27 October 1997

PHYSICS LETTERS A

ELSEMER Physics Letters A 235 (1997) 113-117

Q-Statistics and the master equation Michael Schulz a, Steffen Trimper a, John Kimball b a Fachbereich Physik, Martin-Luther-Vniversitci’t, D-06099 Halle, Germany

b Physics Department, University at Albany, Albany, NY 12222, USA

Received 14 July 1997; accepted for publication 4 August 1997 Communicated by P.R. Holland

Abstract

Master equations of statistical physics can he described using a pseudo-quantum formalism. This formalism is applied

here to operators a and Bt which satisfy the nonstandard commutation relation of a q-statistics ( Zit - q&+ fi = 1). In the

simplest cases, this leads to a master equation which describes unusual dynamics and equilibrium distributions which are a generalization of Poisson statistics. As a physical realization with a temperature dependent q-factor we discuss the exchange of particles in a pore. There results a phase transition at a critical confinement temperature T, below which complete exchange is impossible. @ 1997 Elsevier Science B.V.

PACS: 02.50.E~: 05.20.-y; 05.3O.J~; 64.60.Ht

The master equation of statistical physics can be formulated in terms of Fock space. For the simplest cases, the resulting master equation reflects the struc-

ture of the underlying Fock space. Thus a Bose-like

Fock space naturally describes regions of phase space which can be occupied by an arbitrary number of par-

ticles, and the Fermi-like Fock space is appropriate when phase space cells can be occupied by no more

than one particle [ 11. More recently, two of us have

considered the application of para-Fermi statistics to

statistical mechanics [ 2,3]. Recently, operator pairs ii and ti+ which satisfy the

altered commutation relation of q-statistics [ 41

ira+ - qfi+a = 1 (1)

have been studied in a number of contexts, such as the quantum Hall effect [ 5 1. Q-deformed algebra has also been discussed to study the asymmetric exclusion

process [6] and, very recently, the one-dimensional

KPZ equation [ 71. Another motivation comes from the search for a possible violation of the exclusion principle due to Pauli [ 81. From a more theoretical point of view some authors are interested in the con-

struction of a multiparameter interpolation between para-Bose and para-Fermi statistics of a given order

[ 91. As another model a two parameter deformed mul- timode oscillators was studied recently [lo]. Other

applications are the quantum mechanics of q-deformed

harmonic oscillators [ 111 and the kinetics of bosons and fermions [ 121 within q-statistics. However, the

physical meaning of these commutation relations is not clearly understood.

Here we describe how a simple and natural applica- tion of q-statistics generates a master equations with

new dynamics and new equilibria which are a mod- ification of Poisson statistics. Moreover, we consider a simple physical example where the parameter q de- pends on temperature.

0375-9601/97/$17.00 @ 1997 Elsevier Science B.V. All rights reserved. PII SO375-9601(97)00628-2

114 M. Schulz et al./Physics Letters A 235 (1997) 113-117

To make our formulation as easy as possible, we simplify q-statistics by assuming there is a state IO) which is annihilated by the operator 2,

S/O) = 0.

Then if q > - 1, repeated applications of the commu- tation relations yield results which are consistent with the following assumptions: There is a Hilbert space with a basis In); n = 1,2,3 . . . co, and the following properties

(44 = h?,~ 2 In)+4 = 1, n=o

aln) = zn(q) In - l), fi+ln) = zn+l(q)ln+ l),

(2)

Next we consider a corresponding classical statistical system. This system is characterized by phase space cells which are also indexed by n, and the occupation probability P(n, t) determines the state. The time- dependence of these probabilities is given by a master equation which is a first-order differential equation in the probabilities [ 13,141

zP(n, t> = C M,,,P(m, t). “I

The coefficients M,,, which appear in the master equation guarantee that the sum of the probabilities does not change in time, and that none of the proba- bilities become negative.

The pseudo-quantum picture describes the system in terms of a time dependent state vector IF(t)), and a time-independent reference vector Is) [ 151. The con- nection between these vectors and the probabilities is

(3)

Note that the necessary condition for the existence of (3) is

(nls) # 0, for all states n E {0,1,2,. . .}. (4)

Using this notation, the master equation is viewed as a Schriidinger equation where the Markov matrix plays the role of a pseudo-Hamiltonian A _= -e

&IF(r)) = i I F(t)). (5)

The relation between e and M,,, depends on the choice of the reference state Is), since

= c M,,,P(m, t). (6) m

Thus, each transition probability per unit time (com- pare Ref. [ 141) corresponds to an evolution operator i, defined by a linear transformation using the basis of the Hilbert-Fock space and the reference state. e is defined by

i = c In) (~)Mn.n(mi- n,m

(7)

Because c,“, Irz) (nI = 1, the sum of all probabilities will not change in time if

(s(L(F( t)) = 0.

If this is to be true for all IF(t)), then

(8)

(s@. = 0. (9)

It is at this point that a commutation relation like for in- stance Eq. ( 1) affects our results. Doi (see Ref. [ 1.51, compare also Refs. [ 16,171, was the first to consider the Bose case corresponding to q = 1. More recently an extension to Pauli-operators has been discussed in much detail [ 18-231; see also review [ 241.

Here the case of q-statistics with the relation ( 1) is analyzed. We choose the reference state Is) and the linear operator i to be expressed as simply as possible in terms of the operators 2 and Gt. Up to this point the reference state (sl can be chosen as an available bra-vector of the Hilbert-Fock space, which fulfills the condition (4). A useful representation can

M. Schulz et a/./Physics Letters A 235 (1997) 113-117 115

be obtained, if the reference state is taken to be the generalization of a coherent (or Glauber) state, so

(# = (slk(q). (10)

Hence, the reference state can be expressed with re- spect to the basis 1 n),

(4 = (01 F fi”.. 11 n

(11)

The coherent eigenstate k(q) will be discussed below. Thus to insure conservation of probabilities, e must be of the form

i=(a+-k(q))@. (12)

We then take I@ to have the simplest form possible which can lead to equilibrium. The simplest fi is

B = p - AS. (13)

Remark that Pq. (12) with Eq. (13) represents the most general bilinear form in terms of the basic op- erators a and 2t. With this choice, the corresponding (one-step) master equation (in the classical notation) becomes

+t, r) = -k(q)P(P(n, t) - P(n - 1, t))

- A(z,2P(.,t) - z,:,P(n+ 1,t)). (14)

Now the free value k(q) can be chosen to rescale one of the kinetic coefficients, for instance we use k(q) /I = 1. Physically, the latter equation describes a system where the transition rate to a larger n is con- stant, but the transition rate towards smaller n depends on n. Hence, the elementary transition rates M,,+l,n

and M,-I,, are given by

P(n + 1) + p(n) with M,+i,, = AZ;+,,

P(n - 1) + P(n) with M,,_,,, = 1.

Furthermore, the master equation fulfills the detailed balance condition. The principle of detailed balance is valid, if the transition rates M,,, satisfy the nec- essary condition M,+I,,$~(~ + 1) = Mn,“+lPeq(n) where Pq(n) is the probability of state II being in equilibrium. It is simple to show that the conditions of detailed balance lead to the stationary solution (&P,,(n) = 0) of the master equation ( 14). Hence,

the equilibrium solution of the master equation and the probability distribution, which fulfills the conditions of detailed balance are equivalent, i.e. the principle of detailed balance is guaranteed. In equilibrium, the probabilities are of the form

R l-q P,(n) = P(pr”n -

i=, 1-q’. (15)

Here, PO is determined by the normalization Cz Pes(n) = 1. Note that in the limit q ---, 1 (Bose statistics), the above product term becomes a facto- rial, and Peq(n) is a Poisson distribution.

To illustrate the approach using q-deformed creation and annihilation operators, and in particular Eq. ( 14)) we study a simple model of deposition and evaporation in a pore by thermal activation. Let us consider a single thin pore shaped like a microscopic capillary which is in contact to an infinitely large particle bath. The deposit rate may assumed to be independent of the numbers of particles within the pore, i.e. Mn_l,n = p.

On the other hand the desorption probability depends on the actual number of particles inside the pore. For a large pore diameter we have M,+l ,n = An. In case of an extremely small diameter one obtains M,+,,, = A

because only the particle on top is able to leave the pore. In the intermediate case a particle can pass by another one with the transition rate

q =exp - $ , ( >

(16)

where EA is the characteristic activation (or tun- nelling) energy and T is the temperature. Thus, the total desorption rate becomes

M ,,+~.n=A(l+q+q~+...+q”)=Az,:,. (17)

Hence, this process can be described by Eq. (14) where p is chosen to be unity by time scaling. Con- sequently, the considered process can be mapped on a Fock space with the commutation relation of q-

statistics where q is defined by Eq. ( 16). To continue the general approach let us consider the

average over an arbitrary quantity A, which can be written as

116 M. Schulz et al./Physics Letters A 235 (1997) 113-117

c,, jn)A,(nl is the representation of an operator A

with respect to the basis In). This basis also forms

the set of orthogonal eigenstates. On the other hand, I? = u+, is also an operator with the eigenstates In), i.e. fl Irr) = z,‘(q) In). If the set of eigenfunctions of

the two operators fi and A, respectively are equivalent

there exists a definite functional relation between both

operators. We have

A = d( fi) = c cy,,&” = 1 In)A,(n[. (19) n n

A complete determination of the expansion coeffi- cients (Y,, is possible when the operator A^ is specified by giving it a physical meaning. The average becomes

(A) = c A,,P(n, t)

= &fQ),F(r)) = (A(A)). (20)

Using Eq. (9) the time evolution of the average (A) can be expressed by the equation

&(A) = (sl&lF( t)) = (s@lF( r))

= ($1 L4 Ll IF(t)), (21)

with the conventional commutator [a, b] = ab - ba. Particularly, the evolution operator f, of Eq. ( 12) leads to a closed expression for Eq. (2 1) . Note that e is de-

fit&bye=@+-A?z+d+AP-‘k?--1,seeFqs. (12),

(13) with p = k-’ (q). Thus, taking into account

Eq. ( 10) and the commutation relation ( 1) one ob- tains

(A( &p&t) = p c LY#%+)

= pCa.(s,(:+k?)“d+,F)

= &#?+@h+)“,F) ”

= PC(slk(q) (z+)“lF) n

=k(q)BC%((l+@Y) = (A(1 +q&). ”

In the same manner one finds

(A(fi)R) = (itqfi)).

Finally, the evolution equation for the average (A) reads

&(&A)) = (d( 1 + q&T)) - (d(fi))

+ *( fi,i(Y)) - A(1CrA(&)). (22)

Generally, this equation is the first step of an infinite hierarchical system of equations, e.g. if we choose

A = &‘, the right hand side is now a sum of constants, (fl) and (fi2). Th ere f ore, one must also determine the evolution of (fi2), which now leads to a new aver-

age (fi3) at the right hand side. For a mathematical

treatment such operators A ( A> are of special interest. They lead to a termination of this hierarchical system. In the lowest order (i.e. the hierarchy consists of only

one equation) such an operator A must have the form

G(A) = go(q) go(q) A=

l-(l-q)N aa+-ii+; (23)

where ga( q) is an arbitrary function. Alternatively, the

average over this special operator corresponds to

(G(t)) = go(q) $ --$‘(,, t). (24)

The corresponding equation of evolution is a linear ordinary differential equation,

&ci;, = (; - 1 - A) @) + Ago(q). (25)

The result can be obtained by substitution of A by e in Eq. (22).

To specify the meaning of (6) let us return to the model introduced above where q depends on temper- ature via Eq. ( 16). We assume that the pore is filled up with n particles which should be replaced by other particles with nearly the same properties, for instance isotopes. The time for such a procedure is dominated by the largest time scale where the particle situated at the bottom of the pore has left the capillary. This waiting time is simply related to q-(“+‘). Within this

M. Schulz et al./Physics Letters A 235 (1997) 113-117 117

time all n particles are exchanged. Thus, the averaged exchange time is given by

rex = h-t 2 ‘ikPk. (26) k=a

This mean value corresponds to Eq. (24). The aver- aged exchange time now becomes in equilibrium, see

Es. (239

4 7 - ex- 1 - (1 +A)q’ (27)

There exists a critical value qc = ( 1 + A) -’ for which the averaged exchange time becomes infinite. Using Eq. ( 16) there results a critical temperature

T, = E.4

ln(1 +A)’ (28)

whereas for T > T, a complete exchange of all parti- cles is always possible; however, with increasing ex- change time in case of T < Tc a finite fraction of par- ticles is trapped inside the pore. Remark that a similar situation can occur in the case of traffic jam.

In the present paper we have demonstrated how the Fock space method applied to the study of classical dynamical problems should be modified in case of q-

statistics. The algebraic properties of the underlying operators decisively determine the dynamical and sta- tistical behavior. As a very simple example we have considered the exchange of particles which are al- lowed to enter and to leave a capillary in a stochas- tic manner. Such a situation may surprisingly be de- scribed by assuming a temperature dependent q-factor. For very high temperatures (T + co) the pure Bose commutation rules are retained. Lowering the temper- ature we find a critical temperature T, below which the particles are confined inside the box and complete exchange is impossible. As a further extension of the method one can introduce an additional statical inter- action as was proposed recently [ 251.

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