Available free energy under local phase space diffusion M. J. Hay,1, a) J. Schiff,2 and N. J. Fisch1, 3 1)
Department of Astrophysical Sciences, Princeton University, Princeton, New Jersey 08544 Department of Mathematics, Bar-Ilan University, Ramat Gan, 52900 Israel 3) Princeton Plasma Physics Laboratory, Princeton, New Jersey 08543 2)
arXiv:1604.08573v1 [math-ph] 28 Apr 2016
(Dated: 29 April 2016)
The free energy associated with the birth distribution of alpha particles in DT fusion reactors is an enormous resource that can be tapped with waves. An interesting academic question is the maximum energy that can be extracted from the alpha particle birth distribution in the quasilinear (diffusive) limit of wave-particle interaction. In the nonlocal problem, particles may be diffused from high energy to any lower energy; in the local diffusion problem, particles may only be diffused to incrementally smaller energies. It is shown that the discrete version of the nonlocal problem corresponds to diffusion on the complete graph Kn , whereas the local problem is a solvable case defined on the path graph Pn . We further discuss the extension of the solution to arbitrary subgraphs of Kn . I.
INTRODUCTION
There is a class of diffusion problems that is motivated by opportunities in extracting energy in plasma by waves. Waves can be injected into a fusion reactor such that high energy alpha particles, the byproducts of the fusion reaction, lose energy to the waves, as those alpha particles are diffused by the waves to lower energy.1–3 That extra energy in the waves can then be used, for example, to increase the reactivity of the fuel or to drive electric current.4,5 The wave-particle interaction, however, is by its very nature diffusive; particles in a given location in the 6D phase space of velocity and position are randomly mixed by the wave with particles in another position in the 6D phase space. Thus, the diffusion occurs along a path in velocity-position space. If there is a population inversion in energy along that diffusion path in velocityposition space, then the diffusive process releases energy. If there is only one wave and one diffusion path, the amount of extractable energy can be readily calculated. However, more energy can be extracted when several waves are employed.6,7 When many diffusion paths are possible, it turns out that the order in which these paths are taken affects the energy that can be extracted. Thus, a class of diffusion problems, each involving many paths, will determine the maximum amount of extractable energy under the constraint that the particle distribution function evolves only due to diffusion.8 One formulation of this diffusion problem is to consider the sequence of equalizing the populations of any two energy levels. This is the full-connectivity, or nonlocal, diffusion problem, where diffusion paths can be constructed between any two phase space locations. There are physical reasons why this arrangement is realizable in practice, since, on a sufficiently macroscopic scale, the diffusion of phase space density can appear nonlocal despite the restriction of diffusion to contiguous regions of phase space on the microscopic scale.8 The maximum energy
a)
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extractable under this constraint, or what we call the free energy, has been described previously.9 Optimization problems subject to analogous constraints abound in various fields.10–13 However, the more physical problem is strictly local diffusion, without full connectivity between each volume element of phase space. Rather, waves can only diffuse particles from one phase space position to a contiguous position. In that case, the free energy is less, because there are fewer ways in which the energy might be released. It is the objective here to describe the maximum energy that can be extracted, or the free energy available, in what we call the local diffusion problem. It is what we might call the second interesting academic problem in free energy under the diffusive constraint. The case of nonlocal diffusion can be described as a system of N states, where an energy and a population number is associated with each state. Under nonlocal diffusion, population levels at each step can be equalized in any two states (to be followed by subsequent equalizations in subsequent steps). Under local diffusion, population levels can be equalized in any two states, so long as they are contiguous. A reasonable model of contiguity can be arrived at as follows: first, order all the states according to energy; then allow diffusion only between neighboring states. This model is a simplification; in fact, diffusion occurs in the 6D space of velocity and space, and contiguous regions are described in this 6D space. What this means is that the ordering of states should include many degeneracies, or multiple states with the same energy, with connectivity based on both energy and other considerations (e.g. proximity in position). However, many physical problems of interest are captured by the model of contiguity based on energy only. Other network problems can be defined which capture the spatial element.14 Given then our definition of contiguity, like the nonlocal case, the local diffusion free energy problem has a linear programming solution over the space of accessible states. This work will characterize and enumerate the extreme points of the feasible region under local diffusion. The set of states which can be reached by local
2 diffusive processes is a proper subset of those accessible under nonlocal diffusion. Moreover, the free energy of a system under local diffusion is less than or equal to that of the same system allowing nonlocal diffusion.
This paper is organized as follows. In Sec. II, we review the discrete diffusion model. In Sec. III, we construct the state space accessible with local diffusion operations, emphasizing the location of this space within the general nonlocal space. Sec. IV gives an accounting of the vertices of the local state space polytope which are inherited from the nonlocal problem. Sec. V compares the solution of local and nonlocal diffusion problems and discusses a graph-theoretical framework for treating diffusion-like problems in more complex physical systems.
3
B12
B34
This approach offers some insight into the ‘full’ alpha particle diffusion problem (in both position and energy space). As above, edges correspond to diffusion paths and vertices reference bins associated with the discretization of configuration and energy space. A particularly simple 2-D diffusion problem in x and can be visualized as in Fig. 2, on the composition of path graphs Pm [Pn ], m ≥ n, essentially a grid graph with diagonal edges.16 In this example, diffusion paths are limited to nearest neighbors in either x or . Such an arrangement is characterized by degeneracy in both position and energy bins. A practical alpha channeling scheme would seek to maximize the density at a particular low-energy, large-radius sink node.
13
B
4
In a diffusion problem it is useful to consider the graph with vertices associated with particular energy levels and edges corresponding to the available diffusion operators. Fig. 1 compares the graph representations of the local and nonlocal problems for a four-level system. The representation of the fully nonlocal problem on n levels is isomorphic to the complete graph Kn and likewise the representation of the local problem is isomorphic to the path graph Pn . In the diffusion problem, the distribution function may be flattened along any path on the graph. For example, in Fig. 1(b), the path graph admits the mixing of levels 1, 2, and 3, but not 1, 2, and 4.
B2
Whereas the extreme points of the nonlocal problem can all be constructed by n2 or fewer level mixings in an n-level system, some extreme points unique to the local problem are only reachable by an infinite sequence of operations. (The extreme points inherited from the nonlocal problem can all be reached by only bn/2c.) Curiously, the number of extreme points common to the n-level local and nonlocal problems is the (n + 1)’th term of the Fibonacci sequence. In the local problem, these vertices are complemented by a set of points in the closure of the hull which brings the total number of extreme points to 2n−1 . In fact, the local extreme points have a natural correspondence to the vertices of an (n − 1)-dimensional hypercube. By contrast, the correct enumeration of extreme points in the nonlocal problem is an ongoing effort which owes much to the formalism of R. Stanley.15
B23
2
B14
1
4
(a) Nonlocal diffusion: K4 B12
1
2
B23
3
B34
4
(b) Local diffusion: P4
FIG. 1. Graph representations of two diffusion problems on a four-level system. x
FIG. 2. Diffusion problem on G = P3 [P2 ].
II.
DIFFUSION MODEL
The density vector ρ = (ρ1 , ρ2 , . . . , ρn ) is a probability measure for n system states. A particular example might be the number of electrons in a certain energy P level of an atomic ensemble. ρ is accordingly normalized: i ρi = 1. The objective function f = ρ · w is a particular linear combination of the densities determined by the weights w = (w1 , w2 , . . . , wn ). In our example, these correspond to the level energies. We have assumed ρ1 ≤ ρ2 ≤ · · · ≤ ρn and w1 < w2 < · · · < wn in order to address the most difficult case; if the wi are not strictly increasing, only a subset of the extreme points will exist. The diffusion problem is the minimization of f over ρ in the set K(ρ0 ), the states that can be reached from the initial density vector ρ0 by successive pairwise relaxations of densities. K(ρ0 ) is generally nonconvex but starlike about the uniform distribution ρ = (1/n, 1/n, . . . , 1/n).11,12 Due to the linearity of the objective function f , the diffusion problem has a linear programming solution when taking the convex hull of K, ch(K), as the feasible region.9 We choose the symbol Bij;α to represent a relaxation operation on the states i and j. The Bij;α are examples of the T -transform: T = (1 − α)I + αQ, α ∈ [0, 1], I is the n × n identity matrix, and Q is a permutation of the
3
III.
STATE SPACE UNDER LOCAL DIFFUSION
The state space of a diffusion problem is the trace (free partially commutative) monoid on an alphabet of Bij . In the general, nonlocal problem, the alphabet contains n 2 unique Bij , where n is the number of levels in the system under consideration. Each Bij corresponds to the averaging of any pair of levels i, j such that i 6= j. Likewise, the ‘local’ state space is formed from a strictly smaller alphabet containing only those Bij of the form Bi,i+1 . Clearly, there are n − 1 such operators. The commutation rule in both monoids is familiar from the symmetric group: any two Bij which do not operate on a common level commute. For example B12 and B34 commute, whereas B12 and B23 do not. As noted, this rule induces an equivalence relation between Bij sequences that correspond to the same point in density space.
A.
Nonlocal extreme points
0.5●
▲
▲
n2
identity matrix which exchanges only two rows.17 Like all doubly stochastic matrices, P P T -transforms are measurepreserving: ab Tab ρb = b ρb = 1. It is important to distinguish between the cases of 0 ≤ α ≤ 1/2 and 1/2 < α ≤ 1; the latter case corresponds to moving density from a less-populated state to a morepopulated state. Thus in treating the diffusion problem we will consider only α ≤ 1/2 transforms. In fact, it is sufficient in this case only to consider those Bij;α with α = 1/2, which we refer to henceforth without ambiguity as the Bij .9 For the nonlocal case, Ref. 12 presented a recursive algorithm to identify the extreme points of ch(K) by applying different sequences of these Bij . It was noted in Ref. 9 that this algorithm generates reduced (minimal length) decompositions of the elements in the symmetric group Sn .18 That is, the algorithm identifies every possible minimum-length way to generate each of the n! permutations of a length-n word using only adjacent transpositions σi = (i i + 1). It turns out that each nonlocal extreme point is in bijection with a particular inequivalent reduced decomposition, i.e. an equivalence class of decompositions obtainable from each other by the commutation relation σi σj = σj σi , |i − j| > 1.19
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0.25
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0
0.25 n1
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FIG. 3. Comparison of convex hulls for local and nonlocal state spaces given the initial data ρ = (0, 2/7, 5/7). (Due to normalization, the third coordinate is ignorable.) The extreme points for the local space are labeled with black circles; the extreme points for the nonlocal space are labeled with blue triangles.
Alternately, sequences comprised of local operators Bi,i+1 generate a proper subset of the extreme points of ch(Knl ). These extreme points necessarily carry over to the local problem, because the local state space is in the closure of the nonlocal state space.
B.
Asymptotic local extreme points
In addition to the extreme points inherited from the nonlocal problem, there is a new set of extreme points unique to the local problem. These ‘local’ extreme points are necessarily interior points in the nonlocal convex hull ch(Knl ). They result from the mixture of a contiguous set of levels by an infinitely long sequence of operations Bij . For example, one extreme point for the local state space beginning with ρ0 = (ρ1 , ρ2 , ρ3 ) is the uniform distribution ρ1 + ρ2 + ρ3 ρ1 + ρ2 + ρ3 ρ1 + ρ2 + ρ3 e= , , 3 3 3 = (ρ1 , ρ2 , ρ3 ) = (1/3 , 1/3 , 1/3 ),
One immediate observation is that the local state space is entirely contained within the closure of the nonlocal state space. This follows from the monotonicity of the convex hull operator ch: for two sets X and Y with X ⊆ Y , ch(X) ⊆ ch(Y ). In particular, the trace monoid on the local Bij , Kl , is a proper subset of the trace monoid on all possible Bij , Knl . Therefore ch(Kl ) ⊆ ch(Knl ). Figure 3 compares the convex hulls of the two state spaces in a three-level system.
by normalization. We will prefer the overline notation to denote averaging of states due to its compactness. In fact, the uniform distribution is only one example of the new class of extreme points. There are additional extreme points formed by the mixture of a subset of k > 2 contiguous levels such that they have identical populations. The extreme points containing k = 2 mixtures have already been identified, as these are extreme
4 muting 2-tuple of the local Bij : B12 and B34 . Because these Bij do not operate on a shared level, the states reached by applying them in either order are identical. Both correspond to the same inequivalent reduced factorization, namely {(1, 3), (3, 1)}. More generally, the problem becomes counting the commuting k-tuples among the local Bij . With n = 4 and n = 5, only commuting 2-tuples are possible. This changes with n = 6, when a commuting 3-tuple appears: (B12 , B34 , B56 ). In fact, an n-level system contains only k-tuples satisfying k ≤ bn/2c. Clearly, as n grows, the number of commuting tuples with k n becomes large and direct counting becomes tedious, if not difficult. Fortunately, a general formula is available. Appropriating the notation of Ref. 21, denote the number of commuting k-tuples in a set of cardinality n as Ak (n). Recalling that there are n−1 local Bij in the n-level problem, the number of extreme points for n > 2 levels is FIG. 4. Convex hull of a four-level local diffusion problem represented in R3 . Extreme points are denoted with red circles.
points in the nonlocal problem as well. In the appendix, we prove that convex hull of the local state space has 2n−1 extreme points in bijection with the power set of {1, 2, . . . , n − 1}. Then it is evident that the extreme points for the nlevel problem have the structure of an (n−1)-dimensional hypercube,20 with 2n−1 vertices and n−1 edges from each vertex, in correspondence with the n − 1 sets differing from that vertex’s subset by the inclusion or exclusion of one element (see Appendix). Figs. 3 and 4 illustrate the 2- and 3-cube hulls, respectively, for three- and four-level local problems.
IV. ENUMERATION OF NONLOCAL EXTREME POINTS
All nonlocal extreme points are in bijection with inequivalent reduced factorizations of the permutations in Sn . Those retained as local extreme points therefore remain in one-to-one correspondence with various factorizations. However, there is an interesting and illuminating way to enumerate the local extreme points without reference to this correspondence. Beginning with n = 2, there are but two extreme points: the initial distribution ρ0 and the distribution obtained by averaging the densities in the two levels: ρ0 B12 . With n = 3, there are two local Bij : ρ0 , ρ0 B12 , and ρ0 B23 are all extreme points. Only with n = 4 do we get a flavor of the true problem at hand. There are four extreme points of the types just considered: ρ0 , ρ0 B12 , ρ0 B23 , and ρ0 B34 . Additionally, there is a unique extreme point associated with a com-
1 + (n − 1) + A2 (n) + A3 (n) + · · · + Abn/2c (n),
(1)
where the leading 1 corresponds to the initial distribution ρ0 . We can now attack the Ak (n) in turn. Mapping each Bi,i+1 to the symbol i, A2 (n) is the number of twoelement subsets of {1, 2, . . . , n − 1} which do not contain consecutive numbers. The total number of two-element subsets is n−1 and the number of subsets containing 2 consecutive numbers is n − 2. Therefore n−1 A2 (n) = − (n − 2) = Tn−3 (2) 2 where Tn is the nth triangular number (recall that the expression is restricted to n > 2). Proceeding analogously, A3 (n) is seen to correspond to the tetrahedral numbers. In general, Ak (n) can be identified with the set of regular k-polytopic numbers.22 With the proper offsets, the formula for the number of extreme points for n > 2 is n−2 n−3 n+ + + · · · = Fn+1 , (3) 2 3 where Fn is the nth term of the Fibonacci sequence: 0, 1, 1, 2, 3 . . . with F0 = 0. The identity is the statement that shallow diagonals of Pascal’s triangle sum to Fibonacci numbers:23 bn/2c
X
k=0
n−k k
= Fn+1 .
(4)
(Note that n0 + n−1 = n.) 1 This result might have been anticipated because there are Fn+2 unique subsets of {1, . . . , n} which do not contain consecutive numbers.24 For example, there are three (= F4 ) such subsets of {1, 2}: {∅, {1}, {2}}. The problem of enumerating the extreme points is analogous: the
5
FIG. 5. Comparison of minimal energy states for the diffusion problem with initial data ρ(w) ∝ ew and w = (1, 2, 3, 4). The blue circles label the level densities for the nonlocal problem and likewise the yellow squares the local problem. Whereas the nonlocal operations recovered 68% of the Gardner limit,9 the local operations managed only 50%.
n-level system contains n − 1 operators, leading to Fn+1 Fibonacci subsets. The Fibonacci numbers are the solution to a similar problem in graph theory: Fn+1 is the number of matchings in a path graph with n vertices.25
V.
DISCUSSION
The local diffusion problem admits a general solution for ρ(w) increasing. Whereas the nonlocal problem required the evaluation of a super-exponential number of solutions in the number of levels n, in the local problem the solution is always the uniform distribution ρf (w) = const.9 In some sense, the reason for this behavior is the absence of a local extreme point with the decreasing permutation {n, n−1, . . . , 1}. For illustration, we compare the solution of ρ(w) ∝ ew in the local and nonlocal state spaces, Fig. 5. The case where ρ(w) is non-increasing educes a generalization of the local problem. As before, there are n − 1 operators available, but these are generally distinct from the increasing case discussed so far. For example, the initial data ρ0 = (1, 2, 3, 4)/10 and w = (2, 1, 4, 3) give rise to a diffusion problem with operators B12 , B14 , and B34 . Then extreme points such as (ρ1 , ρ2 , ρ3 , ρ4 ) and (ρ1 , ρ2 , ρ3 , ρ4 ) are inaccessible. Only strict subsets of the inherited nonlocal and unique local extreme points remain. However, these systems may contain new extreme points. We can generalize to other restricted versions of the fully nonlocal problem. An immediate example is the selection rules associated with atomic transitions. Imagining a process which can mix levels joined by e.g. a dipole transition (such as spatially incoherent light at the appropriate frequency), interesting questions arise concerning the maximum possible concentration of the atoms of an ensemble into a specific state.
The simplest system which differs from the nonlocal problem (in the nonrelativistic case) is helium. Fig. 6 contains a truncated level diagram for parahelium (S = 0). Due to level splitting, each energy level is associated with a unique energy. We suppose that the available operations correspond to the allowed dipole transitions. For example, it is possible to average the number of atoms in 2s and 3p, but it is not possible to do the same for 2s and 3s. Four operators are allowed in this five-level system: B1s,2p , B1s,3p , B2s,3p , and B3s,2p . We gain a clearer picture by expressing the energy level diagram of Fig. 6 as a graph G(V, E) with vertices V = {1s, 2s, 3s, 2p, 3p} and edges E = {{1s, 2p}, {1s, 3p}, {2s, 3p}, {2p, 3s}}. The vertices could also be labeled by the energy eigenvalues, and likewise the edges correspond to allowed transitions. In this context, we can educe an analogous diffusion problem with operators B13 , B15 , B25 , and B34 corresponding to edges. Fig. 7 illustrates the graph for the helium problem thus defined, which as discussed is distinct from the nonlocal problem detailed above. In general, one might consider diffusion problems on arbitrary subgraphs of the complete graph for V = {w}, of which the local problem is one solvable case (the path graph with V = {w}). These other problems will inherit some extreme points from the complete graph on the same number of vertices and should also have some unique ones depending on the particular graph structure. There may be rich physical significance for diffusion problems on more or less ‘bottlenecked’ graphs generally (in the sense of Cheeger), or e.g. the wheel graphs, k-regular trees, and complete bipartite graphs.
ACKNOWLEDGMENTS
The authors thank Professor D. Tannor for critical discussions at an early stage of this work. Work supported by DOE Contract No. DE-AC02-09CH11466 and DOE NNSA SSAA Grant No. DE274-FG52-08NA28553. One of us (NJF) acknowledges the hospitality of the Weizmann Institute of Science, where he held a Weston Visiting Professorship during the time over which this work was initiated. 1 N.
J. Fisch and J.-M. Rax, “Interaction of energetic alpha particles with intense lower hybrid waves,” Phys. Rev. Lett. 69, 612–615 (1992). 2 N. J. Fisch and J.-M. Rax, “Current drive by lower hybrid waves in the presence of energetic alpha-particles,” Nucl. Fusion 13, 549 (1992). 3 N. J. Fisch and M. C. Herrmann, “A tutorial on alphachanneling,” Plasma Phys. Control. Fusion 41, A221 (1999). 4 N. J. Fisch and M. C. Herrmann, “Utility of extracting alphaparticle energy by waves,” Nucl. Fusion 34, 1541 (1994). 5 N. J. Fisch, “Theory of current drive in plasmas,” Rev. Mod. Phys 59, 175 (1987). 6 N. J. Fisch and M. C. Herrmann, “Alpha power channeling with two waves,” Nucl. Fusion 35, 1753 (1995). 7 M. C. Herrmann and N. J. Fisch, “Cooling energetic alpha particles in a tokamak with waves,” Phys. Rev. Lett. 79, 1495 (1997).
6
3p
3s
2p 2s
1s FIG. 6. Truncated parahelium energy level diagram with all possible electric dipole transitions represented by arrows, ↔. Energy scale is arbitrary. The s − p splitting is due chiefly to the partially screened Coulomb repulsion of the nucleus (a larger effect in p orbitals vs. s orbitals).26
3
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B34
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B
Appendix: The SA are the complete set of extreme points
B1
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1
B25
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FIG. 7. Graph representation for the diffusion problem on parahelium, cf. Fig. 6. Vertices are labeled by the rank of the corresponding energy eigenvalue (increasing). Although the graph is isomorphic to P5 , the solution of the diffusion problem is distinct from the local case.
8 N.
of Majorization and Its Applications, 2nd ed. (Springer, New York, 2009). 18 OEIS Foundation Inc., “The On-Line Encyclopedia of Integer Sequences,” http://oeis.org/A246865 (2014), [Online; accessed 13-Aug-2015]. 19 G. Berkolaiko and J. Irving, “Inequivalent factorizations of permutations,” J. Combin. Theory Ser. A 140, 1–37 (2016). 20 E. W. Weisstein, “Hypercube. From MathWorld—A Wolfram Web Resource,” (n.d.), last visited on 4/24/2016. 21 P. Erd¨ os and E. Straus, “How abelian is a finite group?” Linear and Multilinear Algebra 3, 307–312 (1976). 22 E. Deza and M. M. Deza, Figurate Numbers (World Scientific, Hackensack, NJ, 2012). 23 C. Stover and E. W. Weisstein, “Pascal’s Triangle. From MathWorld—A Wolfram Web Resource,” (n.d.), last visited on 1/27/2016. 24 L. Comtet, Advanced Combinatorics (D. Reidel, Dordrecht, The Netherlands, 1974). 25 H. Prodinger and R. Tichy, “Fibonacci numbers of graphs,” Fibonacci Q. 20, 16–21 (1982). 26 F. Schwabl, Quantum Mechanics, 4th ed. (Springer, Berlin, 2007).
J. Fisch and J.-M. Rax, “Free energy in plasmas under waveinduced diffusion,” Phys. Fluids B 5, 1754 (1993). 9 M. J. Hay, J. Schiff, and N. J. Fisch, “Maximal energy extraction under discrete diffusive exchange,” Phys. Plasmas 22, 102108 (2015). 10 F. Horn, “Attainable and non-attainable regions in chemical reaction techniques,” in Proceedings of the 3rd European Symposium on Chemical Reaction Engineering (Pergamon, 1964) pp. 1–10. 11 C. Zylka, “A note on the attainability of states by equalizing processes,” Theor. Chim. Acta 68, 363 (1985). 12 D. Thon and S. W. Wallace, “Dalton transfers, inequality and altruism,” Soc. Choice Welfare 22, 447 (2004). 13 M. C. Levy, S. C. Wilks, M. Tabak, S. B. Libby, and M. G. Baring, “Petawatt laser absorption bounded,” Nat. Commun. 5, 4149 (2014). 14 A. I. Zhmoginov and N. J. Fisch, “Flux control in networks of diffusion paths,” Phys. Lett. A 372, 5534–5541 (2008). 15 R. P. Stanley, “On the number of reduced decompositions of elements of Coxeter groups,” Europ. J. Combinatorics 5, 359–372 (1984). 16 E. W. Weisstein, “Graph Composition. From MathWorld—A Wolfram Web Resource,” (n.d.), last visited on 4/28/2016. 17 A. W. Marshall, I. Olkin, and B. C. Arnold, Inequalities: Theory
The local state space is the set of states accessible from the initial state ρ0 , after an arbitrary finite sequence of transformations Bi,i+1 has been applied. We claim that the convex hull of (the closure of) this space has 2n−1 extreme points in bijection with the power set of {1, 2, . . . , n − 1}. Denote these points SA , where the index A runs over all subsets of {1, 2, . . . , n − 1}. The bijection has a simple description. The empty set ∅ corresponds to the initial state ρ0 . The point SA corresponding to the set A has the property that for each i ∈ A the i’th and (i + 1)’th components of SA are equal, and its components are obtained by averaging over subsets of components of ρ0 . So, for example, in a seven-level system, S{1,2,3,6} is the point (x, x, x, x, y, z, z) where x is the average of the first four components of ρ0 , y is the fifth component, and z is the average of the sixth and seventh components. In greater generality, whenever A contains a sequence of k consecutive integers i, i + 1, . . . , i + k − 1 then the k + 1 components of SA from i to i + k are equal to the average of the corresponding components of ρ0 . Lemma 1. The SA are contained in the closure of the local state space. Proof. Fix a subset A ⊆ {1, 2, . . . , n − 1} corresponding to a particular SA . Suppose i, i + 1, . . . , i + k − 1 is a maximal sequence of k consecutive integers in A, i.e. all these are in A, but i − 1 and i + k are not, and consider the quantity ρi+k − ρi . This is non-negative, and, if it is non-zero, strictly decreases when the operator Bi+k−1,i+k . . . Bi+1,i+2 Bi,i+1 is applied to the state ρ. Furthermore it is unchanged when the corresponding operator for a different maximal sequence of consecutive integers is applied to ρ. It follows that by repeated application of such operators the initial state ρ0 can be brought arbitrarilly close to the extreme point SA .
7 Thus the points SA are in the closure of the local state space. Lemma 2. No point SA can be written as a nontrivial convex combination of the others. P Proof. Consider aA SA , with P a convex combination aA ≥ 0 and aA = 1. Let ρA,i denote the component P densities of the point SA . Suppose aA SA = S, with densities ρi . Because an arbitrary sequence of local operators Bi,i+1 cannot achieve a population inversion, we have ρA,i ≤ ρA,i+1 for all A and i. Therefore ρi = ρi+1 if and only if ρA,i = ρA,i+1 for all A with aA > 0. Thus any convex combination yielding the point S has only one summand, S itself. Lemma 3. Any point in the local state space can be written as a convex combination of the SA . Proof. We wish to show that every accessible state ρ can be written as a convex combination of the extreme points, i.e. that we can write X X ρ= λA SA , where λA ≥ 0 , λA = 1. A
A
Every accessible state can be obtained by applying an arbitrary finite sequence of Bi,i+1 to the initial state ρ0 = S∅ . We proceed by induction on the number of Bi,i+1 operators to be applied. To prove the inductive step it is necessary to check that for every A and for every i, SA Bi,i+1 can be written as a convex combination of extreme points. If i ∈ A, then SA Bi,i+1 = SA is an extreme point. If i ∈ / A then the i’th and (i + 1)’th components of SA may be different. SA then takes the form (. . . , x, . . . , x, x, y, y, . . . y, . . .) . where there is a string of k > 0 x’s ending in the i’th position and a string of l > 0 y’s starting in the (i + 1)’th position. The entries on the left of the string of x’s and on the right of the string of y’s remain fixed and identical for all the points that appear in the forthcoming calculation, and thus do not play any role. It is assumed that the entry immediately on the left of the x’s (if there is such) is strictly less than x, and the entry immediately on the right of the y’s (if it exists) is strictly greater than y. Applying Bi,i+1 we have SA Bi,i+1 = (. . . , x, . . . , x,
x+y x+y , , y, . . . , y, . . .) 2 2
where now the strings of x’s and y’s are of length k−1 and l − 1 respectively. It should be emphasized that except in the case k = l = 1 this is not an extreme point, as although the values in the i’th and (i+1)’th positions are equal, they have been determined by components of the initial density ρ0 from outside these positions, and full averaging over the relevant subset has not been achieved.
However we will now show it is a convex combination of four extreme points of the following form: S1 = (. . . , X, . . . X, X, X, X, . . . X, . . .), with k + l X’s, S2 = (. . . , X1 , . . . , X1 , X1 , X1 , Y1 , . . . , Y1 , . . .), with k + 1 X1 ’s and l − 1 Y1 ’s, S3 = (. . . , X2 , . . . , X2 , Y2 , Y2 , Y2 , . . . , Y2 , . . .), with k − 1 X2 ’s and l + 1 Y2 ’s, and S4 = (. . . , X2 , . . . , X2 , Z, Z, Y1 , . . . , Y1 , . . .), with k − 1 X2 ’s and l − 1 Y1 ’s. (These four points are only distinct on the assumption k, l > 1. The cases k = l = 1, k = 1, l > 1, and k > 1, l = 1 should be considered separately but are simpler — in particular in the case k = l = 1 all four points coincide and the resulting point is extreme. We omit the details of the cases k = 1, l > 1 and k > 1, l = 1.) The quantities x, y, X, X1 , Y1 , X2 , Y2 , Z appearing here are not independent, as they are obtained from averaging over certain entries of ρ0 . Denote the average value of entries i − (k − 1), . . . , i − 1 of ρ0 as R1 , the value of entry i as R2 , the value of entry i + 1 as R3 and the average of entries i + 2, . . . , i + l as R4 . Then (k − 1)R1 + R2 k R3 + (l − 1)R4 = l (k − 1)R1 + R2 + R3 + (l − 1)R4 = k+l (k − 1)R1 + R2 + R3 = k+1 = R4 = R1 R2 + R3 + (l − 1)R4 = l+1 R2 + R3 = 2
x= y X X1 Y1 X2 Y2 Z
Our aim is to find λ1 , λ2 , λ3 , λ4 ≥ 0 such that SA Bi,i+1 =
4 X
λi Si and
i=1
4 X
λi = 1
i=1
Solving these linear equations gives three constraints between the λi , which can be written in the form λ1 = C1 λ4 + D1 λ2 = C2 λ4 + D2 λ3 = C3 λ4 + D3 where C1 , C2 , C3 , D1 , D2 , D3 are complicated expressions involving k, l and R1 , R2 , R3 , R4 . However, it follows from R1 ≤ R2 ≤ R3 ≤ R4 that C1 > 0 and C2 , C3 < 0.
8 (Expressions for C1 , C2 , C3 are given later.) Thus for nonnegativity of the λi we need
n − 1 sets differing from A by the inclusion or exclusion of a single element. Expressions for Ci and Di
D1 D2 D3 max 0, − ≤ λ4 ≤ min − ,− C1 C2 C3
(A.1)
D2 D3 1 Expressions for − D C1 , − C2 , − C3 are given later, from D3 2 which it can be seen that − D C2 , − C3 are both nonneg1 ative and not less than − D C1 . Thus the inequalities (A.1) 1 can always be satisfied. Note that the quantity − D C1 is of indeterminate sign. When it is negative it is possible to take λ4 = 0 and thus write SA Bi,i+1 as a convex combination of just three extreme points. However, in general, four are needed.
Theorem 1. The convex hull of the closure of the local state space has 2n−1 extreme points in bijection with the power set of {1, 2, . . . , n − 1}. Proof. By Lemma 1, the SA are all contained in the closure of the local state space and thus in its convex hull. Thus the convex hull of the points SA is contained in the convex hull of the closure of the local state space. By Lemma 2, no point SA can be expressed as a nontrivial convex combination of the others. Thus the convex hull of the points SA has all the 2n−1 points SA as extreme points. By Lemma 3, any point in the local state space can be expressed as a convex combination of the SA . Thus the local state space is contained in the convex hull of the points SA , which is a closed convex set, so must therefore also contain the convex hull of the closure of the local state space. Combining these results we conclude that the convex hull of the closure of the local state space can be identified with the convex hull of the points SA and these are a complete set of extreme points. Corollary 1. The extreme points of the local state space for the n-level diffusion problem form an (n − 1)dimensional hypercube. Proof. By Theorem 1, the total number of extreme points is 2n−1 ; these correspond to the vertices of an (n−1)-dimensional hypercube.20 Choose a particular vertex corresponding to the subset A ⊆ {1, 2, . . . , n − 1}. Because each vertex corresponds to an element of the power set of {1, 2, . . . , n − 1}, there are n − 1 edges from the vertex associated with A in correspondence with the
C1 =
(p2 + 2 p3 ) (p2 + 2 p1 ) (k + l) 2 ((k − 1)p1 + kp2 + (k + 1)p3 ) 1 × (l + 1)p1 + lp2 + (l − 1)p3
C2 = −
(p2 + 2 p3 ) (k + 1) 2 ((k − 1)p1 + kp2 + (k + 1)p3 )
C3 = −
(p2 + 2 p1 ) (l + 1) 2 ((l + 1)p1 + lp2 + (l − 1)p3 )
Here p1 = R2 − R1 p2 = R3 − R2 p3 = R4 − R3 with pi ≥ 0. Also −
D1 = (k − 1)lp1 p2 + klp22 C1 + k(l − 1)p2 p3 − 2(l + k)p1 p3 1 × (p2 + 2 p3 ) (p2 + 2 p1 ) kl
−
D2 (k − 1)lp1 + klp2 + k(l − 1)p3 = C2 (p2 + 2p3 ) kl
−
D3 (k − 1)lp1 + klp2 + k(l − 1)p3 = C3 (p2 + 2p1 ) kl
To see the inequalities (A.1) can be satisfied observe that D3 2 −D C2 , − C3 > 0 and D1 = C1 2p1 ((k − 1)p1 + kp2 + (k + 1)p3 ) >0 k (p2 + 2p3 ) (p2 + 2p1 ) D3 D1 − − − = C3 C1 2p3 ((l + 1)p2 + lp2 + (l − 1)p3 ) >0 l (p2 + 2p3 ) (p2 + 2p1 )
−
D2 − C2
−
