Statistical models of clustering matter

Physica A 197 (1993) 629-666 North-Holland SDI:

0378-4371(93)E0029-E

Statistical

models of clustering matter

A.A. Shanenko Laboratory of Theoretical Physics, Joint Institute for Nuclear Research, Dubna, Russian Federation

P.O. Box 79,

E.P. Yukalova and V.I. Yukalov Department Canada

of Mathematics and Statistics, Queen’s

Received 21 December

University, Kingston,

Ontario K7L 3N6,

1992

A unified approach is suggested for treating various systems composed of particles which can form clusters. The approach is based on an effective cluster Hamiltonian. The principle of thermodynamic equivalence of systems with exact and effective Hamihonians is formulated, which yields the conditions restricting the form of the dependence of a cluster Hamiltonian on thermodynamic variables. Cluster operators, cluster statistics and interactions between clusters are analyzed. This approach allows us to describe different phase transitions connected with the processes of evaporation-condensation and deconfinement-confinement.

1. Introduction

In many regions of physics it is necessary to describe systems consisting of several “elementary” particles being able to congregate into clusters. It often happens that with a change in temperature and density, the state in which unbound “elementary” particles predominate is replaced by a phase in which also clusters occur. In this case one says that clustering has occurred in a system. The inverse process is called declustering. Well-known examples of clustering matter are mixtures of vapour and water droplets; of molecules, ions and electrons; of different substances which can enter into a chemical reaction, etc. The rapid progress of high energy physics has allowed us to know new, exotic, kinds of clustering matter, the so-called non-Abelian gauge systems [l]. In the quarkless case these systems are made of gluons being able to cluster into glueballs. If exotic clustering matter consists of quarks and gluons, then its cluster component includes not only gluon clusters (glueballs) but also quark clusters (mesons, nucleons, etc.). The declustering in such exotic systems is 0378.4371/93/$06.00

0

1993 - Elsevier Science Publishers B.V. All rights reserved

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A.A.

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et al. I Statistical models of clustering matter

named deconfinement. So, it is possible and convenient to use for declustering in any system the name “deconfinement”. As is known, the description of clustering matter dealing with an exact Hamiltonian of corresponding “elementary” particles has many difficulties. Therefore, there exists a statistical model approach which essentially simplifies the description of clustering matter. For example, let us mention both droplet models [2-81 and statistical models of quark-gluon deconfinement [9-181. The main shortcoming restricting the usage of models from the first group consists in the disregard of cluster interactions. The majority of statistical approaches from the second group do not take into account the possibility of the coexistence of clusters (the so-called hadrons) and unbound “elementary” particles (unbound quarks and gluons, or quark-gluon plasma). Allowing for the mentioned coexistence the remaining models from the second group are characterized either by a very rough treatment of the interaction of unbound “elementary” particles with clusters [16,17] or by the assumption, ab initio, of thermodynamic advantage of the heterophase mixture including both manyparticle configurations and unbound “elementary” constituents of matter [ 181. The mentioned shortcomings of the models from the second group have not allowed a complete description of the deconfinement phase transition in non-Abelian gauge theories (see discussions in refs. [19,20]). In this paper we present a statistical model of clustering matter which under the same suppositions describes not only the usual clustering systems but also the exotic gluon-glueball and quark-hadron mixture. In the proposed model both the interaction of clusters and the possibility of the coexistence of clusters and “unbound” particles are allowed for in a consistent way. The main part of the paper is devoted to model fundamentals and applications to usual systems. A brief discussion of features of gluon-glueball systems is given at the end of the article. A more detailed consideration of the exotic variants of clustering matter has been published in papers [20-261.

2. Cluster Hamiltonian As is said in the introduction, the study of thermodynamics of clustering matter based on an exact Hamiltonian is impossible in reality. This fact has stimulated a great popularity of statistical models dealing with effective cluster Hamiltonians. So it is expedient to explore the main peculiarities of constructing a cluster Hamiltonian. Let us treat a clustering system with an exact Hamiltonian H(4), where (1, denotes the field operator of “elementary” particles. The problem of the clustering description can be solved in terms of the cluster quasiparticle

A. A. Shanenko et al. I Statistical models of clustering matter

631

operators

(1) the relation connecting the field operator written in the form

of n-particle configurations

$n(l) = 1 A(12. . . n + 1) e(2) J1(3). . . t+b(n+ 1) d(23.. +

I

B(12..

+,, with +

. n + 1)

. n + 1) $+z + 1). . . jr(3) G(2) d(23..

.n+l)+C,(l). (2)

In expression (2) numbers in parentheses, for brevity, denote coordinates of particles; A(. . .), B(. . .) and C,(. . .) are nonoperator functions. Note that I/+ corresponds to unbound “elementary” particles which are convenient to be called one-particle clusters. Usually, both in the vacuum and in a medium the cluster statistics is the same. However, there exist situations when the vacuum statistics of a cluster differs from its statistics in matter. In particular, a fermion inserted into a boson crystal acquires the boson properties and vice versa [27]. Calculating commutators and anticommutators of two-particle cluster operators one may be convinced of an unreasonable form of anticommutator constructions. On the contrary, we have for commutators the completely conceivable relations [&(l),

Gz(2)]_ = 21 [A(134) A*(254) - B(154) B*(254)]

[&Cl),Ib;WL=+W34)

W54)-

x[G(3),k5L

W154) W34)l

d(345).

This fact testifies to the boson properties of n-particle clusters at even n. Note that the “+” sign of the square brackets in the right-hand sides of the above-mentioned expressions corresponds to the case when “elementary” particles are bosons; the “-” sign corresponds to the case when “elementary” particles are fermions. The clusters are separated by passing from an exact Hamiltonian to an effective cluster Hamiltonian,

A. A. Shanenko

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et al. I Statistical

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that can depend on thermodynamic variables. Here, $C= { +,,, . . . , $,, , . . .} is the set of cluster field operators; NC= {N,, . . . , N,, , . . .}, the set of mean cluster numbers; 0 and V are, correspondingly, the temperature and the volume of clustering matter. The dependence of H, on 0, V and NC is specified by the principle of thermodynamic equivalence of systems with exact and effective Hamiltonians. This principle may be formulated in the following form: iim _: [F(H( 0, V, NJ - F(H,I 0, V, NJ] = 0 , (4) ;im ;

[dF(HIO,

V, NJ

- dF(H,l@,

V, NJ]

= 0,

besides, we must require that computations of thermodynamic using different statistical ensembles yield the same results. In the relations (4)

quantities

by

F(H) 0, V, NC) = - 0 In Tr exp

(5)

and Tr is computed in the space of states with fixed cluster numbers. As V+ ~0 we denote the procedure of the transition to the thermodynamic limit, V*a,

N,-+m,

p,+

N

n =const, V

(6)

p, being the density of n-particle configurations. Let us consider the consequences of the principle of thermodynamic equivalence. If the conditions (4) are valid, then having computed in the canonical ensemble the free energy F(H,IO, V, NC), we are able, with the help of the well-known thermodynamic equalities dF=-SdO-pdV+&,,dN,,,

n

F=E-OS,

(7)

to find the main thermodynamic quantities of clustering matter: the entropy S, the pressure p, the mean energy E and the chemical potential of n-particle clusters, p,. In particular, we obtain from (7) the expression dF aN,

= PII .

(8)


633

But using the grand canonical ensemble we must derive the same relation too. In this case F= -OlnTrexp

(

-

H, - c, PA 0

where Z?,, is the number operator

> + P,,N, >

of n-particle

configurations,

and Tr is computed in the space of states with arbitrary numbers of clusters. Treating p,, as a function of 0, V and N, and taking into account expression (9) we arrive at the result

(11)

Pn> .

Here, ( I~HJ~N,,) denotes the statistical average of the operator Comparing (8) and (11) we get the equality C9H f=

(aNn>

0.

aHJaN,,.

(12)

The equivalence

of the microcanonical

(H,)=E.

and the canonical ensembles provides (13)

Using expression

(5) and the relation

we derive E=(HJ+O(z).

Hence,

(14)

we may conclude that aH

(ca@ >

=()

.

(15)

Strictly speaking, the validity of (15) seems to be unnecessary at zero temperature. But to formulate the whole thermodynamics of a system, we have to assume the continuity of (aH,/a@ ) with respect to temperature.

634

A. A. Shanenko

Moreover, dF _=-

av


it follows from (7) that

(16)

p.

Respectively, we must obtain the same equality with the help of the isobaric ensemble. The use of this ensemble provides the expression H,+pv

F=-OlnTrexp

o

-PV,

(17)

where $‘= Ci V,p;, and pi is the operator of the projection onto the space of system states with the volume Vi; Tr is calculated in the space of states with arbitrary values of the volume. Considering the pressure as a function of 0, V and N, we find 2l3H ( aV >

dF z=-p+

(18)

and, hence, aH

( av > c

=()

(19)

Very important relations (12), (15) and (19) resulting from the principle of thermodynamic equivalence (4) can be called the conditions of thermodynamic correctness of a cluster Hamiltonian. If H, depends on temperature and cluster densities p, = {p,, . . . , p,, . . .}, these conditions are presented as follows:

($)=o,

(2)=0.

(20)

Usually, the separation of clusters and the passage to a cluster Hamiltonian may be performed only in simplified models. In reality it is necessary to invoke some assumptions concerning the form of a cluster Hamiltonian. For instance, being based on the mean field approximation, which will be used further, we can write

H = z z j i,,(~, SJ [E,(-iv)

+ a,(@,

P,)I#~(x, sn) - B(@, P,) V . (21)

Here, S, stands for the cluster inner degrees of freedom; energy of the interaction of an n-particle configuration

‘%,,(O, p,) is the with a medium;


e,(k) = k2/2M, or e,(k) = vm and M, is the cluster system of units in which fi = 1 = c. Besides, B(O, p,) is the that is necessary in the mean field approximation. In multiplication A 1 * A, of operators A 1 and A, is replaced A&42)

+

(AdA

-

635

mass. We use the correction function this approach the by the expression

(A,)(A,) ,

see ref. [28]. The mentioned substitution results in the nonoperator part in expression (21). We assume that clusters including an even number of “elementary” constituents are bosons. An n-particle configuration with an odd IZ is a boson in the case of Bose “elementary” particles, and a fermion in the opposite situation. Field operators corresponding to different kinds of clusters are commutable. The conditions of thermodynamic correctness result in the dependence of B( 0, p,) on %,(O, p,). To find this dependence, let us write the free energy of the system having the Hamiltonian (21),

F(0,

v, NJ

=

cy

rk’ln[l-i”exp(-y)]dk

n

0

(22) where CL,is defined by the equality

pE=-&

rk’[exp(v)-i”]ldk.

(23)

0

and the spectrum of n-particle clusters is defined by (24) In (22) and (23), &, denotes the number of the inner degrees of freedom of an n-particle cluster; 5 = 1 for the variant of “elementary” particles: bosons; f = - 1 in the opposite case. Differentiating F( 8, V, NJ with respect to N,, and using formulae (23) and (24) we derive the equality

which allows us to obtain the equation


636

(25) Investigating

the relation

we can get one more important

expression,

(26) provided

we use the equality

So, the conditions of the thermodynamic correctness of the cluster Hamiltonian (21) are presented by eqs. (25) and (26). If %,(O, p,) and B(0, p,) have continuous second-order derivatives, then (25) yields (27) To obtain formula (27), it is necessary to differentiate expression (25) first with respect to the variable p, and then with respect to the variable pt. Comparison of the results gives relation (27) provided the equalities

azou, _ a2%?! am+

ahap, ’

are taken into consideration. a% n

a@

=o .

d2B -=-

a2B

ap,ap, ap,ap, Moreover,

expressions

(25) and (26) lead to

(28)

As is seen, the conditions of thermodynamic correctness of the cluster Hamiltonian (21) stipulate the independence of Q,, and w, on temperature: %,( p,), wn( p,). Hence, the potential energy of the system must only depend on cluster densities. We mention that while exploring the properties of a cluster Hamiltonian, we did not consider, for brevity, the possibility of different species of an n-particle


637

configuration for every IZ. For example, a cluster may be both in the ground state and excited states. Besides, there exist situations when clustering matter consists of several kinds of “elementary” particles. The conditions of thermodynamic correctness in this case may be obtained by analogy with the abovementioned procedure. As regards excited states, in surroundings it is usually more thermodynamically profitable that clusters which are in ground states survive. And taking into account excited clusters, as a rule, does not essentially perturb results. To make the subject perfectly clear, we shall restrict our consideration to investigating the situation of unexcited clusters including “elementary” particles of one sort. Only at the end of our paper, treating deconfinement in the non-Abelian gauge theories we shall give up this scheme for obtaining more accurate evaluations. So, in this section of the article we have considered the important peculiarities of constructing a cluster Hamiltonian. Since effective Hamiltonians usually depend on thermodynamic variables, we have explored this case carefully. As has been proved, the mentioned dependence cannot be arbitrary. Its character is specified by the conditions of thermodynamic correctness of a cluster Hamiltonian. If these conditions are not fulfilled, the main thermodynamic relations can be disturbed. The results of this disturbance and principles of quasiconsistent methods need a separate consideration.

3. Conditions

of heterophase

equilibrium

Up to now we have considered the ideal situation when it is possible to fix cluster numbers in the mixture. However, from the physical view-point other variants of treating clustering matter are interesting. In the case of usual clustering systems, such as the mixture of vapour and droplets, etc., the temperature 0 and the full density of “elementary” particles P =

c npn n

(29)

are convenient thermodynamic variables. Cluster collisions cause the reactions of cluster disintegration and formation, so that with a change in 0 and p, cluster concentrations being defined by the expression (30)

can vary too, W, being the concentration of n-particle configurations. Calculating the w, for every 12, 0, p it is possible to answer the question about

638


clustering in a usual system. To obtain more complete information, find the free energy of the considered system f(0,

we have to

H-pi9

p) = - $ In Tr exp -

o

>

+ PP 7

where the chemical potential of “elementary” equality

particles p is determined

by the

(&)=N=pV,

and &’ is the operator expression

of the. number

of “elementary”

particles.

Using the

(31) we derive for the asymptotically

f(O,p)=-

large N and V the relation H, - C

i

$lnTrexp

Hence, to calculate cluster concentrations (23), (24), (29) and (30) the expression CL, =

np&”

+CnpN,, n

.

(32)

we must add to the set of equations

w .

(33)

If we know the values of CLfor various 0 and p, we can evaluate other thermodynamic functions. The relations defined by (33) are named the conditions of the heterophase equilibrium [28,29] since in an arbitrary state of usual clustering matter for which the equality (33) is valid we have

W(@, V, Nc)lB,V.N=const =

c pn SN,,= 0 . n

The validity of the relation (33) is stipulated by the equality 6N=&dN,=O. n

In the gluon systems particle collisions cause the generation of additional gluons and glueballs. Hence, the fixation of the full gluon density is not coordinated with the real situation. Similar systems by themselves, because of the thermodynamic advantage, choose the necessary value p(O) for every 0,


639

the following relation being valid:

Therefore we get p = 0. It is not too difficult to conclude that the conditions of the heterophase equilibrium in this case have the form p,=O.

(34)

To calculate the cluster concentrations (23), (24), (2%

in this situation we must explore eqs.

(30) and (34).

4. High temperatures

and small densities

In the previous section of the paper we have discussed the general scheme of investigation of clustering and deconfinement in the usual clustering systems. It is expedient to illustrate our method with concrete results. Let us restrict our consideration to the variant when %” ( p,) is a differentiable function for every n and p,. In this case (Vn, VP, VP,: T n& = P> P, E PC) .

]%(P,)I - %(&(@YPI)‘0 n Then, taking into consideration

(Vn > 1).

the equality

PI) g-jix=Tr4(@) exp (%(P,[@ n@ 1,

(46)

_ %(Pc[@,PI) 0

(0)

>’

which follows from (37), and the character of the dependence temperature at e,(k) = k2/2M,, i.e.

Z,(O) =

z, * 03’* )

rr’exp(-

of Z,(O) on the

&)d,,

(47)

0

we obtain the relation

$$ix-

(O-0,

p=const,

n>l).

Because of p1 6 p, this relation can be rewritten

w,-+O

(O-0,

p = const,

as

n > 1).

The derived expression is at variance with our initial assumption. necessary condition for clustering is proved.

Thus, the


643

The sufjicient condition for clustering. Let us assume that the system of equations (29), (30), (33) and (37) h as a solution. Then, it is sufficient for the solution to yield w1 + 0 when either 0 + 0 ( p = const) or p + CC(0 = const # 0) so that the relation (45) is valid for any pC. Proof.

With the help of (37) and using the inequality mp, c p we can obtain

which leads to the following inequality: l/m

m ~+llm P

I,(@) wr =Gz;m(@)

exp (

%A,) m@

%(PC) >. 0

If &,(k) = k2/2A4,,, then (48) may be rewritten

(48)

in the form

l/m

w1

s

zl

(> 2

p-1-~lm03(l-l/m)12+0,

L

for 0 * 0 ( p = const) and p -+ CO (0 = const), provided sufficient condition of clustering. And if e,(k) = j/m, mation Z,(O)-

G

exp(-

$)

(49)

is valid at sufficiently small temperature w1

at 0-O (p = const). Moreover, as w1 < const x p-l-l’m for all 0, p (see (48)), then w1 + 0 for p- ~0 (0 = const # 0) in the relativistic variant of the choice of cluster spectra at every temperature. Hence, the sufficient condition for clustering is proved.

6. Potentials of cluster interaction The formulation of the necessary and sufficient conditions for clustering did not require the specification of the dependence of cluster spectra upon cluster densities. But this specification is compulsory for a detailed investigation of

A. A. Shanenko

644


models

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matter

clustering. In reality, there usually exists an infinite set of cluster sorts for any clustering matter; therefore it seems to be difficult, if not to say impossible, to get a reasonable approximation for the mentioned dependence of spectra. However, owing to a recurrent relation between cluster potentials, which will be obtained in this part of the paper, this problem can be solved. To derive the mentioned recurrent relation, let us consider the fusion of two clusters in the system (see fig. 1) including three configurations provided the potential of the interaction of II- and m-particle clusters obeys the condition /iJ$ CBn,,(r)=O.

(50)

The energy balance of the reaction is e,(k) + C(P) + e,(4) =E

+ @Jr,,)

+

.+m(P’)+ dk’) + @n+m.,(r)

(51)

3

where p = 1pi, q = 141, k = Ikl and p, q are the momenta of fusing clusters, k is the momentum of the configuration-“observer” which does not take part in the reaction, Qn,,(. ) is the potential of the interaction of IZ- and m-particle clusters; mm, r,!, rm, are the distances between particles before the reaction; p’ denotes the momentum of the configuration being the fusion product, k’ is the after the reaction; r is the distance momentum of the cluster-“observer” between the fusion product and “observer”. Provided that the relation T,,,,,G

-> Fig.

1. Geometric

characteristics

of cluster

fusion.

A. A. Shanenko


models of clustering

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645

rnl, rt?dis valid, we obtain r = r,,, = r,,,,. Besides, the system including the fusing clusters and the system consisting of the configuration-“observer” may be considered as conservative systems at sufficiently large r,,, and rm,. Hence, we have c(P) + 4q) + @nm(r,,)= G+,(P’)

T

(52)

E,(k) = E,(F) .

Taking account of (51) and (52) we get

@L(r)+ @Jr) = %+A3 .

(53)

This leads us to the relations

(13,,(r) = @m-l.nW + %(r) = @m-z,n(r) +2%(r) = m@,,,(r), %k-) = n@l,(r)y which allow us to obtain the simple formula

Gm,(r)= mn@ll(r) As is seen, the relation

=

Yf

Gjj(r) .

(54)

(50) is necessary for deriving the recurrent

(54). The important question arises: what densities provide (54) to be valid. It appears that p has to obey

formula

the approximation

pGl/m3,

(55)

and (T being the solution of the equation q~(cr)=o.

(56)

Indeed, the energy conservation law corresponding and m-particle configurations is

to the fusion of n-particle

(57) where rr: is the our consideration of the beginning M, = 1~77with q=

fusion radius of corresponding clusters. It is to investigating clusters which have stopped of the reaction: k = p = q = 0. Then, using const following from thermodynamic reasons

correct to limit at the moment the evaluation and the expres-


646

sion (57), we get

q&f;)

= 0.

Note that the result does not depend on the form of e,(k) (relativistic or nonrelativistic). To estimate rf:, we use expression (54). This provides the equation

Then r:A = u (Vn, m). We have been convinced of the approximation (54) being correct only when r -> m. Hence, we have to consider the densities obeying the relation (58) for the mean distance between clusters (C, p,,)-I”. TP,cTnp,

As far as

y

then taking account of (29) one may see that (58) follows from (55). 7. Possible variants of cluster spectra Considering the cluster mixture which includes configurations action potentials (54) in the Hartree approximation we obtain

=C $n,,(O) p, = n%(O)C mp, = n&(O) P m m

I,

where &n’,,(O) is the value of the Fourier momentum

As is seen, the interesting “u,o_ n

%m(P)c m

arises from formula (54).

,

with inter-

(59)

image of @,,nm(r) at the zero

relation (Vn, m)

(60)

A. A. Shanenko et al. ! Statistical models of clustering matter

647

However, the Hartree approximation is often not sufficient; taking into account the short-range particle correlations is important for a correct consideration of clustering. As is known, to consider particle correlations is not so easy even for systems composed of particles of one type. As regards the systems made of particles of many species, this situation is much more complicated. But the recurrent relation (54) allows us to simplify the problem. It appears that at the first step we can take into consideration particle correlations in the system including only unbound “elementary” particles. And then, with the help of the obtained energy of the interaction of one-particle clusters, %( pi), we can derive the quantity ‘%,( p,). Indeed, it is correct to assume that in the case of short-range particle correlations the relation (60) should not be disturbed. Then, using (27) and (60), one can get

m

-$ Wt(P,)l =n -$ [%(PJl * n

(61)

m

This allows us to specify the dependence

of %, on cluster densities,

%(PJ = “u,(P)= 1. “u,(P). In fact, as far as

then, in accordance

with (62),

or 8% _ -_-.-

aP” Remember,

a%,ap ap ap,. that n = aplap,.

Due to the property (63)

we can obtain %,(p,) = n(C,p” - C,p’). To illustrate this scheme and to describe the effects occurring through the short-range correlations, it is necessary to specify the function ‘%(pl). Let us

A. A. Shanenko

648

choose the Lennard-Jones

et al. / Statistical

models of clustering

matter

form (64)

for the interaction potential of “elementary” particles. In (64), E, A and Y (A > V) are constants, whose values depend on the sort of a clustering matter. In this case we have (see refs. [31,32])

where S(T) is a smoothing function that can be written [33,34] as S(Y)=exp [ -2 (a)o-2)‘2]) ; provided

a=llpp”‘3)

Q>,,(r) is defined by (64). Expressions

“u(P*) =

C,PT

-

C,Pt3

’

(66) (64)-(66)

lead to the result (67)

Because of the approximate character of the smoothing function we have to write the relations connecting C,, C, with (Y and p in the form of the approximate equalities

where

Taking account of (63) and (67) we get Ql,(P,)

Despite qualitative the form

= n(c,Pa

-

C,P?.

(68)

the approximation being rather rough, one can get a reasonable picture of clustering with the help of (68). In this case eq. (29) has


649

(69) Introducing x=exp

the definition

(

C,p”

- c

-

pP

@ p

)exp($

we can rewrite (69) as (70) with nZ,(@) > 0, Vn. In accordance with the Descartes rule, eq. (70) has a positive solution. Hence, at every 0 and p there is the set p, being the solution of (29), (37) and (68). Further, owing to (68), 011,(p,) is a differentiable function of cluster densities. Therefore, the clustering matter obeying (68) is realized as the gas of unbound “elementary” particles at small values of p and high magnitudes of 0. Respectively, due to the sufficient condition for clustering we arrive at w1 --, 0 both as 040 (p = const) and as p+ CC (0 = const # 0); in the exploring example at large densities the configurations predominate which are greatest in the investigated cluster set. Indeed, as x( 0, p) * CO(p + ~0, 0 = const), where x( 0, p) is the positive solution of (70), and P, = Z,(O) xn, we can get W

2

_

wtn-

nZn(@) mZ,(O)

n--m+m

x

0 =

(P-m,

const)

at 12> m. In the nonrelativistic case of the choice of e,(k) the greatest clusters at low temperature predominate. In the relativistic variant, only the clusters which possess the minimal value of M,ln survive at low temperatures. But this peculiarity corresponds, usually, to the greatest configurations too. To obtain more detailed information on the character of clustering in the case of interest, let us treat the mixture made of unbound “elementary” particles and two-particle clusters. For simplicity it is convenient to limit oneself to the nonrelativistic form of cluster spectra. Then, eq. (29) may be rewritten as I, 03’* exp $ r>

+ 2Z203’* exp z (

where /Li = p - c,pa

+ Cpp@ .

>

= p ,

(71)


650

Solving (71) we get

r:cp

(-1+&-ggJ

w1=q

(72)

1

and w,=l-w,. To investigate other thermodynamic quantities, we must calculate the correcting function which corresponds to the cluster interaction energies defined in form (68). Eqs. (25), (26) and expression (68) result in fqp,)

=

2

pa+’

-

f&

pp+’

+c.

(73)

Because B( p,) + 0 as p + 0 (the limit of noninteracting particles), it appears that C = 0. Using equality (22) we get the pressure of the mixture p(0,

p) = @p(1+

which can be rewritten

p(O,p)=@

[

pn+l - -PCP p+1 p+P ’

W,) + 2 in the form

p+&; 0H2’2(-1+

+ aC, P o+l -- PCP cu+1 p+P

J_)] p+1

(74)

The equality 1-t

SZ,p 1:03/2

-“’ >

I

+ “C,P”

- pc,pp

(75)

results from (74). From (75) it follows that there is a temperature-density range in which a~(@, p)lap < 0 at C, # 0. If C, = 0, then the system is unstable at large densities. If C, # 0, then the mixture pressure has features similar to the peculiarities of the pressure in the region of the liquid-vapour transition. In this case clustering is accompanied by a stratification, and the system is divided into two parts, in which unbound “elementary” particles coexist with their bound states. But in the first part unbound “elementary” particles predominate, which is opposite to the situation in the second part. At C, = 0 the pressure is an increasing function with p. Hence, the stratification is the effect of the attractive character of Q,,(r) at large distances between interacting particles, which is natural. As is shown, the clustering system obeying relation (68) is a good model for the mixture of a vapour and liquid

A. A. Shanenko


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matter

651

droplets. At sufficiently high temperatures the stratification is absent, and one can describe clustering with the help either of the Hartree approximation or of the approach that takes into consideration finite sizes of clusters. Some peculiarities of the excluded volume method will be considered in a subsequent part of the paper. We emphasize that the most important results of this part are the relation (54) connecting the cluster potentials with each other and approximations (60), (61) and (68) following from (54). These results allow us to demonstrate the essential dependence of the clustering character in classical systems on the choice of %,( p,) and to show the importance of including short-range particle correlations. One can describe the qualitative features of a vapour condensation only with the help of these short-range correlations. 8. Excluded

volume

method

In the statistical model approach for describing clustering matter the method of taking into account the particle interactions is very popular, which invokes the idea of Dupre, Hurn and Van der Waals (see refs. [35,36] and a discussion in ref. [37]). For example, we can mention the models [12,13,16,17] used in high energy physics. In this method, expressions connecting cluster numbers, entropy and thermodynamic potentials with cluster chemical potentials can be obtained if in the expressions corresponding to the gas of noninteracting configurations one substitutes V, = V- C, u,N,, for V, where u, is the proper volume of the corresponding cluster. In particular, the number of n-particle configurations obeys the relation

Nn+

rk2exp(_

En(k;-P+k.

(76)

0

The quantity V, is called the free volume of the cluster motion; C, u,N, is the excluded volume. Very often this variant of taking into account the particle interactions is called the excluded volume method. To demonstrate the correctness of this description let us consider the clustering in the system in which clusters are solid spheres. In this case eq. (29) can be rewritten in the form (77) Using the relation u,ln = u1 (Vn) we rewrite (77) as (78)

A. A. Shanenko

652



where x=exp

(r*-> 0

.

Exploring eq. (78) by analogy with (70) we conclude that (78) has a positive solution at any 0 and p < 1 /vi. Hence, one can calculate cluster concentrations with the help of eqs. (30), (76) and (78). The analysis of (30)) (76) and (78) shows that as p + 1 lu, the system tends to the state of configurations which are the greatest in the set of cluster sorts investigated. Respectively, at asymptotically small densities and high temperatures only unbound “elementary” particles survive. At low temperatures, in the system considered either the greatest configurations (if e,(k) = k2/2M,) or the clusters with the minimal value of M,ln (if e,(k) = j/q) predominate. Clustering should not be attended by stratification in accordance with the above-mentioned statement (see section 7) about the applicability of the excluded volume approximation. However, having evident advantages the method has the following shortcoming. This approach appears to be thermodynamically inconsistent. Indeed,

where

iqo,v,p.,>=-@c~n~ ”

(79)

/_L,= {CL,} being the set of cluster chemical potentials. The mentioned inconsistency is not so crucial. In spite of the fact that the well-known relations should be fulfilled for thermodynamic quantities, yet in some regions they can be well approached by functions which do not obey all thermodynamic relations. The degree of thermodynamic inconsistency decreases with improving approximation. Therefore, effective quasiconsistent approaches are able to provide good results for thermodynamics as well as consistent ones. To avoid possible divergences and difficulties in such quasiconsistent methods, one must use restricting rules for the differentiation of thermodynamic potentials. In the case of describing the cluster interaction by analogy with the approach of Dupre, Hirn and Van der Walls this is performed as follows. First, one defines the auxiliary function

&(o, v, p,,

X) = -

$

Vx

T

1 0

k2

expc-

“‘(“~-

“)

dk .

(80)

A. A. Shanenko


653

Second, the equalities

are postulated.

Then, any contradictions

vanish, in particular,

In addition to the excluded volume method other variants of the quasiconsistent description of clustering matter can be employed. For instance, the case may be mentioned in which the energies of interaction of clusters with a medium are approximated by temperature functions [38]. Note that it is impossible to calculate the correcting function when %, = Qu,(0) because the set of equations (25), (26) has no solution. In this case the restricting rules of the differentiation of a thermodynamic potential are formulated so that one may ignore the dependence of cluster interaction energies on the temperature. We will consider questions concerning the applicability of models with the restricting rules in subsequent publications.

9. Correction

of excluded volume method

It is very interesting to know whether the excluded volume method can be converted into a thermodynamically consistent approach with such positive features of the quasiconsistent approach as simplicity and obviousness. To simplify our reasons, let us consider the classical gas including only n-particle clusters. Supposing the analogy of replacing V by V, with taking account of the cluster interaction in the mean field approximation we can write the free energy of the system as

F,z(@, V, N,> = -

+m

@Y,V(l- U,P,) I 2n2

k2

exp( - E”(ki-

“)

dk

0

- B(@,

P,) V + ~3,

>

(82)

where EL,is defined by eq. (76). Allowing for a F,,IdN, = P,, and B(@, P,,) --;, 0 (p,+O) we obtain B(O, p,) = -BP,

-

!

ln(l - u,p,).

633)


654

Then, using (83) we can find other thermodyamic the pressure

p,(O, P,) = -

$ Ml

characteristics,

for example,

- u,P,).

(84)

”

It is interesting that (84) is similar to the expression for the pressure of the lattice gas [39]. We emphasize that there is one more modification [40] of the excluded volume method, in which the grand thermodynamic potential has the form a,=-

l

?!!$’ 71riexp((k)+;Pn-K)dk. 0

In this case @P,

pn=1-u,p,

.

Both the modifications yield consistent approaches, but they have a shortcoming absent in the quasiconsistent method. It happens that at asymptotically high temperatures the pressure behaviour does not correspond to the ideal gas. To overcome the mentioned difficulty, it is reasonable to give up formal modifications of the excluded volume method and to make an attempt of finding the energy of the cluster interaction with a medium which corresponds to replacing V by V, and, hence, obeys the relation exp(-

y)=l-v,p..

This relation appears to be contradictory as exp[ - %,( p,)lO] + Therefore, the assumption of stipulating the substitution V+ V, by interaction is not valid at all temperatures. Nothing remains but that there is a limited range of temperatures in which the cluster can be allowed for with the help of replacing V by V,, and temperature value 0, in this range which obeys the equality exp(-

1 (0 + ~0). the particle to suppose interaction there is a

y)=l-vnpn,

Then, a,(~,)

=

-@,ln(l-

u,P,).

(85)


655

In the case of several cluster species we can get

qn(pn) =

-@,ln(l -IX ump_) ,

036)

m

by analogy with (85). With the help of condition (27) and expression obtain the relation 0, = -@??I u, vt?I ’

(86) we

(87)

which stipulates the notation

(88) At small densities expression

(86) can be rewritten

as

c

“u,(P,)= @n UmPm. m Comparing

this relation with (59) we derive

Taking into account arrive at the result

arbitrariness

OnUrn=

nmql(o)

V, = n

K(O) z

and independence

of cluster densities we

)

or

J

= nv 1 .

(89)

Note that with the disregard of the particle attraction one should use &typ’ instead of &,*(O); in the case of the Lennard-Jones potential we have

So, allowing for (89) we can obtain %,(p,)

= -rzu,

ln(l-

pu,)

Wn) .

(90)

656

A. A. Shanenko



It is important to emphasize that the relation (60) is fulfilled in the case considered. The correcting function corresponding to (90) has the form B(p,) = -zu,p Using expressions ~(0, p,) =

- 2 ln(l-

pu,) .

(91)

(22), (90) and (91) we get the equality

C Op, n

-

zu,p

-

z ln(l

-

pu,)

.

(92)

It is evident that in this construction the term C, Op, predominates at high temperatures, which corresponds to an ideal gas of clusters. Therefore, we could modify the excluded volume method, the modified approach being thermodynamically consistent, being simple and obvious and having no mentioned difficulty at high temperatures. In conclusion of this section let us mention some facts concerning clustering in the system obeying relation (90). Based on the Descartes rule we find that the solution of eqs. (29), (30), (37) and (90) exists and is unique. Taking into consideration expression (60), the sufficient condition for clustering, and the fact that, usually, the largest configurations hold the minimal value of M,ln, we are convinced that the largest configuration predominates at low temperatures and at densities p - 1 lu, (p < 1 /u,). Moreover, as %,, (Vn) is a differentiable function of cluster densities, only unbound “elementary” particles survive at asymptotically high temperatures and small densities. Hence, clustering occurs in the system with a decrease in the temperature and density. In this case clustering should not be accompanied by stratification.

10. Influence of quantum

effects upon clustering

In the previous sections of the paper we have demonstrated the essential dependence of the clustering character on the interaction of clusters, only systems of classical particles being considered. As pointed out, exploring classical clustering systems is justified and is important because in many cases clustering occurs in the temperature-density region in which the quantum nature of particles is unessential. However, there are situations when the quantum effects influence clustering significantly. This influence is conditioned by the fact that clusters may be bosons or fermions; bosons can pass into the Bose-Einstein condensate, which diminishes the free energy of a system, and fermions cannot do this. To show the dependence of the quantum effects on a clustering system explicitly, let us consider the mixture of unbound “elemen-


657

tary” particles, fermions, and m-particle configurations, bosons (even m), at zero temperature, and let %,,, be equal to m%,. (i) Let eI(k) = k2/2M,, e,(k) = k*/2M,,,. Then, at 0 = 0 the cluster densities can be represented [41,42] in the form

{~M,[P., - %(P)I)~‘*

(PI 2 %(PN >

(PI < “u,(P))

9

(0) Pm=

;i

(CL, = %n(P)) (Pm < “u,(P))

(93)

) *

Here ~2’ denotes the density of m-particle clusters being in the Bose-Einstein condensate. Eq. (29) is rewritten as y

- 0&(p)]3’2

= p )

(94)

in the case p = “u,(p) and provided we take into consideration the condition of the heterophase equilibrium p,,ln = p (n = 1, m). Respectively, at p < “u,(p) we derive

$2{~M,[P - %(P)II~‘*=

(95)

P .

Analysis of (94) and (95) allows us to conclude that in this case only m-particle clusters survive in the system at zero temperature. This is in full agreement with the behaviour of the corresponding classical system in which W, = 1 (VP, 0 = 0). So, for a system made up of unbound “elementary” particles, fermions, and m-particle configurations, bosons, the qualitative investigation of clustering can be performed in classical approximation. Note that this conclusion is valid for any mixture of clusters provided the configuration which is largest in the considered cluster set is a boson. (ii) Let El(k) = j/m, c,(k) = vm. At zero temperature the cluster densities can be written [41,42] in the form

lb1 - “u,(~)l*- M:j3’*

(~12 Q,(P) + M,) > (PI
0; (2) A@ = 0.

on temperature

for a gluon-glueball

662

A. A. Shanenko


Fig. 3. The schematic dependence of pi&,_ on temperature for a gluon-glueball mixture: (1) A@ < 0, [AW&(O)[ Q 1; (2) A@ CO, IA@‘/@&)l= 1. pIpsB on temperature is schematically shown for the various values of A@, pss being the pressure of the ideal gluon gas. Results of comparison of our calculations of E /esB, the relative energy, and p/pSB with lattice evaluations of these quantities in the W(2) case are given in fig. 4. Our computations correspond to the set (Y = 0.62, &(O) = 5 x 10v3 MeVP2 and C1’(3n+1) =

I

.

300

200

.

.

.

.

.

.

.

II

LOO

.

. .

500

OlMeVI

.

e/eSB a?d pIpsB versus temperature in the SU(2) case: solid lines, results of our calculations at (Y = 0.62, 4,(O) = 5 X 10m3 MeV2, C1’(3a+‘) = 175 MeV, circles and squares, the lattice data from ref. [18].

Fig.

4. The quantltles


663

1.00L

0

m

0

50

100

..:.. 150

I. 200

250

. . .I.... 300

350

QIMeVI

versus temperature Fig. 5. The quantities cJc,,, 5 X lo-’ MeV-‘, C”(3a+1) = 175 MeV.

in the SU(2) case at a = 0.62, &(O) =

175 MeV. The lattice data (circles and squares) are taken from paper [18] and are placed in the figure under the condition O,,, = 210 MeV, this value of the deconfinement temperature is coordinated with the estimate of O,,, in the W(2) theory. In fig. 5 the dependence of cylcy,sB, where cV,sB is the heat capacity of the ideal gluon gas, on temperature is shown at (Y= 0.62, &(O) = 5 x lo-3 MeV-2, ,93a+l) = 175 MeV. The form of the curve testifies to the fact that in the SU(2) gluon system deconfinement is either a second-order phase transition (according to Ehrenfest) or a transition close to that of the second order. In fig. 6 the concentration of unbound gluons is presented for the W(2)

0 OlMeVl

Fig. 6. The dependence of wg on temperature lo-’ MeV-=, C”Oof” = 175 MeV.

in the SU(2) case at (Y= 0.62, &JO) =5 X

A. A. Shanenko

664


models

of clustering

matter

case. Let us emphasize that an appreciable number of unbound gluons are present in the system below the transition point. Moreover, the glueball concentration is significant at 0 > O,,,. However, the glueball role in the mixture is appreciable in a wide temperature region, O,,. < 0 < 20,,, (wo 2 0.1, wo being the glueball concentration). As to the gluon plasma, the concentration of unbound gluons becomes negligible at 0 < O,,, - A@, where A@/@,,, ~0.1. For example, at (Y= 0.62, &,(O) = 5 X 10m3MeVe2 and C “(3a+1) = 175 MeV, we have wg - 10e3 if 0 = 180 MeV, wg - lo-* provided 0 = 150 MeV. In fig. 7 comparison to our computations for E/Q, and pIpsB with the lattice data for these characteristics is given for the SU(3) variant. The results of our approach correspond to the parameters CY= 0.62, GZ2(0) = 2 X lo-3 MeV-2, ,53a+1) = 225 MeV, with this choice the deconfinement temperature being equal to 225 MeV. The results for the 163 X 4 lattice [45] (circles and triangles) are given in the figure besides data for the 123 x 4 (rectangles, rhombuses) lattice taken from paper [46]. The lattice data are presented under the condition O,,. = 225 MeV [19]. As is seen, deconfinement in the SU(3) gluon system is a weak transition of the first order. It is interesting to note that the value of the parameter (Yin the SU(2) case is very close to the magnitude of (Yin the SU(3) variant.

1.00 Fl E/E*,,SU13) :2

P/P,,,SUi31

0.83:

p-2 0 0

of321

I

,....l....l........, Loo 300

0 0

100

A

200

51

EIlMeVl Fig. 7. The quantltles e/esB and pIpse versus temperature in the SU(3) case: solid lines, results of our calculations at (Y = 0.62, &,(O) = 2 x 10m3 MeV-‘, C “0a+‘) = 225 MeV, circles and triangles, the lattice data from ref. [45]; rectangles and rhombuses, the lattice data from ref. [46].


665

12. Conclusion

A statistical approach describing not only the usual clustering systems, but also exotic clustering matter (gluon-glueball, quark-hadron), was formulated. Peculiarities of constructing a cluster Hamiltonian were investigated, which allowed us to obtain the conditions of thermodynamic correctness. The performed consideration of different examples of clustering matter shows that clustering strongly depends on the character of the particle interaction. Moreover, at low temperatures the quantum effects can influence clustering. The “scenario” of clustering may be of various types. In the usual systems clustering can be accompanied by stratification or can be a gradual process. In the SU(2) gluon system deconfinement is a second-order phase transition (according to Erenfest). In SU(3) gluon matter deconfinement is a first-order phase transition. Finally, in quark-hadron matter declustering is a gradual crossover (strictly speaking, at the zero baryon density and at low temperatures). From our viewpoint, important applications of our method could be the investigation of phase transitions in hot and compressed nuclear matter, the consideration of vapour condensation and, respectively, processes in real gases, equilibrium chemical reactions, for instance, the influence of gas admixtures upon the ozone concentration in air.

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