing the birth of nonlinear science. This set of lectures does not even attempt to cover all areas where nonlinearity has proved to be of importance in modern physics. I will however try to describe some of the basic concepts mainly from the angle of condensed matter / statistical mechanics, an area which provided an impressive list of nonlinearly governed phenomena over the last fifty years starting with the Fermi-Pasta-Ulam numerical experiment and its subsequent interpretation by Zabusky and Kruskal in terms of solitons (“paradox turned discovery”, in the words of J. Ford). There is widespread agreement that both solitons and chaos have achieved the status of theoretical paradigm. The third concept introduced here, localization in the absence of disorder, is a relatively recent breakthrough related to the discovery of independent (nonlinear) localized modes (ILMs), a.k.a. “discrete breathers”. Since neither the development of the field nor its present state can be described in terms of a unique linear narrative, both the exact choice of topics and the digressions necessary to describe the wider context are to a large extent arbitrary. The latter are however necessary in order to provide a self-contained presentation which will be useful for the non-expert, i.e. typically the advanced undergraduate student with an elementary knowledge of quantum mechanics and statistical physics.
Konstanz, June 2006
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1
Background: Hamiltonian mechanics
Consider a mechanical system with s degrees of freedom. The state of the mechanical system at any instant of time is described by the coordinates {Qi (t), i = 1, 2, · · · , s} and the corresponding velocities {Q˙ i (t)}. In many applications that I will deal with, this may be a set of N point particles which are free to move in one spatial dimension. In that particular case s = N and the coordinates are the particle displacements. The rules for temporal evolution, i.e. for the determination of particle trajectories, are described in terms of Newton’s law - or, in the more general Lagrangian and Hamiltonian formulations. The more general formulations are necessary in order to develop and/or make contact with fundamental notions of statistical and/or quantum mechanics.
1.1 Lagrangian formulation of dynamics The Lagrangian is given as the difference between kinetic and potential energies. For a particle system interacting by velocity-independent forces L({Qi , Q˙ i }) = T − V (1.1) s X 1 T = mi Q˙ 2i 2 i=1 V
= V ({Qi }, t)
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where an explicit dependence of the potential energy on time has been allowed. Lagrangian dynamics derives particle trajectories by determining the conditions for which the action integral Z t
S(t, t0 ) =
dτ L({Qi , Q˙ i , τ })
(1.2)
t0
has an extremum. The result is
d ∂L ∂L = dt ∂ Q˙ i ∂Qi which for Lagrangians of the type (1.2) becomes ¨i = − mi Q
∂V ∂Qi
(1.3)
(1.4)
i.e. Newton’s law.
1.2 Hamiltonian dynamics 1.2.1 Canonical momenta Hamiltonian mechanics, uses instead of velocities, the canonical momenta conjugate to the coordinates {Qi }, defined as ∂L Pi = . (1.5) ∂ Q˙ i
1
1 Background: Hamiltonian mechanics In the case of (1.2) it is straightforward to express the Hamiltonian function (the total energy) H = T + V in terms of P ’s and Q0 s. The result is H({Pi , Qi }) =
s X Pi2 + V ({Qi }) . 2mi
(1.6)
I=1
1.2.2 Poisson brackets Hamiltonian dynamics is described in terms of Poisson brackets {A, B} =
¾ s ½ X ∂A ∂B ∂A ∂B − ∂Qi ∂Pi ∂Pi ∂Qi i=1
(1.7)
where A, B are any functions of the coordinates and momenta. The momenta are canonically conjugate to the coordinates because they satisfy the relationships
1.2.3 Equations of motion According to Hamiltonian dynamics, the time evolution of any function A({Pi , Qi }, t) is determined by the linear differential equations dA ∂A A˙ ≡ = {A, H} + . dt ∂t
(1.8)
where the second term denotes any explicit dependence of A on the time t. Application of (1.8) to the cases A = Pi and A = Qi respectively leads to P˙i Q˙ i
= {Pi , H} = {Qi , H}
(1.9)
which can be shown to be equivalent to (1.4). The time evolution of the Hamiltonian itself is governed by µ ¶ dH ∂H ∂V = = . (1.10) dt ∂t ∂t
1.2.4 Canonical transformations Hamiltonian formalism important because the “symplectic”structure of equations of motion (from Greek συµπλ²κω = crosslink - of momenta & coordinate variables -) remains invariant under a class of transformations obtained by a suitable generating function (“canonical”transformations). Example, transformation from old coordinates & momenta {P, Q} to new ones {p, q}, via a generating function F1 (Q, q, t) which depends on old and new coordinates (but not on old and new momenta - NB there are three more forms of generating functions - ): Pi pi K
∂F1 (q, Q, t) ∂Qi ∂F1 (q, Q, t) = − ∂qi ∂F1 = H+ ∂t =
2
(1.11)
1 Background: Hamiltonian mechanics new coordinates are obtained by solving the first of the above eqs., and new momenta by introducing the solution in the second. It is straightforward to verify that the dynamics remains form-invariant in the new coordinate system, i.e. p˙i q˙i
= {pi , K} = {qi , K}
(1.12)
and
∂K(p, q, t) dK(p, q, t) = . (1.13) dt ∂t Note that if there is no explicit dependence of F1 on time, the new Hamiltonian K is equal to the old H.
1.2.5 Point transformations A special case of canonical transformations are point transformations, generated by X F2 (Q, p, t) = fi (Q, t)pi ; (1.14) i
New coordinates depend only on old coordinates - not on old momenta; in general new momenta depend on both old coordinates and momenta. A special case of point transformations are orthogonal transformations, generated by X F2 (Q, p) = aik Qk pi (1.15) i,k
where a is an orthogonal matrix. It follows that X aik Qk qi = k
pi
=
X
aik Pk
.
(1.16)
k
Note that, in the case of orthogonal transformations, coordinates transform among themselves; so do the momenta. Normal mode expansion is an example of (1.16).
1.3 Hamilton-Jacobi theory 1.3.1 Hamilton-Jacobi equation Hamiltonian dynamics consists of a system of 2N coupled first-order linear differential equations. In general, a complete integration would involve 2N constants (e.g. the initial values of coordinates and momenta). Canonical transformations enable us to play the following game:1 Look for a transformation to a new set of canonical coordinates where the new Hamiltonian is zero and hence all new coordinates and momenta are constants of the motion.2 Let (p, q) be the set of original momenta and coordinates in eqs of previous section, 1 Hamilton-Jacobi
theory is not a recipe for integration of the coupled ODEs; nor does it in general lead to a more tractable mathematical problem. However, it provides fresh insight to the general problem, including important links to quantum mechanics and practical applications on how to deal with mechanical perturbations of a known, solved system. 2 Does this seem like too many constants? We will later explore what independent constants mean in mechanics, but at this stage let us just note that the original mathematical problem of integrating the 2N Hamiltonian equations does indeed involve 2N constants.
3
1 Background: Hamiltonian mechanics (α, β) the set of new constant momenta and coordinates generated by the generating function F2 (q, α, t) which depends on the original coordinates and the new momenta. The choice of K ≡ 0 in (1.11) means that ∂F2 ∂F2 ∂F2 + H(q1 , · · · qs ; ,···, ; t) = 0 ∂t ∂q1 ∂qs
.
(1.17)
Suppose now that you can [miraculously] obtain a solution of the first-order -in general nonlinear-PDE (1.17), F2 = S(q, α, t). Note that the solution in general involves s constants {αi , i = 1, · · · , s}. The s + 1st constant involved in the problem is a trivial one, because if S is a solution, so is S + A, where A is an arbitrary constant. It is now possible to use the defining equation of the generating function F2 βi =
∂S(q, α, t) ∂αi
(1.18)
to obtain the new [constant] coordinates {βi , i = 1, · · · , s}; finally, “turning inside out”(1.18) yields the trajectories qj = qj (α, β, t) .
(1.19)
In other words, a solution of the Hamilton-Jacobi equation (1.17) provides a solution of the original dynamical problem.
1.3.2 Relationship to action It can be easily shown that the solution of the Hamilton-Jacobi equation satisfies dS =L , dt or
Z
(1.20) t
S(q, α, t) − S(q, α, t0 ) =
dτ L(q, q, ˙ τ)
(1.21)
t0
where the r.h.s involves the actual particle trajectories; this shows that the solution of the Hamilton-Jacobi equation is indeed the extremum of the action function used in Lagrangian mechanics.
1.3.3 Conservative systems If the Hamiltonian does not depend explicitly on time, it is possible to separate out the time variable, i.e. S(q, α, t) = W (q, α) − λ0 t
(1.22)
where now the time-independent function W (q) (Hamilton’s characteristic function) satisfies ¶ µ ∂W ∂W ,···, = λ0 H q1 , · · · qs ; ∂q1 ∂qs
,
(1.23)
and involves s − 1 independent constants, more precisely, the s constants α1 , · · · αs depend on λ0 .
4
1 Background: Hamiltonian mechanics
1.3.4 Separation of variables The previous example separated out the time coordinate from the rest of the variables of the HJ function. Suppose q1 and ∂W ∂q1 enter the Hamiltonian only in the combination ³ ´ φ1 q1 , ∂W ∂q1 . The Ansatz 0
W = W1 (q1 ) + W (q2 , · · · , qs )
(1.24)
in (1.23) yields à H
µ ¶! 0 0 ∂W ∂W ∂W1 q2 , · · · qs ; ,···, ; φ1 q1 , = λ0 ∂q2 ∂qs ∂q1
since (1.25) must hold identically for all q, we have µ ¶ ∂W1 φ1 q1 , = ∂q1 Ã ! 0 0 ∂W ∂W H q2 , · · · qs ; ,···, ; λ1 = ∂q2 ∂qs
;
(1.25)
λ1 λ0
.
(1.26)
The process can be applied recursively if the separation condition holds. Note that cyclic ∂W1 coordinates lead to a special case of separability; if q1 is cyclic, then φ1 = ∂W ∂q1 = ∂q1 , and hence W1 (q1 ) = λ1 q1 . This is exactly how the time coordinate separates off in conservative systems (1.23). Complete separability occurs if we can write Hamilton’s characteristic function - in some set of canonical variables - in the form X W (q, α) = Wi (qi , α1 , · · · , αs ) . (1.27) i
1.3.5 Periodic motion. Action-angle variables Consider a completely separable system in the sense of (1.27). The equation pi =
∂S ∂Wi (qi , α1 , · · · , αs ) = ∂qi ∂qi
(1.28)
provides the phase space orbit in the subspace (qi , pi ). Now suppose that the motion in all subspaces {(qi , pi ), i = 1, · · · , s} is periodic - not necessarily with the same period. Note that this may mean either a strict periodicity of pi , qi as a function of time (such as occurs in the bounded motion of a harmonic oscillator), or a motion of the freely rotating pendulum type, where the angle coordinate is physically significant only mod 2π. The action variables are defined as I I 1 ∂Wi (qi , α1 , · · · , αs ) 1 pi dqi = dqi (1.29) Ji = 2π 2π ∂qi and therefore depend only on the integration constants, i.e. they are constants of the motion. If we can “turn inside out”(1.29), we can express W as a function of the J’s instead of the α’s. Then we can use the function W as a generating function of a canonical transformation to a new set of variables with the J’s as new momenta, and new “angle”coordinates θi =
∂Wi (qi , J1 , · · · , Js ) ∂W = ∂Ji ∂Ji
5
.
(1.30)
1 Background: Hamiltonian mechanics In the new set of canonical variables, Hamilton’s equations of motion are J˙i θ˙i
= 0 ∂H(J) = ≡ ωi (J) . ∂Ji
(1.31)
Note that the Hamiltonian cannot depend on the angle coordinates, since the action coordinates, the J’s, are - by construction - all constants of the motion. In the set of action-angle coordinates, the motion is as trivial as it can get: Ji θi
= const = ωi (J) t + const
.
(1.32)
1.3.6 Complete integrability A system is called completely integrable in the sense of Liouville if it can be shown to have s independent conserved quantities in involution (this means that their Poisson brackets, taken in pairs, vanish identically). If this is the case, one can always perform a canonical transformation to action-angle variables.
1.4 Symmetries and conservation laws A change of coordinates, if it reflects an underlying symmetry of physical laws, will leave the form of the equations of motion invariant. Because Lagrangian dynamics is derived from an action principle, any such infinitesimal change which changes the particle coordinates qi → qi0 q˙i → q˙i0
= =
qi + ²fi (q, t) q˙i + ²f˙i (q, t)
(1.33)
and adds a total time derivative to the Lagrangian, i.e. L0 = L + ²
dF dt
,
(1.34)
will leave the equations of motion invariant. On the other hand, the transformed Lagrangian will generally be equal to L0 ({qi0 , q˙i0 })
= L({qi0 , q˙i0 }) = = =
and therefore the quantity
s · X ∂L
¸ ∂L ˙ L({qi , q˙i }) + ²fi + ²fi ∂qi ∂ q˙i i=1 µ ¶ ¸ s · X ∂L ˙ d ∂L ²fi + ²fi L({qi , q˙i }) + dt ∂ q˙i ∂ q˙i i=1 µ ¶ s X d ∂L L({qi , q˙i }) + fi dt ∂ q˙i i=1 s X ∂L fi − F ∂ q˙i i=1
will be conserved. Such underlying symmetries of classical mechanics are:
6
(1.35)
1 Background: Hamiltonian mechanics
1.4.1 Homogeneity of time L0 = L(t + ²) = L(t) + ²dL/dt, i.e. F = L; furthermore, qi0 = qi (t + ²) = qi + ²q˙i , i.e. fi = q˙i . As a result, the quantity s X ∂L H= q˙i − L (1.36) ∂ q˙i i=1 (Hamiltonian) is conserved.
1.4.2 Homogeneity of space The transformation qi → qi + ² (hence fi = 1) leaves the Lagrangian invariant (F = 0). The conserved quantity is s X ∂L P = (1.37) ∂ q˙i i=1 (total momentum).
1.4.3 Galilei invariance The transformation qi → qi − ²t (hence fi = −t) does not generally change the potential energy (if it depends only P on relative particle positions). It adds to the kinetic energy a term −²P , i.e. F = − mi qi . The conserved quantity is s X
mi qi − P t
(1.38)
i=1
(uniform motion of the center of mass).
1.4.4 Isotropy of space (rotational symmetry of Lagrangian) Let the position of the ith particle in space be represented by the vector coordinate ~qi . Rotation around an axis parallel to the unit vector n ˆ is represented by the transformation ~qi → ~qi + ²f~i where f~i = n ˆ × ~qi . The change in kinetic energy is ²
X
˙ ~q˙ i · f~i = 0
.
i
If the potential energy is a function of the interparticle distances only, it too remains invariant under a rotation. Since the Lagrangian is invariant, the conserved quantity (1.35) is s s X X ∂L ~ · fi = mi ~q˙ i · (ˆ n × ~qi ) = n ˆ · I~ , ∂ ~q˙ i=1
i
i=1
where I~ =
s X
mi (~qi × ~q˙ i )
i=1
is the total angular momentum.
7
(1.39)
1 Background: Hamiltonian mechanics
1.5 Continuum field theories 1.5.1 Lagrangian field theories in 1+1 dimensions Given a Lagrangian in 1+1 dimensions, Z L = dxL(φ, φx , φt )
(1.40)
where the Lagrangian density L depends only on the field φ and first space and time derivatives, the equations of motion can be derived by minimizing the total action Z S = dtdxL (1.41) and have the form
d dt
µ
∂L ∂φt
¶ +
d dx
µ
∂L ∂φx
¶ −
∂L =0 ∂φ
.
(1.42)
1.5.2 Symmetries and conservation laws The form (1.42) remains invariant under a transformation which adds to the Lagrangian density a term of the form ²∂µ Jµ (1.43) where the implied summation is over µ = 0, 1, because this adds only surface boundary terms to the action integral. If the transformation changes the field by δφ, and the derivatives by δφx , δφt , the same argument as in discrete systems leads us to conclude that µ ¶ ∂L ∂L ∂L dJ0 dJ1 δφ + δφx + δφt = ² + (1.44) ∂φ ∂φx ∂φt dt dx which can be transformed, using the equations of motion, to µ ¶ µ ¶ µ ¶ d ∂L ∂L d ∂L ∂L dJ0 dJ1 δφ + δφt + δφ + δφx = ² + dt ∂φt ∂φt dx ∂φx ∂φx dt dx
(1.45)
Examples: 1. homogeneity of space (translational invariance) x → δφ = δφt δφx
= =
δL
=
x+² φ(x + ²) − φ(x) = φx ² φt (x + ²) − φt (x) = φxt ² φx (x + ²) − φx (x) = φxx ² dL dL δx = ² ⇒ J1 = L , J0 = 0 dx dx
.
Eq. (1.45) becomes µ ¶ µ ¶ d ∂L ∂L d ∂L ∂L dL φx + φxt + φx + φxx = dt ∂φt ∂φt dx ∂φx ∂φx dx or
d dt
µ
∂L φx ∂φt
¶ +
d dx
8
µ
∂L φx − L ∂φx
(1.46)
(1.47)
¶ =0
;
(1.48)
1 Background: Hamiltonian mechanics integrating over all space, this gives Z dx
∂L φx ≡ −P ∂φt
(1.49)
i.e. the total momentum is a constant. 2. homogeneity of time t δφ δφt δφx δL
→ t+² = φ(t + ²) − φ(t) = φt ² = φt (t + ²) − φt (t) = φtt ² = φx (t + ²) − φx (t) = φxt ² dL dL = δt = ² ⇒ J0 = L , J1 = 0 dt dt
.
Eq. (1.45) becomes µ ¶ µ ¶ d ∂L ∂L d ∂L ∂L dL φt + φtt + φt + φtx = dt ∂φt ∂φt dx ∂φx ∂φx dt or
d dt
µ
¶ µ ¶ ∂L d ∂L φt − L + φt = 0 ∂φt dx ∂φx
integrating over all space, this gives · ¸ Z ∂L dx φt − L ≡ H ∂φt
;
(1.50)
(1.51)
(1.52)
(1.53)
i.e. the total energy is a constant. 3. Lorentz invariance
1.6 Perturbations of integrable systems Consider a conservative Hamiltonian system H0 (J) which is completely integrable, i.e. it possesses s independent integrals of motion. Note that I use the action-angle coordinates, so that H0 is a function of the (conserved) action coordinates Jj . The angles θj are cyclic variables, so they do not appear in H0 . Suppose now that the system is slightly perturbed, by a time-independent perturbation Hamiltonian µH1 (µ ¿ 1) A sensible question to ask is: what exactly happens to the integrals of motion? We know of course that the energy of the perturbed system remains constant since H1 has been assumed to be time independent. But what exactly happens to the other s − 1 constants of motion? The question was first addressed by Poincar´e in connection with the stability of the planetary system. He succeeded in showing that there are no analytic invariants of the perturbed system, i.e. that it is not possible, starting from a constant Φ0 of the unperturbed system, to construct quantities Φ = Φ0 (J) + µΦ1 (J, θ) + µ2 Φ2 (J, θ)
,
(1.54)
where the Φn ’s are analytic functions of J, θ, such that {Φ, H} = 0
9
(1.55)
1 Background: Hamiltonian mechanics holds, i.e. Φ is a constant of motion of the perturbed system. The proof of Poincar´e’s theorem is quite general. The only requirement on the unperturbed Hamiltonian is that it should have functionally independent frequencies ωj = ∂H0 /∂Jj . Although the proof itself is lengthy and I will make no attempt to reproduce it, it is fairly straightforward to see where the problem with analytic invariants lies. To second order in µ, the requirement (1.55) implies {Φ0 + µΦ1 + µ2 Φ2 , H0 + µH1 } = 0 {Φ0 , H0 } + µ ({Φ1 , H0 } + {Φ0 , H1 }) + µ2 ({Φ2 , H0 } + {Φ1 , H1 }) = 0 . The coefficients of all powers must vanish. Note that the zeroth order term vanishes by definition. The higher order terms will do so, provided {Φ1 , H0 } {Φ2 , H0 }
= =
−{Φ0 , H1 } −{Φ1 , H1 }
(1.56) .
The process can be continued iteratively to all orders, by requiring {Φn , H0 } = −{Φn+1 , H1 } .
(1.57)
Consider the lowest-order term generated by (1.57). Writing down the Poisson brackets gives ¶ ¶ s µ s µ X X ∂Φ1 ∂H0 ∂Φ1 ∂H0 ∂Φ0 ∂H1 ∂Φ0 ∂H1 − =− − . (1.58) ∂θi ∂Ji ∂Ji ∂θi ∂θi ∂Ji ∂Ji ∂θi j=1 j=1 The second term on the left hand side and the first term on the right-hand side vanish because the θ’s are cyclic coordinates in the unperturbed system. The rest can be rewritten as s s X X ∂Φ1 ∂Φ0 ∂H1 ωi (J) = . (1.59) ∂θi ∂Ji ∂θi j=1 j=1 For notational simplicity, let me now restrict myself to the case of two degrees of freedom. The perturbed Hamiltonian can be written in a double Fourier series X H1 = An1 ,n2 (J1 , J2 ) cos(n1 θ1 + n2 θ2 ) . (1.60) n1 ,n2
Similarly, one can make a double Fourier series ansatz for Φ1 , X Φ1 = Bn1 ,n2 (J1 , J2 ) cos(n1 θ1 + n2 θ2 ) .
(1.61)
n1 ,n2
Now apply (1.59) to the case Φ0 (J) = J1 . Using the double Fourier series I obtain Bn(J11,n) 2 =
n1 An ,n n1 ω1 + n2 ω2 1 2
,
(1.62)
which in principle determines the first-order term in the µ expansion of the constant of motion J10 which should replace J1 in the new system. It is straightforward to show, using the same process for J2 , that the perturbed Hamiltonian can be written in terms of the new constants J10 as H = H0 (J10 , J20 ) + O(µ2 ) . (1.63) Unfortunately, what looks like the beginning of a systematic expansion suffers from a fatal flaw. If the frequencies are functionally independent, the denominator in (1.62) will in general vanish on a denumerably infinite number of surfaces in phase space. This however means that Φ1 cannot be an analytic function of J1 , J2 . Analytic invariants are not possible. All integrals of motion - other than the energy - are irrevocably destroyed by the perturbation.
10
2
Background: Statistical mechanics
2.1 Scope Classical statistical mechanics attempts to establish a systematic connection between microscopic theory which governs the dynamical motion of individual entities (atoms, molecules, local magnetic moments on a lattice) and the macroscopically observed behavior of matter. Microscopic motion is described - depending on the particular scale of the problem - either by classical or quantum mechanics. The rules of macroscopically observed behavior under conditions of thermal equilibrium have been codified in the study of thermodynamics. Thermodynamics will tell you which processes are macroscopically allowed, and can establish relationships between material properties. In principle, it can reduce everything everything which can be observed under varying control parameters ( temperature, pressure or other external fields) to the “equation of state”which describes one of the relevant macroscopic observables as a function of the control parameters. Deriving the form of the equation of state is beyond thermodynamics. It needs a link to microscopic theory - i.e. to the underlying mechanics of the individual particles. This link is provided by equilibrium statistical mechanics. A more general theory of non-equilibrium statistical mechanics is necessary to establish a link between non-equilibrium macroscopic behavior (e.g. a steady state flow) and microscopic dynamics. Here I will only deal with equilibrium statistical mechanics.
2.2 Formulation A statistical description always involves some kind of averaging. Statistical mechanics is about systematically averaging over hopefully nonessential details. What are these details and how can we show that they are nonessential? In order to decide this you have to look first at a system in full detail and then decide what to throw out - and how to go about it consistently.
2.2.1 Phase space An Hamiltonian system with s degrees of freedom is fully described at any given time if we know all coordinates and momenta, i.e. a total of 2s quantities (=6N if we are dealing with point particles moving in three-dimensional space). The microscopic state of the system can be viewed as a point, a vector in 2s dimensional space. The dynamical evolution of the system in time can be viewed as a motion of this point in the 2s dimensional space (phase space). I will use the shorthand notation Γ ≡ (qi , pi , i = 1, s) to denote a point in phase space. More precisely, Γ(t) will denote a trajectory in phase space with the initial condition Γ(t0 ) = Γ0 . 1 1 Note
that trajectories in phase space do not cross. A history of a Hamiltonian system is determined by differential equations which are first-order in time, and is therefore reversible - and hence unique.
11
2 Background: Statistical mechanics
2.2.2 Liouville’s theorem Consider an element of volume dσ0 in phase space; the set of trajectories starting at time t0 at some point Γ0 ∈ dσ0 lead, at time t to points Γ ∈ dσ. Liouville’s theorem asserts that dσ = dσ0 . (invariance of phase space volume). The proof consists of showing that the Jacobi determinant ∂(q, p) D(t, t0 ) ≡ (2.1) ∂(q 0 , p0 ) corresponding to the coordinate transformation (q 0 , p0 ) ⇒ (q, p), is equal to unity. Using general properties of Jacobians ¯ ¯ ∂(p) ¯¯ ∂(q, p) ∂(q, p) ∂(q 0 , p) ∂(q) ¯¯ = · = · (2.2) ∂(q 0 , p0 ) ∂(q 0 , p) ∂(q 0 , p0 ) ∂(q 0 ) ¯p=const ∂(p0 ) ¯q=const and ¯ ¶¯ s µ X ∂ q˙i ∂ p˙i ¯¯ ∂D(t, t0 ¯¯ = + ¯ ∂t ¯t=t0 ∂qi ∂pi ¯ i=1
= t=t0
¶ s µ X ∂2H ∂2H − =0 ∂qi ∂pi ∂pi ∂qi i=1
,
(2.3)
and noting that D(t0 , t0 ) = 1, it follows that D(t, t0 ) = 1 at all times.
2.2.3 Averaging over time Consider a function A(Γ) of all coordinates and momenta. If you want to compute its longtime average under conditions of thermal equilibrium, you need to follow the state of the system over a long time, record it, evaluate the function A at each instant of time, and take a suitable average. Following the trajectory of the point in phase space allows us to define a long-time average Z 1 T ¯ A = lim dtA[Γ(t)] . (2.4) T →∞ T 0 Since the system is followed over infinite time this can then be regarded as a true equilibrium average. More on this later.
2.2.4 Ensemble averaging On the other hand, we could consider an ensemble of identically prepared systems and attempt a series of observations. One system could be in the state Γ1 , another in the state Γ2 . Then perhaps we could determine the distribution of states ρ(Γ), i.e. the probability ρ(Γ)δΓ, that the state vector is in the neighborhood (Γ, Γ + δΓ). The average of A in this case would be Z < A >= dΓρ(Γ)A(Γ) (2.5) Note that since ρ is a probability distribution, its integral over all phase space should be normalized to unity: Z dΓρ(Γ) = 1 (2.6) A distribution in phase space must obey further restrictions. Liouville’s theorem states that if we view the dynamics of a Hamiltonian system as a flow in phase space, elements of volume are invariant - in other words the fluid is incompressible: ∂ d ρ(Γ, t) = {ρ, H} + ρ(Γ, t) = 0 dt ∂t
12
.
(2.7)
2 Background: Statistical mechanics For a stationary distribution ρ(Γ) - as one expects to obtain for a system at equilibrium {ρ, H} = 0
,
(2.8)
i.e. ρ can only depend on the energy2 . This is a very severe restriction on the forms of allowed distribution functions in phase space. Nonetheless it still allows for any functional dependence on the energy. A possible choice (Boltzmann) is to assume that any point on the phase space hypersurface defined by H(Γ) = E may occur with equal probability. This corresponds to 1 ρ(Γ) = δ {H(Γ) − E} (2.9) Ω(E) where
Z Ω(E) =
dΓ δ {H(Γ) − E}
(2.10)
is the volume of the hypersurface H(Γ) = E. This is the microcanonical ensemble. Other choices are possible - e.g. the canonical (Gibbs) ensemble defined as ρ(Γ) =
1 e−βH(Γ) Z(β)
where the control parameter β can be identified with the inverse temperature and Z Z(β) = dΓe−βH(Γ)
(2.11)
(2.12)
is the classical partition function.
2.2.5 Equivalence of ensembles The choice of ensemble, although it may appear arbitrary, is meant to reflect the actual experimental situation. For example, the Gibbs ensemble may be “derived”- in the sense that it can be shown to correspond to a small (but still macroscopic) system in contact with a much larger “reservoir”of energy - which in effect holds the smaller system at a fixed temperature T = 1/β. Ensembles must - and to some extent can - be shown to be equivalent, in the sense that the averages computed using two different ensembles coincide if the control parameters are appropriately chosen. For example a microcanonical average of a function A(Γ) over the energy surface H(Γ) = ² will be equal with the canonical average at a certain temperature T if we choose ² to be equal to the canonical average of the energy at that temperature, i.e. < A(Γ) >micro =< A(Γ) >canon if ² =< H(Γ) >canon . ² T T If ensembles can be shown to be equivalent to each other in this sense, we do not need to perform the actual experiment of waiting and observing the realization of a large number of identical systems as postulated in the previous section. We can simply use the most convenient ensemble for the problem at hand as a theoretical tool for calculating averages. In general one uses the canonical ensemble, which is designed for computing average quantities as functions of temperature.
2.2.6 Ergodicity The usage of ensemble averages - and therefore of the whole edifice of classical statistical mechanics - rests on the implicit assumption that they somehow coincide with the more physical time averages. Since the various ensembles can be shown to be equivalent (cf. 2 or
- in principle - on other conserved quantities; in dealing with large systems it may well be necessary to account for other macroscopically conserved quantities in defining a proper distribution function.
13
2 Background: Statistical mechanics above), it would be sufficient to provide a microscopic foundation for the ensemble most directly accessible to Hamiltonian dynamics, i.e. the microcanonical ensemble. The ergodic hypothesis states that 1 lim T →∞ T
Z 0
T
1 dtA [Γ(t)] = Ω(E)
Z dΓ δ {H (Γ) − E} A(Γ)
(2.13)
i.e. that time averages and microcanonical averages coincide. This requires that as a point Γ moves around phase space, it spends - on the average - equal times on equal areas of the energy hypersurface (recall that the phase point must stay on the energy hypersurface because H(Γ) is a constant of the motion. This seems like a strong & rather nonobvious assertion; Boltzmann had a rough time when he tried to sell it as a plausible basis for the emerging theory of statistical mechanics. One of the reasons why (2.13) appears implausible was a theorem proved by Poincar´e which stated that if a Hamiltonian system is bounded, its trajectory in phase space - although not allowed to cross itself - will return arbitrarily close to any point already traveled, provided one waits long enough. Therefore, even statistically improbable microstates may recur. The catch is that Poincar´e recurrence times for rare events in large systems are of order eN and may easily exceed the age of the universe[1]. In fact, ergodicity was later shown by Birkhoff to hold if the energy surface cannot be divided in two invariant regions of nonzero measure (i.e. regions such that the trajectories in phase space always remain in one of them). The energy surface is then called metrically indecomposable. One way this decomposition could occur might be if further integrals of motion are present.
14
3
The FPU paradox
3.1 The harmonic crystal: dynamics Consider a chain of N point particles, each of unit mass. Each of the particles is coupled to its nearest neighbor via a harmonic spring of unit strength; let Qi be the displacement of the ith particle; the Hamiltonian (1.6) is N
H(P, Q) =
N
1X 2 1X 2 P + (Qi+1 − Qi ) 2 i=1 i 2 i=0
,
(3.1)
where the canonical momenta are Pi = Q˙ i and the end particles are held fixed, i.e. Q0 = QN +1 = 0 (NB: N degrees of freedom). The Fourier decomposition r Qi
=
Pi
=
µ ¶ N iπλ 2 X sin Aλ N +1 N +1 λ=1 r µ ¶ N 2 X iπλ sin Bλ N +1 N +1
(3.2)
λ=1
is a canonical transformation (cf. above) to a new set of coordinate and momenta {Aλ , Bλ }. (NB: exercise, check properties, orthogonality, trigonometric sums, boundary conditions satisfied). In this new set of coordinates, the Hamiltonian can be written as H=
N X
Hλ ≡
λ=1
N ¢ 1 X¡ 2 Bλ + Ω2λ A2λ 2
where
½ Ω2λ
(3.3)
λ=1
= 4 sin
2
πλ 2(N + 1)
¾ .
(3.4)
This is a case of a separable Hamiltonian, where Hamilton-Jacobi theory can be trivially applied, i.e. µ ¶2 1 ∂Wλ 1 (3.5) + Ω2λ A2λ = ²λ ∀ λ = 1, · · · , N. 2 ∂Aλ 2 where each ²λ is a constant representing the energy stored in the λth normal mode. The substitution √ 2²λ sin θ¯λ (3.6) Aλ = Ωλ transforms (3.5) to
∂Wλ 2²λ cos2 θ¯λ = Ωλ ∂ θ¯λ
.
(3.7)
The corresponding action variable Jλ =
1 2π
I Bλ dAλ =
15
1 2π
I
∂Wλ dAλ ∂Aλ
(3.8)
3 The FPU paradox can now be evaluated as Jλ =
Z
1 2²λ 2π Ωλ
2π
dθ¯λ cos2 θ¯λ =
0
²λ Ωλ
(3.9)
by integrating over a full cycle of the substitution variable θ¯λ . The Hamiltonian can be rewritten in terms of the action variables H=
X
²λ =
X
λ
Ωλ Jλ
(3.10)
λ
The angle variables conjugate to the action variables can be found from (1.30 θλ =
∂Wλ (Aλ , Jλ ) ∂Jλ
.
(3.11)
It can be shown explicitly that θj = θ¯j . The Hamiltonian equations in action-angle variables are J˙λ θ˙λ
= 0 ∂H = = Ωλ ∂Jλ
,
(3.12)
i.e. the Ωλ ’s are the natural frequencies of the normal modes. Note that we did not need the explicit form of the solution of the Hamilton-Jacobi equation to derive this. More explicitly, the time evolution of the normal mode coordinates is µ Aλ (t) =
2Jλ Ωλ
¶1/2
¡ ¢ sin Ωλ t + θλ0
,
(3.13)
with an analogous expression for the momenta Bλ . In the action-angle representation, the 2N constants of integration are the N action variables {Jλ } and the N initial phases {θλ0 }.
3.2 The harmonic crystal: thermodynamics The average energy of the harmonic chain at any given temperature T is given by the canonical average Z 1 < H >= dΓe−H(Γ)/T H(Γ) , (3.14) Z where Z is the partition function Z Z(T ) =
dΓe−H(Γ)/T
.
(3.15)
It is possible to transform the integrals in both numerator and denominator of (3.14) to action-angle coordinates (cf. previous section). Because of the separability property of the Hamiltonian, the denominator splits into product over all N normal modes Z=
N Y λ=1
16
Zλ
(3.16)
3 The FPU paradox where Z Zλ
Z
∞
=
dJλ 0
2π
dθλ e−Ωλ Jλ /T
0
2πT Ωλ
=
(3.17)
whereas the numerator transforms to is a sum of the form N X Y Zλ0 Nλ λ=1
λ0 6=λ
∞
Z
(3.18)
where Z Nλ
= 0
= It follows that < H >=
N X λ=1
2π
dJλ 2πT 2 Ωλ
< ²λ >=
dθλ e−Ωλ Jλ /T Ωλ Jλ
0
.
(3.19)
N X
Nλ /Zλ =
λ=1
N X
T = NT
,
(3.20)
λ=1
i.e. each the average energy which corresponds to each degree of freedom is equal to T (equipartition property). The “statistical mechanics of the harmonic chain” has a fundamental flaw: although canonical averages are straightforward to obtain, there is obviously no basis for assuming ergodicity - in the presence of N integrals of motion. Now, this might not be a serious problem if one could argue that a tiny generic perturbation, as might arise from e.g. a small nonlinearity of the interactions, could drive the system away from complete integrability, and into an ergodic regime. If this turned out to be the case, one could still argue that the computed canonical averages reflect the intrinsic thermodynamic properties of the harmonic chain, in the “programmatic” sense of statistical mechanics. Fermi, Pasta and Ulam decided to put this implicit assumption to a numerical test.
3.3 The FPU numerical experiment Fermi, Pasta and Ulam (FPU[2]) investigated the Hamiltonian H(P, Q) =
N −1 N −1 N −1 1 X 2 1 X α X 2 3 Pi + (Qi+1 − Qi ) + (Qi+1 − Qi ) 2 i=1 2 i=0 3 i=0
,
(3.21)
where the canonical momenta are Pi = Q˙ i and the end particles are held fixed, i.e. Q0 = QN = 0. Their work - undertaken as a suitable “test” problem for one of the very first electronic computers, the Los Alamos “MANIAC”- is considered as the first numerical experiment. In other words, it is the first case where physicists observed and analyzed the numerical output of Newton’s equations, rather than the properties of a mechanical system described by these same equations. The dynamics of the Hamiltonian (3.21) was studied as an initial value problem; the initial configuration was a half-sine wave Qi = sin(iπ/N ), with N = 32 and all particles at rest; the nonlinearity parameter was chosen as α = 1/4. Energy was thus pumped at the lowest
17
3 The FPU paradox
Figure 3.1: The quantity plotted is the energy (kinetic plus potential in each of the first four √
modes). The time is given in thousands of computational cycles. Each cycle is 1/2 2 of the natural time unit. The initial form of the string was a single sine wave (mode 1). The energy of the higher modes never exceeded 6% of the total. (from [2]).
Fourier mode, λ = 1, in the notation of (). The objective of the experiment was to study the energies stored in the first few Fourier modes, i.e. the quantities Hλ ≡ where
r Aλ =
´ 1 ³ ˙2 Aλ + Ω2λ A2λ 2
µ ¶ N 2 X iπλ sin Qi N i=1 N
(3.22)
(3.23)
as a function of time, i.e. to test the onset of equipartition. Note that the decomposition P of the total energy in Fourier modes is not exact - but as long as α stays small, H ≈ λ Hλ will hold. Fig. 3.1 shows the time dependence of the energies of the first four modes. After an initial redistribution, all of the energy (within 3%) returns to the lowest mode. The energy residing in higher modes never exceeded 6 % of the total. Longer numerical studies have shown the return of the energy to the initial mode to be a periodic phenomenon; the period is about 157 times the period of the lowest mode. The phenomenon is known as FPU recurrence. The results of a more recent numerical study on FPU recurrence[3] are summarized in Fig. 3.2. The Hamiltonian (3.21) is fairly generic. In fact, the original FPU paper describes a further study with quartic, rather than cubic, anharmonicities which exhibits similar behavior. FPU recurrence has been shown to be a robust phenomenon. The upshot of those exhaustive numerical observations is that anharmonic corrections to the Hamiltonian, contrary to the original expectation which held them as agents that might help establish ergodicity, actually appear to generate new forms of approximately periodic behavior. The process of under-
18
3 The FPU paradox
Figure 3.2: FPU recurrence time, divided by N 3 vs a scaling variable R = α(E/N )1/2 N 2 where E/N ≈ [πB/(2N )]2 is the energy density. Typical values used by FPU correspond to R À 1. The asymptotic regime is well described by the relationship Tr /N 3 = R−1/2 (from Ref. [3]).
standing the source of this behavior - also known as the FPU paradox - and relating it to other manifestations of nonlinearity [4] has led to a profound change in theoretical physics.
19
4
The Korteweg - de Vries equation
4.1 Shallow water waves Original context: Wave motion in shallow channels, cf. Scott-Russell1 Mathematical description due to Korteweg and deVries (KdV [6]). The equation arises in wide variety of physical contexts (e.g. plasma physics, anharmonic lattice theory). Hence it counts as one of the “canonical” soliton equations. Long waves (typical length l) in a shallow channel l À h. Small amplitude (¿ h) waves (weak nonlinearity) Two-dimensional fluid flow (motion in lateral dimension of channel neglected) x: horizontal direction, y: vertical direction
4.1.1 Background: hydrodynamics Fluid velocity
~ ≡ uˆ V x + v yˆ
(4.1)
Equations of (Eulerian) incompressible fluid dynamics • continuity equation • Euler equation
~ =0 ∇·V
(4.2)
~ ∂V ~ · ∇)V ~ = − 1 ∇p + ~g + (V ∂t ρ
(4.3)
where ~g = −g yˆ plus • irrotational flow (no vortices) ~ =0⇒V ~ = ∇Φ ∇×V
.
(4.4)
Using vector identity
~ · ∇)V ~ = 1 ∇V 2 − V ~ × (∇ × V ~) (V (4.5) 2 in (4.3) (only first term survives due to (4.4) ), and (4.4) in (4.2) transforms hydrodynamics equations to 1 “I
was observing the motion of a boat which was rapidly drawn along a narrow channel by a pair of horses, when the boat suddenly stopped - not so the mass of water in the channel which it had put in motion; it accumulated round the prow of the vessel in a state of violent agitation, then suddenly leaving it behind, rolled forward with great velocity, assuming the form of a large solitary elevation, a rounded, smooth and well-defined heap of water, which continued its course along the channel apparently without change of form or diminution of speed. I followed it on horseback, and overtook it still rolling on at a rate of some eight or nine miles an hour, preserving its original figure some thirty feet long and a foot to a foot and a half in height. Its height gradually diminished, and after a chase of one or two miles I lost it in the windings of the channel. Such, in the month of August 1834, was my first chance interview with that singular and beautiful phenomenon which I have called the Wave of translation.”[5]
20
4 The Korteweg - de Vries equation 1. continuity 4Φ = 0 2. Euler
,
(4.6)
p ∂Φ 1 + (∇Φ)2 + + gy = 0 ∂t 2 ρ
.
(4.7)
4.1.2 Statement of the problem; boundary conditions The above eqs (4.6) and (4.7) must now be solved subject to the boundary conditions 1. bottom: no vertical motion of the fluid v(x, y = 0) = 0
∀x
(4.8)
2. top: free surface defined as y = h + η(x, t).
(4.9)
Velocity of free boundary coincides with fluid velocity, dy dt
=
v
=
∂η ∂η dx + ∂t ∂x dt ∂η ∂η + u ∂t ∂x
hence (4.10)
holds at the free surface. The solution will involve two steps: first, find a general class of solutions of (4.6) which satisfy the bottom BC (4.8), and then use this general class to determine the height profile (4.9) by demanding that the Euler equation (4.7) be satisfied at the free surface, where p = 0 holds. The Euler equation can then be used to determine the pressure at any point.
4.1.3 Satisfying the bottom boundary condition Consider the general form of an expansion (the height O(h) is small in a sense which will be made precise below) of the type
The conditions as
∂u ∂y
=
u = v =
f (x) + f1 (x)y + f2 (x)y 2 + f3 (x)y 3 + · · · g1 (x)y + g2 (x)y 2 + g3 (x)y 3 + · · · .
∂v ∂x
∂u ∂x
and
(4.11)
∂v = − ∂y imposed by (4.6) can now be written, respectively,
f1 + 2f2 y + 3f3 y 2 = g1x y + g2x y 2
(4.12)
and fx + f1 y + f2 y 2 = −g1 − 2g2 y − 3g3 y 2
(4.13)
from which f1 2f2 3f3
= 0
(4.14)
= =
(4.15) (4.16)
21
g1x g2x
4 The Korteweg - de Vries equation and fx f1x f2x
= = =
−g1 −2g2 −3g3
(4.17) (4.18) (4.19)
follow. Using the second set in the first, results in f1 = 0, 2f2 = −fxx , 2f3 = −1/2f1xx (= 0); it follows that g2 = 0 and g3 = −1/3f2x = 1/3!fxxx . Collecting terms, u v
1 = f − fxx y 2 + O(y 4 ) 2 1 = −fx y + fxxx y 3 . 3!
(4.20) (4.21)
4.1.4 Euler equation at top boundary Set p = 0 in (4.7) and differentiate with respect to x: ∂u 1 ∂ 2 ∂η + (u + v 2 ) + g =0 ∂t 2 ∂x ∂t
.
(4.22)
The problem is now to solve the system of coupled differential equations (4.22) and (4.10) using the expressions (4.20) and (4.21). Key: follow the scale of variation of the physical quantities involved. First note that if the water height is not much different from h (small nonlinearity), it will be useful to set η = ²h¯ η (4.23) Note ² is not a parameter of the problem. It simply serves as a “tag” to let us keep track of scales. At the end we will have to check the consistency of the assumptions and approximations made. According to our assumption, the length scale on which the fluid profile varies along the x direction is of the order l À h. In order to incorporate this assumption in the approximation, I define a rescaled variable via x = l¯ x . (4.24) √ Dimensional consideration determine a natural velocity scale c = gh. The motion should be slow with respect to that scale - in agreement with small amplitude variations of the profile. In other words, we expect u ¿ c. Note that from the leading orders of (4.20) and (4.21)it follows that v is typically of order h/l ≡ δ smaller than u. It is therefore reasonable to rescale f u
= ²cf¯ = ²c¯ u
(4.25) (4.26)
v
= δ²c¯ v .
(4.27)
Finally I use a rescaled time t = t¯ l/c
.
(4.28) 2
With these rescalings, keeping lowest order terms, i.e. of O(²) and O(δ ), the rescaled equations (4.20) and (4.21) become - on the surface u ¯ = v¯ =
1 f¯ − δ 2 f¯x¯x¯ 2 1 −(1 + ²¯ η )f¯x¯ + δ 2 f¯x¯x¯x¯ 6
22
(4.29) ;
(4.30)
4 The Korteweg - de Vries equation accordingly, the top boundary condition (4.10) and the Euler equation (4.22) transform to 1 f¯x¯ + η¯t¯ + ²(f¯η¯)x¯ − δ 2 f¯x¯x¯x¯ = 0 6 ² 1 f¯t¯ + η¯x¯ + (f¯2 )x¯ − δ 2 f¯x¯x¯t¯ = 0 2 2
(4.31) .
(4.32)
First we note that in the absence of nonlinearity (² = 0) and dispersion (δ = 0), free wave propagation with unit velocity (in dimensionless units) occurs; in that (zeroth) order, f¯ = η¯. But of course this is hypothetical because δ and ² are not parameters of the problem - they just help us keep track of things! However, the zeroth order approximation is useful in the sense that it suggests a coordinate transformation which absorbs the fastest time dependence; let ξ τ
x ¯ − t¯ ²t¯ .
= =
(4.33) (4.34)
Keeping terms to first order in ² and δ 2 , we use the property η¯x¯ = η¯t¯ =
η¯ξ −¯ ηξ + η¯τ ²
(4.35) (4.36)
(which holds for f¯ as well) transform the system (4.32) to 1 f¯ξ − η¯ξ + ²¯ ητ + ²(¯ η 2 )ξ − δ 2 η¯ξξξ 6 1 ² 2 ¯ η )ξ + δ 2 η¯ξξξ −fξ + η¯ξ + ²¯ ητ + (¯ 2 2
=
0
=
0
(4.37) .
(4.38)
where we have used the property f¯ = η¯ in terms which contain ² or δ 2 factors. The sum of (4.38) is 3 2 1 2²¯ ητ + ²(¯ η )ξ + δ 2 η¯ξξξ = 0 . (4.39) 2 3 The three terms in (4.39) will be of the same order if δ 2 = O(²), i.e. if the nonlinearity balances the dispersion. We choose ² = δ 2 /6. Note that the choice must be tested at the end to check whether it satisfies the original requirements (small amplitude, long waves). With this choice and the substitution η¯ = 4φ I arrive at the “canonical” KdV form, φτ + 6φφξ + φξξξ = 0
.
(4.40)
4.1.5 A solitary wave At this stage, without recourse to advanced mathematical techniques, it is possible to follow the path of KdV and look for special, exact, propagating solutions of (4.40) of the type φ(s), where s = ξ − λτ . (4.40) becomes −λφs + 3(φ2 )s + φsss = 0
(4.41)
which has an obvious first integral −λφ + 3φ2 + φss = const.
(4.42)
If we are looking for solutions which vanish at infinity (lims→∞ φ(s) = 0 and lims→∞ φs (s) = 0) the constant will be zero, i.e. φss = λφ − 3φ2 =
23
d 1 2 ( λφ − φ3 ) dφ 2
(4.43)
4 The Korteweg - de Vries equation Multiplying both sides by 2φs we can integrate once more, obtaining φ2s = λφ2 − 2φ3
(4.44)
where the integration constant must vanish once again (cf. above). Note that, if a solution exists, the parameter λ must be > 0 and φ < λ/2. Taking the square root of (4.44) and inverting the fractions I obtain dφ ds = ± √ (4.45) φ λ − 2φ which can be integrated directly, resulting in φ(s) =
2 cosh2
λ h√
λ 2 (s
i − s0 )
(4.46)
where s0 is an arbitrary constant. (The plus sign in (4.45) has been chosen for s < s0 and the minus for s > s0 ). Note that the properties of the propagating solution (4.46) - except for its initial position, which is determined by s0 - are all governed by a single parameter. If the velocity λ is given, the amplitude is fixed at λ/2 and the spatial extent at 2λ−1/2 . In other words - in the canonical units of (4.40) - a slow pulse will also have a small amplitude and a large spatial extent.
4.1.6 Is the solitary wave a physical solution? Eq. (4.46 ) is an exact, propagating, pulse-like solution of (4.40). But is it an acceptable solution of the original problem? In other words, is the surface profile of low amplitude and is it a long wave? To do this, we have to go back to the original variables, and convince ourselves that (4.46) generates (some) acceptable solutions for the original problem (Exercise)
4.2 KdV as a limiting case of anharmonic lattice dynamics Consider the 1-d anharmonic chain; atomic displacements are denoted by {un }; neighboring atoms of mass m interact via anharmonic potentials of the type V (r) =
1 2 1 kr + kbr3 2 3
(4.47)
where r is the distance between nearest neighbors. The equations of motion are m¨ qn
= = =
∂ [V (qn+1 − qn ) + V (qn − qn−1 ]) ∂qn k(qn+1 + qn−1 − 2qn ) − kb[−(qn+1 − qn )2 + (qn − qn−1 )2 ] k(qn+1 + qn−1 − 2qn ) − kb(qn+1 + qn−1 − 2qn )(qn+1 − qn−1 ) . −
(4.48)
If the displacements do not vary appreciably on the scale of the lattice constant a, we can use a continuum approximation; keeping terms of fourth order in the lattice constant, m¨ q ≡ qtt = ka2 qxx + ka4
2 qxxxx + kba2 qxx 2aqx 4!
,
where x = na is the continuum space variable; defining c2 = ka2 /m, this can be written as 1 1 2 qtt − qxx = a qxxxx + 2αqx qxx 2 c 12
24
,
(4.49)
4 The Korteweg - de Vries equation where α = ab provides a dimensionless measure of the anharmonicity. I now look for solutions which vary smoothly in space, i.e. over a typical length of many lattice spacings, and where the main time dependence is contained in the wave equation part, i.e. of the form x − ct , δω0 t) , (4.50) q(ξ, τ ) ≡ q(² a p where ω0 = c/a = k/m, ² ¿ 1 and δ ¿ ²; the exact dependence of δ on ² will be fixed later. The relevant derivatives transform according to qx
=
qxx
=
qxxx
=
qtt
=
² qξ a ³ ² ´2
qξξ
a ³ ² ´3 a¡
ω02
qξξξ
¢ ² qξξ − 2²δqξτ + O(δ 2 ) 2
.
Using them in (4.49) gives 2δqξτ +
1 3 ² qξξξξ + 2αqξ qξξ = 0 12
which, after a rescaling qξ (= and setting
a ² qx ) = − aφ ² 4α
,
(4.51)
(4.52)
2
δ=
1 3 ² 24
can be reduced to the canonical KdV form φτ − 6φφξ + φξξξ = 0
.
(4.53)
Note that the rescaling of length, i.e. the value of the small parameter ² is still a matter of free choice, depending on the (initial) conditions of the problem. The above analysis shows that one may legitimately suspect that nonlinear propagating solitary waves will be generic in anharmonic lattices, at least for certain parameter ranges. Again, one has to make sure that the solutions found from solving the KdV equation (4.53) are appropriate for the original problem (4.49) (check consistency of approximations made).
4.3 KdV as a field theory 4.3.1 KdV Lagrangian The KdV equation ut − 3(u2x )x + uxxx = 0
(4.54)
can be derived from the Lagrangian Z L= 2 note
dxL(φ, φt , φx , φxx )
that this guarantees δ ¿ ² as demanded above.
25
(4.55)
4 The Korteweg - de Vries equation where L=
1 1 φx φt − φ3x − φ2xx 2 2
.
(4.56)
Note that because the Lagrangian density depends on the second derivative of the field, (1.42) contain an extra term µ ¶ d2 ∂L . (4.57) − 2 dx ∂φxx Minimization of the action leads to the field equations of motion φxt − 3(φ2x )x + φxxxx = 0
(4.58)
which reduces to (4.54) upon the substitution φx = u .
(4.59)
Continuous symmetries of the Lagrangian will again give rise to an equation like (1.44), with an extra term ∂L δφxx (4.60) ∂φxx on the left-hand side. The above modifications generate an extra contribution ∂L d2 δφxx − 2 ∂φxx dx
µ
∂L ∂φxx
¶ δφ
(4.61)
to the left-hand side of (1.45).
4.3.2 Symmetries and conserved quantities For some infinitesimal transformations (cf. section ) one can verify explicitly that δφxx = d2 δφ/dx2 . If this is the case, the integral over all space of the extra contribution (4.61) can easily be seen to vanish (repeated integration by parts of either of the two terms). In this case, the standard symmetries are reflected in the same standard conservation (with the same densities of conserved quantities), as in section .... . Translational invariance in space Conservation of the total momentum Z Z Z ∞ 1 ∞ 1 ∞ ∂L φx = − dx φ2x = − dx u2 P =− dx ∂φ 2 2 t −∞ −∞ −∞
.
(4.62)
Translational invariance in time Conservation of the total energy µ ¶ Z ∞ ∂L H= dx φt − L ∂φt −∞
¶ 1 2 3 = − dx φ + φx 2 xx −∞ ¶ µ Z ∞ 1 2 3 u +u . = − dx 2 x −∞ Z
26
∞
µ
(4.63)
4 The Korteweg - de Vries equation Conservation of mass The symmetry φ → φ + ² generates δφ = ², and all other variations are zero. From (1.45), conservation of Z ∞ Z Z ∂L 1 ∞ 1 ∞ dx = dx φx = dx u , (4.64) M= ∂φt 2 −∞ 2 −∞ −∞ the total “mass”, is deduced. Galilei invariance The transformation x → x − ²t, φ(x, t) → φ(x − ²t) − ²x (or in terms of the u-field, u(x, t) → u − ², generates (cf. section ....) x → x − ²t δφ = φ(x − ²t) − φ(x) − ²x = −²tφx − ²x δφt = φt (x − ²t) − φt (x) = −²tφxt δφx = φx (x − ²t) − φx (x) − ² = −²tφxx − ² dL dL δL = δx = − ²t ⇒ J1 = −tL , J0 = 0 dx dx
.
(4.65)
Owing to δφxx = (δφ)xx there are no extra terms in the conserved currents. Eq. (1.45) applies. Since δφx = (δφ)x the two last terms in the left-hand side of (1.45) combine to form a total space derivative; similarly, because of δφt = (δφ)t , the first two terms combine to form a total time derivative, i.e. the conserved density is ∂L 1 δφ/² = φx (−tφx − x) , ∂φt 2 or, integrating over all space, and dividing by the total mass M , Z ∞ P u ¯= 1 X t + const. dx x = M −∞ 2 M which expresses the fact that the center of mass moves at a constant velocity.
4.3.3 KdV as a Hamiltonian field theory
27
(4.66)
(4.67)
5
Solving KdV by inverse scattering
5.1 Isospectral property Given the KdV equation ut − 6uux + uxxx = 0
(5.1)
and a well behaved initial condition u(x, 0), which vanishes at infinity, it is possible to determine the time evolution u(x, t) in terms of a general scheme, which is known as inverse scattering theory. The scheme is based on the following particular property of (5.1): Given the linear operator 2 L(t) = −∂xx + u(x, t)
(5.2)
whose parametric time dependence is governed by (5.1), and the associated eigenvalue equation L(t)ψj (x, t) = λj (t)ψj (x, t) , (5.3) it can be shown that
dλj =0 dt
.
(5.4)
5.2 Lax pairs The “isospectral” property can be formulated somewhat more generally: Suppose we can construct a linear, self-adjoint operator B = B † , dependent on u and such that iLt ≡ i
dL L(t + ∆) − L(t) ≡ i lim = [L, B] ∆→0 dt ∆
(5.5)
holds as an operator identity, i.e. iLt f = [L, B]f
∀f
⇔ (5.1) .
(5.6)
The operators L and B are then called a Lax pair. The time evolution of L is governed by L(t) = U (t)L(0)U †
(5.7)
where U = eiBt
.
(5.8)
Consider (5.3) at t = 0, and apply the operator U (t) to both sides from the left, i.e. U (t)L(0)
U † (t)U (t)ψj (0) = λj (0)U (t)ψj (0)
(5.9)
where, in addition I have inserted a factor U † U = 1. It can be recognized immediately that the l.h.s. of (5.9) and (5.3) are identical, provided ψj (t) = U (t)ψj (0)
28
,
(5.10)
5 Solving KdV by inverse scattering and that, in order for the r.h.sides to coincide, I must have λ(t) = λ(0)
∀t
(5.11)
(isospectral property). The form of the operator B in the KdV case is 3 B = 4i∂xxx − 3i (u∂x + ∂x u)
(5.12)
(verify explicitly (5.6).
5.3 Inverse scattering transform: the idea The isospectral property tentatively suggests that it might possible to proceed as follows: • solve the linear problem (5.3) at time t = 0, i.e. determine the eigenvalues {λj } and the eigenfunctions {ψj (x, 0)} from the known u(x, 0). • determine the evolution of the eigenfunctions from (5.10) at a later time t. • try to solve the “inverse problem” of determining the “potential” u(x, t) from the known spectra and eigenfunctions at the time t. In fact, the last step is the well known problem of inverse scattering theory in quantum mechanics, where physicists had tried to extract information on the nature of interparticle interactions from analyzing particle scattering data. The one-dimensional problem (corresponding to a spherically symmetric potentials in 3 dimensions) was completely solved in the 1950’s (Gel’fand, Levitan & Marchenko). I will present the solution below, but before doing that, let me outline some broad features: “Scattering data”in the mathematical sense are the asymptotic properties of the solution of the associated linear problem, i.e. the properties far from the source of scattering, where the potential is effectively zero. What GLM have shown is that you can reconstruct the potential from the scattering data. Furthermore, it turns out that the operator B takes an especially simple form in the asymptotic limit, which allows us to write down an exact, analytic formula for the time evolution of scattering data. Evolution of the scattering data is the easy part of the game. But then if I only need scattering data at time t, and I know how these data evolve in time, all the input I need is the scattering data for the potential u(x, 0). This is exactly the program of the inverse scattering transform (IST). Because it is based only on the asymptotic part of the solution of the associated linear problem, it can be written down in closed form. I summarize the IST program schematically: 1. determine the scattering data S of the linear problem (5.3) at time t = 0, from the known u(x, 0). 2. determine the evolution of the scattering data S(t) at a later time t from the asymptotic from of the operator B. 3. do the inverse problem at time t, i.e. determine the potential u(x, t) from the known scattering data S(t).
5.4 The inverse scattering transform 5.4.1 The direct problem This is just a summary of properties known from elementary quantum mechanics.
29
5 Solving KdV by inverse scattering Jost solutions The linear eigenvalue problem · −
¸ ∂ + u(x) ψ(x) = k 2 ψ(x) ∂x2
(5.13)
has, in general, a discrete and a continuum spectrum, corresponding to imaginary and real values of k respectively. For real k there are in general two linearly independent solutions. Such a linearly independent set is provided by the Jost solutions: f1 (x, k) ∼ f2 (x, k) ∼
eikx x → ∞ e−ikx x → −∞
.
The Jost solutions of (5.13) satisfy the integral equations Z ∞ ikx f1 (x, k) = e − dx0 G(x, x0 )f1 (x0 , k) x Z x −ikx f2 (x, k) = e + dx0 G(x, x0 )f2 (x0 , k)
(5.14)
(5.15)
−∞
where
sin k(x − x0 ) u(x0 ) . (5.16) k Eqs. (5.15) can be analytically continued to the upper half plane of complex k. Some information on the analytic properties can be obtained by considering the lowest iteration, 0 where we substitute f1 (x0 , k) = eikx in the r.h.s. of the first equation. This gives G(x, x0 ) =
Z
f1 (x, k)
∞
0
0
0 eik(x −x) − e−ik(x −x) u(x0 )eikx 2ik x Z ∞ 0 ikx ikx 1 ≈ e −e dx0 {1 − e2ik(x −x) }u(x0 ) 2ik x
≈ eikx −
dx0
(5.17)
which can be thought of as the beginning of a systematic expansion in inverse powers of k. Note that since x0 − x > 0, the exponential will be convergent in the upper-half plane of k; therefore, if the potential vanishes sufficiently rapidly at infinity, I estimate g1 (x, k) ≡ f1 (x, k) − eikx ∼ eikx h(x, k)
(5.18)
where h vanishes as 1/k for high values of k. The property f2 (x, k) = a(k)f1 (−k, x) + b(k)f1 (k, x) .
(5.19)
will be useful. For bound states, corresponding to k = iκ, the Jost solutions are degenerate. Asymptotic scattering data The asymptotic (scattering) data of (5.3) is defined as follows: • discrete spectrum (bound states) λn = −κ2n
n = 1, · · · , N
30
,
(5.20)
5 Solving KdV by inverse scattering where κn > 0; ψn (x) = =
f1 (x, k) ∼ e−κn x x → ∞ Cn f2 (x, k) ∼ Cn eκn x x → −∞
.
(5.21)
I will also need the normalization integral of each bound state Z ∞ Z ∞ 1 2 = dxψn (x) = dxf12 (x, iκn ) αn −∞ −∞
(5.22)
• continuous spectrum (scattering states) λ(k) = k 2
−∞ x; hence, in nonsymmetric matrix form, N X
Aij hj (x, t) = Ci (x, t)
(5.63)
j=1
where Aij (x, t) = δij +
αi (t) −(κi +κj )x e κi + κj
(5.64)
and Cj (x, t) = −αi (t)e−κi x Thus
1 hj = det det A
A11 A21 ·
A12 A22 ·
··· ··· ···
C1 C2 C3 · · ·
.
(5.65)
A1 j+1 A2 j+1 ·
··· ··· ···
.
where the jth column in the matrix A has been substituted by the vector A11 A12 · · · C1 e−κj x A1 j+1 · · · A21 A22 · · · C2 e−κj x A2 j+1 · · · N · 1 X · · · · C3 e−κj x · ··· gˆ1 (x, x) = det · det A j=1 · ·
(5.66)
C; it follows that ;
(5.67)
note however that, since dAij = −αi e−(κi +κj )x = Ci e−κj x dx
,
this is equivalent to gˆ1 (x, x) =
d ln det A . dx
36
(5.68)
5 Solving KdV by inverse scattering At this stage it is convenient to introduce the symmetrized form of the matrix A, obtained by Aˆ = DAD−1 , where Dij = (αi )−1/2 δij , i.e. (αi αj )1/2 −(κi +κj )x Aˆij (x, t) = δij + e κi + κj
,
(5.69)
whereupon
d2 ln det Aˆ (5.70) dx2 where I have reintroduced the time dependence, with the understanding that it arises solely from the αi s. u(x, t) = −2
Application: N = 2, the two-soliton solution In the case N = 2 α1 −2κ1 x α1 −2κ2 x α1 α2 det Aˆ = 1 + e + e + 2κ1 2κ1 4κ1 κ2 or, setting
κ1 − κ2 ≡ e−∆ κ1 + κ2
,
µ
κ1 − κ2 κ1 + κ2
αj ≡ 2κj e2θj +∆
¶2 e−2(κ1 +κ2 )x
,
∆ ∆ det Aˆ = 1 + e−2(κ1 x−θ1 − 2 ) + e−2(κ2 x−θ2 − 2 ) + e−2[(κ1 +κ2 )x−(θ1 +θ2 )]
(5.71) .
(5.72)
Note that now the time dependence is carried by the θj ’s, i.e., θj → θj (t) = θj0 + 4κ3j t
(5.73)
In order to extract the asymptotic behavior of u(x, t) at early and late times, I proceed as follows: Assume κ1 > κ2 without loss of generality. Then as, t → −∞, at sufficiently early times, it is possible to satisfy the double inequality θ1 θ2 ¿ κ1 κ2 It is easy to see that, unless x ≈ κθ11 or x ≈ will be vanishingly small. This is true • for x À
θ2 κ2 ,
θ2 κ2 ,
. the 2nd derivative of the expression (5.72)
because the last three terms vanish, leaving det Aˆ = 1
• for x ¿ κθ11 , because, although the three last terms are all exponentially large, the last one will be dominant. This leaves ln det Aˆ ∝ x and the second derivative vanishes. • for κθ11 ¿ x ¿ κθ22 the second term will be exponentially small, and the third term be much larger than the last. Again, ln det Aˆ ∝ x and the second derivative vanishes. This leaves the cases where x is appreciably near either κθ11 or κθ22 . In the first case, the contributions to (5.72) come from the 3rd and 4th terms, i.e. h i ∆ ∆ , det Aˆ ≈ e−2(κ2 x−θ2 − 2 ) 1 + e−2(κ1 x−θ1 + 2 ) or, ln det Aˆ ≈ −2(κ2 x − θ2 −
· ¸ ∆ ∆ ∆ ) − (κ1 x − θ1 + ) + ln 2 cosh(κ1 x − θ1 + ) 2 2 2
37
,
5 Solving KdV by inverse scattering
κ1=2 κ2=1
θ1 /κ1= -2. 0
θ2 /κ2= -1. 0
0.4 8.000
7.000
6.000
0.2
5.000
t
0.5
4.000
0.0
3.000
t
5
1.950
0
0.0
1.000
-0.2
x
0
-5 -5
0
5 x
Figure 5.1: The two-soliton solution [−u(x, t)] of the KdV equation as a function of space and time. Left panel: a 3-d plot shows the collision of the two solitons. Right panel: a contour plot of the same function; note the asymptotic motion of the local maxima and the phase shifts as a result of the interaction.
and hence u(x, t) ≈ −2κ21 sech2 (κ1 x − θ1 +
∆ ) 2
if
x ≈ θ1 /κ1
Similarly, it can be shown that u(x, t) ≈ −2κ22 sech2 (κ2 x − θ2 −
∆ ) if 2
x ≈ θ2 /κ2
.
Combining the above, and reintroducing the explicit time dependence, I can write that u(x, t) ∼ −2
2 X
κ2j sech2 (κj x − 4κ3j t − θj0 ±
j=1
∆ ) if 2
t → −∞
,
(5.74)
where the upper sign holds for j = 1, and the lower for j = 2. The above analysis can be repeated almost verbatim for asymptotically late times and leads to u(x, t) ∼ −2
2 X
κ2j sech2 (κj x − 4κ3j t − θj0 ∓
j=1
∆ ) 2
if
t→∞
.
(5.75)
The above equations describe the soliton property in a mathematically exact fashion. As we follow the evolution from very early to very late times, we see the larger - and faster - local compression reach the smaller - and slower - , interact with it in an apparently intricate fashion, and then disengage itself and resume its motion with the same velocity. Both waves maintain shape, amplitude and speed. The interaction does however leave a signature. The center of mass of each wave becomes slightly displaced; the fastest by an amount of ∆/κ1
38
5 Solving KdV by inverse scattering (forwards), the slower by an amount of −∆/κ2 (backwards). Note that because the mass of each soliton is proportional to κj , the center of mass of the combined two-soliton system moves at a constant speed before and after the two-soliton collision. This type of elastic, transparent interaction which leaves velocities unchanged and results only in spatial shifts2 is characteristic of soliton bearing systems, and accounts for their remarkable dynamical properties. The analysis can be generalized to the N −soliton solution. It can be shown that phase shifts are pairwise additive, i.e. the total phase shift of any soliton as a result of its interaction with the other N − 1 solitons is the sum of the N − 1 phase shifts resulting from the N − 1 collisions. Fig. 5.1 exhibits graphically the dependence of the two-soliton solution on space and time.
5.6 Integrals of motion It is possible to deal with integrals of motion in a systematic fashion, by following the analytic structure of the scattering data. Recall that the transmission coefficient does not carry any time dependence under the IST, i.e. it can be treated as a constant of the motion!
5.6.1 Lemma: a useful representation of a(k) Given the fact that a(k) (recall that a is the inverse of the transmission coefficient) has simple zeros in the upper half of the complex plane, the following identity holds: ! Ã Z ∞ N 0 2 X 1 k − kj 0 ln |a(k )| ln a(k) = (5.76) dk + ln 2πi −∞ k0 − k k − kj∗ j=1 (cf. appendix ...).
5.6.2 Asymptotic expansions of a(k) The asymptotic expansion ln a(k) ∼
∞ X
Jn n (2ik) n=1
(5.77)
holds for |k| > max{|kj |}. Multiply both sides of (5.77) by k l−1 /(2πi) and integrate over a circle of radius R > max{|kj |} centered at the origin of the complex k-plane. The only term which survives in the sum is that with j = l, hence I 1 −l dk k l−1 ln a(k) ; (2i) Jl = 2πi performing the dk integration in the first term of (5.76) generates a contribution −k 0 l−1 . The second term can be integrated by parts and generates contributions from all poles. This results in Z ∞ N X ¢ 1 1 ¡ ∗l Jl l−1 2 = − dk k ln |a(k)| + kj − kjl . (5.78) (2i)l 2πi −∞ l j=1 2 the
term “phase shifts” is generically applicable.
39
5 Solving KdV by inverse scattering So far this has been general. Applying to the KdV equation, I set kj = iκj ; note that the terms in the discrete sum vanish if l is even. Due to the reflection symmetry |a(k)| = |a(−k)|, the integrals vanish as well for even l. This leaves J2m+1 = −(−1)m 22m+1
Z
∞
dk k 2m ln |a(k)|2 + 2
−∞
N X κ2m+1 j 2m +1 j=1
.
(5.79)
In what follows, I will relate the Jl s - which are by definition constants of the motion, since they only depend on a(k) and bound state eigenvalues - to the family of conserved quantities generated in the previous section, independently of the IST. From the general property of the Jost functions
I deduce that if Imk > 0
f2 (x, k) = a(k)f1 (x, −k) + b(k)f1 (x, k)
(5.80)
¤ £ lim f2 (x, k)eikx = a(k) .
(5.81)
x→∞
This is because in the limit of large positive x the first term in (5.80), f1 (x, −k) ∼ e−ikx is exponentially large and the second f1 (x, −k) ∼ e−ikx is exponentially small. On the other hand, £ ¤ lim f2 (x, k)eikx = 1 . (5.82) x→−∞
holds by definition. Knowledge of the two limits allows me to define σ(x) = with the property
ª¤ f 0 d £ © ln f2 (x, k)eikx = 2 + ik dx f2 Z
(5.83)
∞
dx σ(x) = ln a(k)
.
(5.84)
−∞
Now we can use the fact that f2 is a solution of the associated linear problem, to derive a differential equation for σ in terms of u. To do this I multiply both sides of (5.83) and differentiate with respect to x. This gives f200 + ikf20 = f20 σ + f2 σ 0
,
or uf2 − k 2 f2 + ikf20 = f20 σ + f2 σ 0
;
substituting f20 = (σ − ik)f2 generates σ 0 − 2ikσ + σ 2 − u = 0
(5.85)
a nonlinear ordinary first order differential equation of the Ricatti type. I can now try to generate an asymptotic solution of the Ricatti equation (5.85), σ(x, k) ∼
∞ X σn (x, k) (2ik)n n=1
where I note that, because of (5.77) and (5.84), Z ∞ dx σn (x) = Jn −∞
40
.
(5.86)
(5.87)
5 Solving KdV by inverse scattering Indeed, I note that the asymptotic ansatz (5.86) in (5.85) generates the recurrence relationships n−1 X d σn+1 (x) = σn (x) + σj (x)σn−j (x) , n = 2, 3, · · · (5.88) dx j=1 with σ1 (x) = −u(x). This generates a countable infinity of conserved densities. The first few are σ2 σ3 σ4 σ5
= = = =
ux −uxx + u2 −uxxx + 2(u2 )x −uxxxx + (u2 )xx + u2x + 2uuxx − 2u3
.
(5.89) (5.90) (5.91) (5.92)
Note that the even σn ’s are total derivatives, i.e. they generate trivial, vanishing integrals; we know this, since the corresponding Jn ’s vanish. The first few odd σn ’s generate the mass, momentum, and energy integrals of section ....
5.6.3 IST as a canonical transformation to action-angle variables It can be shown [] that the inverse scattering transform is a canonical transformation from the original field variables to action-angle variables. The scattering data of the IST are in essence action-angle variables. This demonstrates the KdV Hamiltonian system is completely integrable.
41
6
Solitons in anharmonic lattice dynamics: the Toda lattice
The Toda lattice [11] is a unique example of a nonlinear discrete particle system which is completely integrable. Although the property of complete integrability is certainly a singular feature due to the peculiarity of the lattice potential, the model has been extremely useful as a theoretical laboratory for the exploration of a number of novel concepts and phenomena related to loss-free supersonic pulse propagation.
6.1 The model The Hamiltonian H=
X ½ p2
n
2m
n
where φ(r) =
¾ + φ(qn − qn−1 )
ª a © −br e + br − 1 b
(6.1)
(6.2)
describes a chain of N particles with equal mass m, which interact via nearest-neighbor repulsive potential of exponential form. The range of the potential is given by 1/b and its strength by a/b. The linear term in the potential represents an external force which is necessary to achieve confinement. Important limiting cases of (6.2) are: • the harmonic limit a → ∞, b → 0, ab → k which leads to φ(r) =
1 2 kr 2
;
• the hard-sphere limit b→∞ (i.e. the range approaches zero) with a finite, which leads to φ(r)
=
0
=
∞
if r > 0 if r < 0
.
I will, from now on, set a = 1, b = 1, m = 1. Units will be reintroduced when appropriate. The equations of motion are q˙n
=
pn
p˙n
=
e−(qn −qn−1 ) − e−(qn+1 −qn )
42
(6.3)
6 Solitons in anharmonic lattice dynamics: the Toda lattice
6.2 The dual lattice Consider the variables rn = qn − qn−1 which describe the difference in displacements of neighboring sites. Differentiating both sides with respect to time gives r˙1 r˙2 r˙j
= q˙1 − q˙0 = q˙2 − q˙1 = q˙j − q˙j−1
.
Summing left and right sides separately gives allows me to express the velocity coordinates as j X q˙j = r˙l , (6.4) l=1
where I have assumed q˙0 = 0. The total kinetic energy is à n !2 N 1X X r˙l T = 2 n=1
.
l=1
If I now define new momentum variables, conjugate to the rn coordinates, via sj =
∂T ∂ r˙j
= r˙1 + · · · + r˙j + r˙1 + · · · + r˙j+1 + ··· + r˙1 + · · · + r˙N ,
the sj ’s will satisfy sj − sj+1 = r˙1 + · · · r˙j = q˙j
(6.5)
and therefore I can rewrite the kinetic energy as N
T =
1X 2 (sj+1 − sj ) 2 j=1
Since the total potential energy is clearly only a function of the rn ’ s, X
φ(rn )
,
n
this process has defined a new canonically conjugate set of variables. Following Toda [11] we view this set as describing a new lattice, “dual” to the original. The equations of motion are ∂H = −φ0 (rj ) ∂rj ∂H = 2sj − sj+1 − sj−1 ∂sj
s˙ j
= −
r˙j
=
(6.6)
and describe - by construction - the same dynamics as the original equations of motion (6.3).
43
6 Solitons in anharmonic lattice dynamics: the Toda lattice
0.0
-0.2
α=1
δ/α=0
-3.5 -7.25
-0.4
rn 10
-0.6
0.1 1E-3 1E-5
-0.8
-rn
1E-7 1E-9
0
2
4
-1.0 -10
-5
0
n
5
n
6
8
10
10
Figure 6.1: The local compression corresponding to a Toda soliton (6.13). The value of α is equal to 1. The three curves represent different choices of the phase δ. Inset: the dependence of −rn vs n on a logarithmic scale for the case δ = 0.
Now it is possible to eliminate either the sn ’s or the rn ’s from (6.6). In the first case I obtain r¨n = 2e−rn − ern+1 − ern−1 (6.7) and in the second
1 + S¨n = e−2Sn +Sn+1 +Sn−1
where
Z
t
Sn =
(6.8)
dt0 sn (t0 ) .
(6.9)
0
Note that owing to (6.6), qn = Sn − Sn+1
.
(6.10)
6.2.1 A pulse soliton A special solution of (6.8) is Sn (t) = ln cosh(αn ∓ βt − δ)
(6.11)
where α > 0, δ is an arbitrary constant, and β = sinh α. Differentiating with respect to time, I obtain sn (t) = ∓β tanh(αn ∓ βt − δ) (6.12) and therefore e−rn = 1 +
β2 cosh2 (αn ∓ βt − δ)
.
(6.13)
This special solution corresponds to a supersonic, compressional pulse soliton moving with the velocity β sinh α v=± =± . α α The form of the pulse is shown in Fig. 6.1.
44
6 Solitons in anharmonic lattice dynamics: the Toda lattice Mass of the soliton The total mass carried by the soliton can be shown - using (6.11) - to be M=
X
rj = lim (qn − q−n ) = −2α n→∞
j
.
(6.14)
Momentum of the soliton The total lattice momentum carried by the soliton can be shown - using (6.12) - to be P =
X j
q˙j = lim (sn − s−n ) = ∓2β = M v n→∞
.
(6.15)
Energy of the soliton The total energy of the soliton is given by X¡ ¢ X 1X (sn+1 − sn )2 + e−rn − 1 + rn 2 n n
.
The sum of the first two terms can be shown to be sinh 2α; the third sum we recognize as the soliton mass. Thus E = sinh 2α − 2α (6.16)
6.3 Complete integrability Define new coordinates in terms of the original positions and momenta an
=
bn
=
1 − 1 (qn −qn−1 ) e 2 2 1 − pn . 2
(6.17)
Using the original equations of motion, I obtain 1 b˙ n = − p˙n = 2(a2n+1 − a2n ) 2
(6.18)
and ln(2an ) = a˙ n an a˙ n
= =
1 − (qn − qn−1 ) 2 1 − (pn − pn−1 ) 2 an (bn − bn−1 ) .
(6.19)
Note that decaying boundary conditions at (plus or minus) infinity correspond to an → 1/2, bn → 0. This allows for a constant value of the displacement q (cf. the pulse solution of the previous section). Now one can directly verify that the set of equations is equivalent to the condition dL = [B, L] , (6.20) i dt
45
6 Solitons in anharmonic lattice dynamics: the Toda lattice where
L=
and
B = i
··· ··· ··· ··· ··· ···
··· ··· ··· ··· ··· ···
··· bn−1 an 0 0 ···
··· an bn an+1 0 ···
··· 0 −an−1 0 0 ···
··· 0 an+1 bn+1 an+2 ···
··· an−1 0 −an 0 ···
··· 0 0 an+2 bn+2 ···
··· 0 an 0 −an+1 ···
··· ··· ··· ··· ··· ···
··· 0 0 an+1 0 ···
··· ··· ··· ··· ··· ···
(6.21)
(6.22)
are tridiagonal matrices which form a Lax pair. In other words, the Toda lattice with decaying boundary conditions can be completely integrated using the inverse scattering transform. Details can be found in [11]. This means that there are multisoliton solutions, and that Toda solitons have all the nice properties of exact solitons which we encountered in the KdV example (e.g. elastic scattering which only results in phase shifts etc).
6.4 Thermodynamics The partition function of the Toda chain ! Z ÃY N dpi dqi e−βH Z=
,
i=1
where β is the inverse temperature, can be factorized into two contributions, ZK and ZP , coming from the kinetic and potential energy respectively. The integration over momentum variables gives a product of N identical integrals, µZ ∞ ¶N µ ¶N/2 2π −βp2 /2 ZP = dp e = β −∞ whereas the integration over position coordinates gives ! Z ∞ ÃY N PN ZK = dqi e−β i=1 φ(qi −qi−1 ) −∞
Z
∞
=
i=1
ÃN Y
−∞
µZ
! dri
e−β
i=1
∞
=
dr e
PN
φ(ri )
¶N −βφ(r)
−∞
¶N
=
µ Z eβ
=
£ β −β ¤N e β Γ(β)
∞
i=1
dy y β−1 e−βy
0
−r
(6.23)
where the substitution y = e has been made. Combining terms I obtain the free energy per site 1 1 β 1 1 ln Z = −1 + ln β − ln Γ(β) + ln (6.24) f =− Nβ β 2β 2π
46
6 Solitons in anharmonic lattice dynamics: the Toda lattice At low temperatures, β À 1, one can use the Stirling approximation to the gamma function µ ¶ √ 1 Γ(z) ∼ e−z z z−1/2 2π 1 + + ··· (6.25) 12z and obtain f∼
1 β 1 ln − + ··· β 2π 12β 2
(6.26)
where the first term is identified as the free energy per site of a harmonic chain, and the second is the leading term of a systematic asymptotic expansion in powers of the temperature.
47
7
Chaos in low dimensional systems
7.1 Visualization of simple dynamical systems 7.1.1 Two dimensional phase space Linear stability analysis Consider the following general dynamical system consisting of two coupled differential equations. ~x˙ = F~ (~x) , (7.1) where F1 (x1 , x2 ), F2 (x1 , x2 ) are arbitrary, in general nonlinear functions of x1 , x2 . Note that (7.1) does not necessarily represent a mechanical system. It could for example represent a coupled system of prey-predator species with populations x1 and x2 respectively, for which F1 (x1 , x2 ) F2 (x1 , x2 )
= rx1 − kx1 x2 = −sx2 + k 0 x1 x2
(7.2)
(Lotka-Volterra equation). In the absence of interaction, the prey and predator populations will, respectively, grow and die off, at exponential rates (Malthus model of population biology). Interaction creates new possibilities. Note first that for x∗1 = s/k 0 , x∗2 = r/k, the right-hand side of (7.2) vanishes. The two populations may coexist stably at these levels. Suppose however that you are dealing with fish populations, and some outside agent, without the power to modify the biological parameters, simply removes a part of one - or both populations. If the perturbation is large, we would have to solve the full system (7.1) with the new set of initial conditions. For small perturbations however, it is possible to make some general statements about the system’s behavior in terms of linear stability analysis. Consider a state of the system near the fixed point, i.e. ~x = ~x∗ + δ~x(t) .
(7.3)
If δ~x(t) is sufficiently small, we may expand (7.1) around the fixed point, and obtain d δ~x(t) = M δ~x(t) dt where
µ Mij =
∂Fi ∂xj
(7.4)
¶ . ~ x=~ x∗
The ansatz δ~x(t) = exp(λt)f~ leads to the eigenvalue equation Mf~ = λf~ ,
(7.5)
A perturbation which has a nonzero component along an eigenvector with positive eigenvalue will grow exponentially. On the other hand, if both eigenvalues are negative, the system will be stable in all directions around the fixed point. The various possibilities are summarized as follows:
48
7 Chaos in low dimensional systems • λ1 , λ2 real. If λ1 λ2 < 0 we have a saddle (stable in one direction, unstable in the other); in the special case λ1 + λ2 = 0, the saddle is called a hyperbolic fixed point. If λ1 λ2 > 0 we have a node. A node is stable if both eigenvalues are negative, and unstable if both eigenvalues are positive. • If λ1 , λ2 are complex conjugates we have a focus. A focus will be stable or unstable according to whether the real part of λ is negative or positive, respectively. If λ1 , λ2 are pure imaginary we have an elliptic fixed point. The undamped harmonic oscillator
q˙ = p˙ =
p −ω02 q
(7.6)
Elliptic fixed point at p = 0, q = 0. Eigenvalues are λ1,2 = ±iω0 . Because there is a conserved quantity (Hamiltonian), orbits in phase space are one dimensional (ellipses). The damped harmonic oscillator
q˙ = p p˙ = −ω02 q − γp
(7.7)
The fixed point at p = 0, q = 0 is either a stable focus (if γ < 2ω0 ), or a stable node (if γ = 2ω0 ). There is no conserved quantity; orbits in phase space have a spiral form. The pendulum
H(p, q) = q˙ = p˙ =
1 2 p − ω02 cos q 2 p −ω02 sin q
(7.8)
(7.9)
There are fixed points at p = 0, q = kπ, where k = 0, ±1, ±2, · · ·. The points at even k are elliptic, the ones at odd k are hyperbolic. Orbits in phase space are again one-dimensional, due to energy conservation. They are either bounded (near a fixed point), or unbounded. A special orbit (separatrix) separates the two types of motion. The separatrix connects two hyperbolic fixed points. The bistable potential
H(p, q) = q˙
1 2 1 p + (1 − q 2 )2 2 2
=
p˙ =
(7.10)
p (1 − q 2 )q
(7.11)
There are fixed points at p = 0, q = 0 (hyperbolic), and p = 0, q = ±1 (elliptic). Motion is bounded but has a different topology according to the value of the energy. The different types of motion are separated by a particular orbit (separatrix).
49
7 Chaos in low dimensional systems
7.1.2 4-dimensional phase space The dynamics of a Hamiltonian system with two degrees of freedom H=
1 2 (p + p2y ) + V (q1 , q2 ) 2 x
(7.12)
is formulated in terms of a system of 4 coupled differential equations which are first order in time: q˙1 q˙2
= =
p˙1
=
p˙2
=
p1 p2 ∂V (q1 , q2 ) − ∂q1 ∂V (q1 , q2 ) − ∂q2
.
(7.13)
In a generic Hamiltonian system only the energy is conserved. If for some reason there is a further constant of motion, orbits will lie on a 2-dimensional torus. In the generic case, phase space orbits will be on the energy shell, i.e a 3-d hypersurface. It is possible to visualize this with the help of Poincar´e surfaces of section, i.e. by projecting the energy hypersurface on the q1 = 0 plane. A Poincar´e surface of section - briefly, Poincar´e cut -, consists of points (q2 , p2 on a plane, taken at q1 = 0, p1 > 0. A cut of a 2-d torus would be a continuous curve. A cut of a generic 3-d hypersurface would fill a portion of the plane. Evidence of such “filling” in systems which are perturbed away from an integrable limit, is interpreted as “breaking of the torus”. This is what Hamiltonian chaos is all about. The Henon-Heiles Hamiltonian 1 2 1 1 (px + p2y ) + (x2 + y 2 ) + x2 y − y 3 , (7.14) 2 2 3 originally proposed as a model for integrable behavior in galactic motion [12], was a milestone in the study of Hamiltonian chaos. The equipotential surfaces, shown in Fig. 7.1, suggest its usefulness as a model for triatomic molecules. H=
2.0
1.0 10.00
1.5 8.500
0.8
7.000
1.0
0.6
5.500
0.5
4.000
0.4
y
2.500
0.0
y
1.000
0.2
0.1667
-0.5
0.06250
0.0
0
-1.0 -0.5000 -2.000
-1.5
-1
0
1
-0.4 -0.6 -1.0
-2.0 -2
-0.2
2
-0.5
0.0
0.5
1.0
x
x
Figure 7.1: Left: equipotential surfaces of the Henon-Heiles Hamiltonian; right: details of the region of bounded motion, E = 1/30, 1/15, 1/10, 2/15, 1/6 (outer surface).
Fig. 7.2 shows Poincar´e cuts obtained at increasing energies. At E = 1/12 - which is not a small energy! - the motion is almost entirely regular (note however the seeds of irregularity
50
7 Chaos in low dimensional systems in the immediate vicinity of the separatrix). As the energy increases further, the various tori begin to disappear. Widespread chaos ensues. Note that the scattered points all belong to the same trajectory in phase space.
0.6
0.5
0.4
0.4
0.3
0.4
0.3 0.2 0.2 0.1 py
0.2
0.1 py
0.0
py
0.0
0.0
-0.1
-0.1
-0.2
-0.2 -0.2 -0.3 -0.3 -0.4 -0.4
-0.4
-0.4 -0.5 -0.2
0.0
0.2
0.4
-0.6 -0.4 -0.2
y
0.0
0.2
0.4
0.6
-0.5
y
0.0
0.5
1.0
y
Figure 7.2: Poincar´e cuts for the Henon-Heiles system. Left, E=1/12; center E=1/8; right, E=1/6. The percentage of area covered by such scattered points provides a measure of chaos.
7.1.3 3-dimensional phase space; nonautonomous systems with one degree of freedom A nonautonomous Hamiltonian system with one degree of freedom H(p, q, t) =
1 2 p + V (q, t) 2m
(7.15)
is described by the equations q˙
=
p˙ = H˙
=
p m ∂V (q, t) − ∂q ∂V (q, t) . ∂t
(7.16)
Phase space is now in general 3-dimensional. There are no conserved quantities to reduce it. However, if the system is externally driven by a periodic force of period T , one may attempt to visualize its behavior by using stroboscobic plots, i.e. plotting pairs pn , qn obtained at times tn = nT . As an example, consider the plots obtained for the bistable oscillator with m = 2 in a periodic field V (q, t) = −2q 2 + q 4 + ²q cos ωt
.
(7.17)
In the absence of a driving field, the trajectories in 2-dimensional phase space are shown in Fig. 7.3 (left panel). The equations of motion have 3 fixed points, two of them (at q = ±1) elliptically stable, and one (at q = 0) hyperbolically unstable. If the total energy is low (near -1), the particle performs low-amplitude oscillations at the bottom of the left or the right well. The limiting natural frequency of oscillation is ω0 = 2.
51
7 Chaos in low dimensional systems The other two panels of Fig. 7.3 show what happens when a periodically varying field is turned on. The frequency of the field ω = 1.92 is chosen to lie near ω0 . At low amplitudes of the driving field and reasonably low energies, a stroboscopic plot of motion is not fundamentally different from the corresponding plot at the conservative limit ² = 0; the particle stays confined near the top of the potential well. As the driving amplitude increases, the particle escapes the well and performs a chaotic motion in the vicinity of the separatrix of the conservative limit (right panel).
2
2
1
2
1
p
2.0 2.0 1.5 1.0 0.50 0 -0.50 -1.0
0
-1
-2
1
p
p 0
0
-1
-1
ϖ=1.92 ε=0.01
-2 -1
0
1
-1
0
x
ϖ=1.92 ε=0.1
-2
1
-1
x
0
1
x
Figure 7.3: Stroboscobic plot of the dynamics of (7.17) for ω = 1.92 and 0 < t < 2000 (after [13]). The left panel shows the contours of phase space trajectories of the unperturbed, conservative system; note the separatrix at E = 0, which separates bounded from unbounded motion. Initial conditions were p = 0 and q = 0.24, corresponding to E = −0.112, an energy near the top of the potential well. The middle panel, at ² = 0.01, shows that the particle remains trapped in the well. The right panel, at ² = 0.1, illustrates the escape from the well, and the “breaking of the torus” which occurs near the separatrix.
7.2 Small denominators revisited: KAM theorem Recall there was a problem of small denominators; if you start with an integrable Hamiltonian H0 (J1 , J2 ) and functionally independent frequencies ωi = ∂H0 /∂Ji and perturb it with a small perturbation µH1 (J1 , J2 , θ1 , θ2 ) then Poincar´e showed that there are no analytic invariants of the perturbed system H0 + µH1 . Is chaos inevitable? The answer is more or less yes. Is chaos imminent and overwhelming, even for a small perturbation? The answer is no as we have seen from circumstantial evidence in the Henon-Heiles Hamiltonian. Kolmogorov, Arnold and Moser (KAM) showed that, if the Hessian of the unperturbed Hamiltonian is nondegenerate, i.e. ¶ µ ¶ µ 2 ∂ωi ∂ H0 det = det 6= 0 , (7.18) ∂Ji ∂Jj ∂Jj a torus of the H0 Hamiltonian with frequencies ωi survives, slightly deformed, in the perturbed system, provided |n1 ω1 + n2 ω2 | ≥
K(²) α ||n1 | + |n2 ||
∀ n1 , n2
(7.19)
where α > 2 and K(²) depend on the particulars of the problem. Tori which do not fulfill this condition may break up. The destroyed tori constitute a dense set. Yet they have a very small measure. Most tori survive. It is possible to understand this using an analogy
52
7 Chaos in low dimensional systems with the length obtained by excluding from the line continuum a small neighborhood, say ²/n3 , around every rational number m/n (recall that the rationals form a dense set). The measure of the continuum deleted is ∞ X n ∞ X X ² ² π2 ² . = = 3 2 n n 6 n n=1 m=1
(7.20)
Although irrationals do not form a dense set, they make most of the measure of real numbers. In this sense, almost all tori survive the addition of a small (in practice: even a moderately large) perturbation. Eventually however, as the perturbation grows, chaos ensues. Note I have used the language of systems with two degrees of freedom just for simplicity. In fact, the KAM theorem holds for an arbitrary number of degrees of freedom, under the conditions described above.
7.3 Chaos in area preserving maps 7.3.1 Twist maps The twist map allows direct visualization of a Hamiltonian system with two degrees of freedom, moving on a torus. Let J1 , J2 be the action coordinates, and θ1 , θ2 the corresponding angle coordinates. Make a Poincar´e cut each time θ2 = 0 mod 2π. This will by definition be √ every τ = 2π/ω2 seconds, where ω2 = ∂H0 /∂J2 . Then plot the coordinates ρ = 2J1 and φ = θ1 on a plane. The points will lie on a circle. I can express the successive values of the angle coordinate on the cut by the sequence φn+1 = φn + ω1 τ or, more generally, in terms of the winding number w = ω1 /ω2 ρn+1 φn+1
= ρn = φn + 2πw(ρn )
(7.21)
where I have explicitly allowed all possible J1 ’s and hence all possible radii. For a given energy this fixes J2 , so that the winding number is only a function of ρ. In shorthand notation this will be µ ¶ µ ¶ ρn+1 ρn = T0 , (7.22) φn+1 φn where T0 stands for the unperturbed twist map. Now if the winding number can be expressed as a rational fraction r/s, the cut will be composed of s points (s-cycle). If not, we have quasiperiodic motion; the cut fills the circle densely. We would like to find out what happens under a perturbation. This is described below (Poincar´e-Birkhoff theorem). For the moment, let me just describe what a perturbed map will look like - and how to get it. In general, ρn+1
= ρn + ²f1 (ρn , φn )
φn+1
= φn + 2πw(ρn ) + ²f2 (ρn , φn )
(7.23)
where I must choose the functions f1 and f2 such that the map represents a Hamiltonian flow, i.e. it should be a canonical transformation. This can be achieved by using an appropriate
53
7 Chaos in low dimensional systems generating function F (φ1 , φ2 ) such that µ ρn+1
=
− µ
φn+1
=
∂F ∂φn
∂F ∂φn+1
¶ φn+1
¶
.
(7.24)
φn
A class of such perturbed maps can be obtained by the generating function F (φn , φn+1 ) =
1 2 (φn − φn+1 ) + ²V (φn ) . 2
(7.25)
The maps have the form ρn+1 φn+1
ρn + ²V 0 (φn ) φn + ρn+1 ;
= =
(7.26)
The above map equations (7.26) can also be derived by demanding that the “action” W =
m X
F (φn , φn+1 )
(7.27)
n=0
should be an extremum with respect to any of the m internal coordinates φ1 , · · · , φm (i.e. the end coordinates are φ0 , φm+1 are held fixed). F can thus be interpreted as a discrete Lagrangian. Later in the course I will show that this has important applications in an entirely different context - determining energy minima and studying prototypes of amorphous solids; in other words, spatial rather than temporal chaos.
7.3.2 Local stability properties The local stability properties of fixed points are governed by the tangent map (cf. continuous ~ ∗ = (ρ∗ , φ∗ ) is a fixed point of the map T , dynamics). Thus if X ~ ∗ = T (X ~ ∗) X
(7.28)
the tangent map of T is defined via a linearization procedure around the fixed point: ~n = X ~ ∗ + δX ~n X
(7.29)
~ n+1 = M (X ~ ∗ )δ X ~n δX
(7.30)
where in general
à ~ n) = M (X
∂ρn+1 ∂ρn ∂φn+1 ∂ρn
∂ρn+1 ∂φn ∂φn+1 ∂φn
! .
(7.31)
Since the map T is area preserving, the eigenvalues of M will satisfy the relationship λ1 λ2 = 1. There are two distinct cases • both roots are imaginary; they must be of the form λ1,2 = e±iδ
(7.32)
(elliptic fixed point), or • both roots are real |λ1 | > 1
|λ2 | < 1
(7.33)
(hyperbolic fixed point if both positive, hyperbolic with reflection if both negative).
54
7 Chaos in low dimensional systems ~ ∗, X ~ ∗, · · · , X ~ ∗ ) is represented by fixed points of Periodic motion (s− cycles in the form X s 1 2 s the T map, ~ j∗ = T s (X ~ j∗ ) j = 1, 2, · · · , s . X (7.34) The stability of the s-cycle (7.34) is governed by the eigenvalues of the product matrix ∗ ~ s∗ )M (X ~ s−1 ~ 1∗ ) M (s) = M (X ) · · · M (X
.
Note that, since the determinant of each one of the terms in the above product is unity, det M (s) = 1. The classification of stability properties is therefore exactly the same (elliptic vs hyperbolic cycles) as in the case of fixed points (cf. above).
7.3.3 Poincar´ e-Birkhoff theorem The unperturbed twist map with a rational winding number w = r/s will generate an s-cycle whose points lie on a circle C. This will happen no matter where one starts on the circle. In this sense, every point the circle will be a fixed point of the unperturbed Tos map, Tos C = C
.
(7.35)
Note that this differs from the generic situation of an irrational winding number; the circle with a radius which corresponds to an irrational winding number maps onto itself - but its points are not fixed points of any finite repeated application of the map. What happens under the influence of a perturbation? In order to see this, consider two neighboring circles, C + , with a slightly larger, irrational winding number w+ , and C − with a slightly smaller, irrational winding number w− . Under application of the same unperturbed twist map, C + will be slightly twisted - with respect to C -in the positive (counterclockwise) direction, since w+ > w; similarly C − will be slightly twisted in the negative (clockwise) direction, since w− < w. These relative opposite twists of the circles survive under the perturbed twist map T²s - although their form may be distorted. By a continuity argument it is possible to construct a “zero twist” curve R. If I now apply the map T²s to R, the resulting curve will be distorted with respect to R only in the radial direction (zero twist). Because the map is area preserving, there should, in general, be an even number of intersections, 2ks (exceptions are possible in cases where the curve T²s R might tangentially touch the curve R). These intersections are the only fixed points which survive from the original invariant circle C in the presence of a perturbation. Of the 2ks fixed points, half are elliptically stable and half hyperbolically unstable; elliptic and hyperbolic fixed points come in pairs and they alternate. This is the Poincar´e-Birkhoff theorem.
7.3.4 Chaos diagnostics Power spectra Given a suitably averaged time-dependent quantity f (t), it is possible to define its power spectrum Z ∞ 1 I(ω) = dteiωt f (t) . (7.36) 2π −∞ If the “signal” is periodic in time, i.e. if f (t) = f (t + T ), it is possible to express it as a Fourier series ∞ X αn e−inΩt (7.37) f (t) = n=−∞
55
7 Chaos in low dimensional systems
Figure 7.4: Illustration of the Poincar´e-Birkhoff theorem. (a) upper left: the unperturbed map: a circle C with a rational winding number w, along with neighboring circles C + , C − with irrational winding numbers w+ (positive twist), w− (negative twist). (b) upper right: the perturbed map; outer and inner curve represent, respectively, the slightly deformed versions of C + , C − . The intermediate curve R is a zero-twist curve obtained by the requirement of continuity. (c) lower right: tR(continuous curve) and its T²s map (dashed curve). In this case s = 2. There is no twist under the action of the map, just pulling and pushing along the radial direction. There is a total of 4 intersections, corresponding to a stable and an unstable 2-cycle. Following the arrows, it is possible to determine which points are elliptic and which are hyperbolic. Note that the small arrows outside R are all pointing in the outward direction (positive twist), and those inside R in the negative direction (negative twist). (d) a more abstract view of the elliptic and hyperbolic 2-cycles.
where Ω = 2π/T . It follows that the spectrum I(ω) =
∞ X
αn δ(ω − nΩ)
(7.38)
n=−∞
will be composed of a series of δ-peaks situated at the fundamental frequency and its higher harmonics. One can generalize this to the case of a multiply periodic motion - which would be more apt to describe motion on on a torus. In this case of a doubly periodic motion f (t) is
56
7 Chaos in low dimensional systems described by a double Fourier expansion ∞ X
f (t) =
∞ X
αn1 ,n2 e−i(n1 Ω1 +n2 Ω2 )t
(7.39)
αn1 ,n2 δ(ω − n1 Ω1 − n2 Ω2 ) ,
(7.40)
n1 =−∞ n2 =−∞
and the spectrum I(ω) =
∞ X
∞ X
n1 =−∞ n2 =−∞
forms peaks at all sum and difference frequencies. Under ideal conditions (cf. Fig. 7.5) it
10
10
-4
1x10
-5
10
-6
10
-7
10
-8
1x10
-5
10
-6
Power spectra
10
-6
1x10
Power spectra
-5
Power spectra
1x10
-7
-8
0.0
0.1
0.2
0.3
0.4
0.0
0.1
ϖ/2π
0.2
0.3
0.4
10
-7
10
-8
0.0
0.1
0.2
0.3
0.4
ϖ/2π
ϖ/2π
Figure 7.5: Power spectra of py (t) for quasiperiodic (left and center panels) and chaotic (right panel) trajectories of the Henon-Heiles system at energy E = 1/8. In the case of quasiperiodic motion (left) it is possible to make a detailed identification of the five peaks in terms of two fundamental torus frequencies at f1 = 0.16 and f2 = 0.12, their second harmonics, and the difference f1 − f2 = 0.04. A similar assignment can be made in the case of the center panel. Chaotic spectra (right panel) are characterized by broader, noisier features.
should of course be possible to distinguish regularity from chaos by its spectral signatures. In the former case the spectrum is periodic or quasiperiodic, in the latter case there is a lot of noise, perhaps accompanied by broad peaks. In practice however, the intrinsic limitations of obtaining useful power spectra from finite numerical (or experimental) data, renders spectral information somewhat limited as a sole criterion of deciding whether a given process is chaotic or not. Lyapunov exponents Lyapunov exponents quantify the usual defining property of deterministic chaos, which is the sensitive dependence on initial conditions. Consider a certain trajectory of the - not necessarily area-preserving - N -dimensional map T ~ j+1 = T (X ~ j) X
j = 0, · · · n − 1,
(7.41)
~ 0 + δX ~ 0 . The difference between the two and a “neighboring” trajectory, which starts at X trajectories after the first iteration can be expressed in terms of the tangent map: ~ 1 = M (X ~ 0 )δ X ~0 δX
57
;
7 Chaos in low dimensional systems after the second iteration it will be ~ 2 = M (X ~ 1 )δ X ~ 1 = M (X ~ 1 )M (X ~ 0 )δ X ~0 δX
,
and after n iterations ~n δX
~ n−1 )M (X ~ n−2 ) · · · M (X ~ 0 )δ X ~0 M (X ~ 0, · · · , X ~ n−1 )δ X ~0 , Λn (X
= =
(7.42)
where the N × N matrix Λ is the nth root of the product of all n tangent maps involved in the trajectory; in general, Λ will have N eigenvalues λα (n), α = 1, · · · N , which will depend on the order of iteration n. The Lyapunov exponents are defined as σi = lim ln |λα (n)| α = 1, · · · , N. n→∞
(7.43)
Note that in general there are as many Lyapunov exponents as the dimensionality of the map. If the map is area preserving, they come in pairs, i.e. for each positive exponent, a negative exponent with the same magnitude must occur. This corresponds to expanding and shrinking directions; It is obvious from (7.42) that, if we order Lyapunov exponents in decreasing order σ1 > σ2 > · · · σN (7.44) the largest (positive) exponent will eventually dominate the right hand side of (7.42). This ~ 0 in the direction of the will happen even if there is a vanishingly small component of δ X ~ eigenvector of Λ which corresponds to σ1 . The norm ||δ Xn || will grow exponentially as eσ1 n . This is exactly the physical content of “sensitive dependence on initial conditions”. Lyapunov exponents provide a measure of just how sensitive this dependence is. Note: here I have defined Lyapunov exponents in the context of maps. If time permits, I will present the definitions - and computational procedures - for dynamical systems governed by differential equations, i.e. Hamiltonian or dissipative dynamics.
7.3.5 The standard map kicked pendulum, kicked rotator Consider the nonautonomous Hamiltonian system defined by a kicked pendulum, where gravity acts in bursts, every τ seconds: ∞ X p2 K H= − cos(2πq) δ(t − nτ ) 2 (2π)2 n=−∞
(7.45)
where p is the angular momentum and 2πq the angle, referred to the perpendicular direction (cf. H-atom in electric field.) The equations of motion are p˙ =
∞ X ∂H K δ(t − nτ ) − =− sin(2πq) ∂q 2π n=−∞
q˙
∂H ∂p
=
.
The first equation implies that p is constant, except at times t = nτ , when it changes by a discrete step. Defining pn = lim p(nτ − ²) , ²→0
58
7 Chaos in low dimensional systems I can integrate in the neighborhood of t = nτ , set τ = 1 and obtain what is known as the standard map pn+1
=
qn+1
=
K sin(2πqn ) 2π qn + pn+1 .
pn −
(7.46)
Eqs. 7.46 belong to the general class (7.26) of area preserving twist maps. In the following, the coordinates pn , qn will be understood as mod1, unless stated otherwise. Fixed points The map (7.46) has two fixed points: • p∗ = 0, q ∗ = 0, which is elliptic, and • p∗ = 0, q ∗ = 1/2, which is hyperbolic . (NB: the published literature has adopted a variety of conventions; one of them has a different sign in the Hamiltonian; this amounts to a shift of q by 1/2) Summary of results At small values of K (cf. Fig. 7.6) there is no sign of chaos. We observe the tori which surround the elliptic fixed point, which extend up to the separatrix which leaves off the hyperbolic fixed point. Furthermore, we observe a large number of “horizontal” tori - meaning that they run all the way from left to right; these tori are the slightly deformed versions of the original irrational tori of the unperturbed system, which have survived the perturbation. Finally, the structure which emanates from the period 2 cycle, around the center of the picture, is visible. This resonant torus is broken; according to the Poincar´e-Birkhoff theory, we observe a period 2 island chain, and the hyperbolic fixed points nested between them. K=0.5
K=0.8
1.0
1.0
0.8
0.8
0.6
0.6
p
p 0.4
0.4
0.2
0.2
0.0 0.0
0.2
0.4
0.6
0.8
1.0
0.0 0.0
0.2
0.4
q
0.6
0.8
1.0
q
Figure 7.6: Trajectories of the standard map at K = 0.5 (left panel), K = 0.8 (right panel). As the perturbation increases, more and more near-resonant horizontal tori break up. Chaos develops around the separatrices of the leading resonances (hyperbolic fixed point
59
7 Chaos in low dimensional systems
K=0.9716354
K=1.1716354
1.0
1.0
0.8
0.8
0.6
0.6
p
p
0.4
0.4
0.2
0.2
0.0 0.0
0.2
0.4
0.6
0.8
0.0 0.0
1.0
0.2
0.4
0.6
0.8
1.0
q
q
Figure 7.7: Trajectories of the standard map at K = Kc = 0.9716354 (left panel), K = 1.17 (right panel).
and in the crossings between period 2 island chain). The survival of a torus depends on “how irrational”- its winding number is. In order to see what this means, look at the continued fraction representation of an irrational number w = a0 +
1 a1 +
1 a2 +···
≡ {a0 ; a1 , a2 , · · ·}
,
(7.47)
where the integers {ai } satisfy a0 ≥ 0 and ai > 1 ∀i ≥ 1. An n-th order approximation w = rn /sn can be generated by the sequence rn sn
= an rn−1 + rn−2 = an sn−1 + sn−2
(7.48)
with r−2 = 0, r−1 = 1, s−2 = 1, s−1 = 0. Eq. (7.48) implies that sn+1 > an+1 sn It follows that |w −
.
rn 1 1 |< < sn sn sn+1 an+1 s2n
(7.49) .
(7.50)
Thus, if an+1 is large, the nth approximation is a good one. An example is π = {3; 7, 15, 1, 292, · · ·} which leads to π = 3.14159265 · · · ≈ r3 /s3 = 355/113 = 3.14159292, good to 7 digits. Conversely, the representation √ 2 = {1; 2, 2, 2, 2, · · ·} √ leads to 2 = 1.414213 · · · ≈ r3 /s3 = 17/12 = 1.41666 · · ·, which has an error in the 4th digit. In this sense, the golden mean √ 5+1 = {1; 1, 1, 1, 1, · · ·} (7.51) 2 and its inverse
√
5−1 = {0; 1, 1, 1, 1, · · ·} 2
60
(7.52)
7 Chaos in low dimensional systems can be considered as the “most irrational” numbers. Therefore, the non-resonant torus with a winding number equal to the inverse golden mean, is expected to be the last to break. The disappearance of the last, “golden mean” torus at K = Kc = 0.9716354 (cf. Fig. 7.7, left panel)is a key event in the nonlinear scenario. It signals the transition from local to widespread chaos. The following aspects deserve special attention: • breaking of analyticity: As K approaches the critical value Kc , the deformation of the torus increases dramatically. The following procedure [14] makes it possible to follow the torus’ shape and detailed properties. First observe, following Greene [15], that an instability of a torus with irrational winding number w can be associated with the instability of an sn → ∞ cycle, where sn is defined in terms of the sequence rn /sn used to approximate w. Thus, rather that try to construct a torus directly, it is possible to determine successive cycles and their thresholds of instability. It useful to introduce the Moser representation (parametrization) [14] qj = tj + u(tj ) ,
(7.53)
appropriate to any cycle with a rational winding number w; here tj = jw = jr/s. The property qj = qj+s implies u(tj ) = u(tj + 1)
mod 1
.
(7.54)
Note that the periodic function u(t) - which can be shown to be odd - is only defined on a rational set t = tj = jr/s, but this set becomes more and populous as s is increased. Fig. 7.8 shows the dependence of u(t), evaluated for an s = 4181 cycle which approximates the torus with a golden mean winding number, as a function of K. Note how the function becomes less and less smooth as Kc is approached. • self similarity: Fig. 7.8 shows the shape of the KAM golden-mean torus at two noncritical K’s and at K = Kc . Note the detailed view of the non-smooth function. The detailed numerics [14] allows the conjecture of self-similarity; in other words, the valleys and hills of the curve repeat themselves at all possible scales of numerical observation. In this sense, KAM torus disappearance resembles a critical phenomenon. Self-similarity is very well demonstrated in the frequency spectra. The odd function u(t) can be represented in a Fourier series u(t) =
∞ X
Af sin 2πf t
f =1
The product f Af is shown in Fig. 7.9 as a function of f , for the same values of K as in Fig. 7.8. Note the presence of more and more peaks as the critical value of K is approached. At K = Kc self similar behavior occurs, with primary peaks occurring at the Fibonacci numbers. • Arnold diffusion: A picture of the widespread chaos which occurs at higher values of the nonlinear parameter K > Kc is given in Fig. 7.7) (right panel). Unless a point starts in the immediate neighborhood of the elliptic fixed point, or the very few islands, it will typically generate a chaotic orbit which may diffuse over a large two-dimensional area of phase space. This diffusive behavior can be quantitatively characterized as follows. Suppose we relax the mod 1 condition on the momentum p in (7.46). In other words we allow the phase space to be a cylinder of perimeter 1 and look at the quantity # " 2 (pj+n − pn ) , (7.55) D = lim n→∞ 2n
61
7 Chaos in low dimensional systems 0.0465
0.70
0.06 0.0460
0.676914
0.65
0.674538 0.120
0.125
0.130
0.04
p u 0.60
0.676912
0.674536 0.02
p 0.55
0.676910
0.0
0.2
0.4
0.6
0.8
0.674534
0.337
1.0
0.338
0.339
0.340
0.00 0.0
0.1
0.2
0.3
q
q
0.4
0.5
t
Figure 7.8: The torus with√ a winding number approximately equal to the inverse of the golden
mean W ∗ = ( 5 − 1)/2, at K = 0.5, 0.9, Kc . The curves shown are actually sets of discrete points belonging to cycles of order s = 4181 with a rational winding number w = r/s = 2584/4181 which differs from W ∗ by less than 3 × 10−8 . I. Left panel: the torus in the (p, q) plane. II. Center panel: a detailed view of the same torus in the cases K = .9 (right y-scale) and K = Kc (left y-scale). III. Right panel: the function u(t) which describes the torus of the standard map in parametric form. Again, the curve shown is actually obtained from an 4181-cycle which approximates the irrational winding number W ∗ . Note how the function changes from smooth at K = 0.5, to somewhat bumpy at K = 0.9, to very bumpy at K = Kc . The inset shows a detailed view of the critical curve, which suggests self-similar behavior (After [14]).
0.05
0.05
0.05
0.04
0.04
0.04
0.03
0.03
0.03
f|A(f)|
f|A(f)|
f|A(f)|
0.02
0.02
0.02
0.01
0.01
0.01
0.00
0.00 2
8
32
128
f
512 2048
0.00 2
8
32
128
512 2048
2
f
8
32
128
512 2048
f
Figure 7.9: Fourier coefficients of the function u(t), which describes parametrically the torus with with a golden mean winding number, at K = 0.5, 0.9, Kc . The quantity plotted is f |Af |. The curve at K = Kc (right panel), with primary contributions at the Fibonacci numbers, suggests self-similar behavior (after [14]).
which describes the coefficient of diffusion in momentum space. As long as K < Kc , the existence of even a single torus presents a topological barrier to diffusion1 . D should vanish. 1 this
is no more the case in higher dimensions! Arnold diffusion is generic in higher dimensionality because tori can be bypassed.
62
7 Chaos in low dimensional systems In the opposite limit K À Kc , we can estimate the diffusion coefficient as follows. From (7.46) j+n−1 K X pj+n = pj − sin(2πql ) (7.56) 2π l=j
and hence
µ (pj+n − pj )2 =
K 2π
¶2
j+n−1 X
sin(2πql ) sin(2πql0 ) .
(7.57)
l,l0 =j
Now, since qj+1 = qj + pj+1 mod 1, if pj+1 is large, qj+1 is essentially random, i.e. uncorrelated to qj . The only correlations which survive are from terms l0 = l. On the average, the double sum will therefore be equal to n/2 (the 1/2 factor from the average value of sin2 ). Therefore µ D≈
K 4π
¶2 if
K À Kc
.
(7.58)
In the case where K slightly exceeds Kc , Chirikov has estimated 2.56
D ∝ (K − Kc )
.
(7.59)
• Cantori: (7.59) makes clear that, even beyond the stochasticity threshold, diffusion does not proceed uninhibited. At values slightly above Kc the diffusion constant is in fact very close to zero. It appears that some barriers to diffusion persist after all KAM tori have been broken. Resistance to diffusion can be related to a particular class of orbits with irrational winding numbers, which do not fully cover a one-dimensional curve (Fig. 7.10), but leave a countable set of open intervals empty - i.e. they form a Cantor set. Because of this property they were named cantori by Percival. The existence of cantori as isolated regular orbits embedded in a sea of chaos is remarkable. We will deal with them again in Chapter 9, in the context of solid state theory. An excellent review of the transport properties of Hamiltonian maps has been written by Meiss [16].
7.3.6 The Arnold cat map The area preserving map xn+1 yn+1
= xn + yn mod 1 = xn + 2yn mod 1
has a tangent map
µ M=
1 1
1 2
(7.60)
¶ (7.61)
which does not depend on the coordinates. The eigenvalues of the map are λ1,2 =
√ ´ 1³ 3± 5 2
(7.62)
and the Lyapunov exponents σ1 = ln λ1 = −σ2 . There is a single, hyperbolic fixed point at x∗ = y ∗ = 0. Neighboring trajectories everywhere diverge exponentially. All cycles are unstable. What happens to a cat thus mapped is shown in Fig. 7.11.
63
7 Chaos in low dimensional systems
0.8
0.8
0.7
0.7
p
p 0.6
0.6 0.62 0.61 0.60
0.5
0.5
0.59 0.58
0.58
0.60
0.62
0.4
0.4 0.0
0.2
0.4
0.6
0.8
1.0
0.0
0.2
0.4
q
0.6
0.8
1.0
q
Figure 7.10: Standard map cantori with a golden-mean winding number, obtained at K −Kc = 0.01 (left panel) and K − Kc = 0.3 (right panel). 1.0
1.0
1.0
0.8
0.8
0.8
0.6
0.6
0.6
y
y
y
0.4
0.4
0.4
0.2
0.2
0.2
0.0 0.0
0.2
0.4
0.6
0.8
0.0 0.0
1.0
0.2
0.4
0.6
0.8
1.0
0.0 0.0
x
x
0.2
0.4
0.6
0.8
1.0
x
Figure 7.11: The fate of a cat under two iterations of the map (7.60).
7.3.7 The baker map; Bernoulli shifts The map xn+1 yn+1 xn+1 yn+1
= =
= =
2xn 1 2 yn
2xn − 1 1 1 2 yn + 2
¾
1 2
if
0
≤ xn
