P @@2E@4E OCR Output

~I\\\|\\\\\||\\\||\\\l\\\|\\|\\l\|\\||\\\|\\l|lllllQERN LIBRHRIES. BENEVQ

particles or of degree of antisymmetrisation or symmetrisation of the wave

to the parameter rc the meaning of degree of indistinguishability of identical

statistical distributions different from the FD and BE ones. We attribute

diate value, except zero, between the extreme values -1 and +1, we obtain

for rz = 1 the Bose—Einstein (BE) distribution. When rc assumes an interme

(FD) distribution, for is : 0 the Maxwell-Boltzmann (MB) distribution and

ourselves to brownian particles, we derive exactly for A : -1 the Fermi—Dirac

tion, depending on the value we give to the parameter rz. When we limit

equation we obtain a general expression of the stationary distribution func

ance the effect of the full or partial validity of EIP. After deriving the kinetic

contains a parameter rs, whose values range between -1 and +1 and can bal

transition rates by means of an inhibition or an enhancement factor, which

(EIP). This exclusion or inclusion principle is introduced into the classical

function of particles obeying a generalized exclusion-inclusion Pauli principle

D—dimensional continuous or discrete space, in order to study the distribution

(EP). In the present work, on the same grounds, we extend this kinetics to

tion for the distribution function of particles obeying an exclusion principle

In a previous work we have derived a non-linear 1—dimensional kinetic equa

Abstract

in coothsé or- eu sue:-vc rm! mo mw; . {.>g;\,·_ g

(March 10, 1994)

Istituto Nazionale di F isica Nueleare, Sezioni di Torino e di Cagliari

Consorzio Interuniversitario Nazionale di Fisica della Materia, Unitd del Politecnico di Torino

Dipartimento di Fisica del Politecnico, Corso Duca degli Abruzzi 24 10129 Torino, Italy

G. Kaniadakis , P. Quarati

Classical model of bosons and fermions

/ S Us B =~i· A '%

fat swosa

Typeset using REVTEX OCR Output

02.50.Ga,, 05.30.Fk, 05.40.+], 05.20.Dd

function ofthe particle system.

Let us clarify the above arguments with this example. lf we consider only two quantum OCR Output

entering the same state.

first fermion entering the state. But for n : 1 a second fermion is strictly inhibited from

indistinguishability requirements. lf n = O there is no inhibition of the probability for the

smaller by an inhibition factor of l~—n than it would be if there were no quantum mechanical

of fermions in a quantum state is n, the probability of one more fermion joining them is

quantum mechanical indistinguishability requirements. lf the occupation number of a system

joining them is larger by an enhancement factor of 1 + n than it would be if there were no

functions. If there are already n bosons in a quantum state, the probability of one more

for deriving classical and quantum Fermi—Dirac (FD) and Bose-Einstein (BE) distribution

known, at thermal equilibrium, where the detailed balancing postulate is the starting point

relations between the quantum transition rates and the classical transition rates are well

For quantum indistinguishable particles, bosons or fermions, we can remark that the

be in that state.

ular quantum state very delinitely influences the chance that another identical particle will

the effect of indistinguishability of quantum particles is that the presence of one in a partic

that another identical and distinguishable particle will be in that state. On the other hand,

guishable particle in some particular energy state in no way inhibits or enhances the chance

It is well known that, at thermal equilibrium, the presence of one classical and distin

probability rates and therefore, as a consequence, in the kinetic equation.

of a system of indistinguishable particles; the quantum effects are contained in the transition

particularly we shall derive and solve a non-linear kinetic equation describing the evolution

indistinguishability of identical particles, force significant changes in the classical procedures;

identical particles. In this work we shall see how quantum considerations, particularly

identical particles in a system, non classical effects arise from the indistinguishability of

ln quantum physics, if there is appreciable overlapping of the wave functions of two

by means of a factor which contains,fully or partially, the inhibition or the enhancement OCR Output

deriving from the indistinguishability of identical particles, we give an expression of them

we consider first of all the transition rates and, taking into account the quantum effects

paper, which was limited to 1—dimensional fermionic particles At the basis of this work

In this work we follow and generalize the approach already explained by us in a previous

Our approach is quite different.

obtains distributions close to the FD one, however a full agreement is not at all reached.

a way that fermions do not collapse to the lowest level By means of this method one

treated by some authors inserting a repulsive potential into the diffusion equations, in such

rates from one occupancy site to another one. In the case of fermions the problem has been

task. These equations should contain particular constraints on the transition probability

evolution in time towards equilibrium from classical kinetic equations is a very stimulating

any time towards equilibrium. The research of quantum statistical distribution and of their

numerical solution of the FP equation, the distribution functions of fermions and bosons at

has ever been set in the transition probability rates to obtain, directly from the analytical or

derivation of this equation.To our knowledge, up to now, no inhibition or enhancement factor

distribution function n(t,v) and tl1e transition probability rates are at the basis of the

The usual, linear Fokker-Planck (FP) equation describes the time dependent, classical

distributions in a very simple way.

to derive the classical Maxwell—Boltzmann (MB) distribution and the quantum FD and BE

principle. The application of these relations to the detailed balancing postulate allows us

zero to one, the inhibition factor ranges from one to zero, in agreement with the exclusion

classical particles multiplying the classical rates by 1 — it inhibition factor; n ranges from

principle). The fermionic transition rates r{2,rg1 can be expressed in terms of the rates for

one to ever larger values (in agreement, let us say, with a principle that we call inclusion

1+11. As n ranges from zero to larger and larger values, the enhancement factor ranges from

obtain their values simply multiplying the classical rates r§2, rg, by the enhancement factor

states 1 and 2 and the two bosonic transition rates rl{2,rg1 between the two states, we can

The research of statistical distributions interpolating between the FD and BE ones and OCR Output

of the exclusion Pauli principle and of the inclusion principle.

existence is also related to the research of small violations of the FD and BE statistics, i.e.

addition to the FD an the BE distributions, also a family of different distributions, whose

an arbitrary dimensional space. In contrast to the above quantum case, we can deduce, in

and fermion systems are possible. The non—linear kinetic equations we use are defined in

space. In one and two dimensions it is shown that also intermediate cases between boson

straint. This is however only the case if the particles move in a three or higher dimensional

a natural way from the formalism, without having to be introduced as an additional con

The restriction on wave functions, to be either symmetric or antisymmetric then appears in

restriction, corresponding to the symmetrization postulate, is put on the state functions.

the particles is taken into account in the definition of the configuration space, no additional

straints on the wave function need to be postulated. ln fact, since the indistinguishability of

This fact is used in the quantized theory and has, as a consequence, that no symmetry con

is not the Cartesian product of the single particle spaces, but rather an identification space.

fected by indistinguishability of the particles. The configuration space of a N—particle system

of identical particles and shown that the classical description of a many-particle system is af

subject by Leinaas and Mirheim They have examined the classical configuration space

lating between FD and BE statistics. We wish to recall here the early work related to this

In the last years a great effort has been spent in studying intermediate statistics interpo

and +1, except the value zero.

the parameter rc contained in the inhibition or enhancement factor has values between -·1

see, we may obtain also statistical distributions different from the FD and BE ones, when

could derive numerical solutions for the time dependent distribution functions. As we shall

the equilibrium, as time goes to infinite, can be deduced in a simple way; in other cases one

kinetic equation. In some case, like in the brownian particle case, analytical solutions at

whose values range between -1 and +1. Then, as a consequence, we derive a non—linear

effect. The graduality of the inhibition or the enhancement is assured by a parameter rc

' ` T

the degree of antisymmetrisation or of symmetrisation, respectively. Depending on the value OCR Output

has the meaning of degree of indistinguishability of fermions or of bosons, corresponding to

cases when A assumes one of the intermediate values between -1 and +1. The parameter rc

and the FD case rz : -1, in addition to the classical MB case rc : O and to all intermediate

therefore, as we shall see, varying the value of rc, we can consider both the BE case rc : +1

distribution function n(t,v). The values of the parameter is ranges between -1 and +1,

As a consequence the kinetic equation becomes non-linear because the factor contains the

the inhibition or enhancement factor 1 + 1m(t,v) into the transition probabilities rates.

Complete antisymmetrisation means complete indistinguishability. We shall introduce

know that pairs of fermions are not completely bosons, but only approximately so.

particle-particle pairs of nucleons can be coupled to angular momentum two, although we

of fermions, the fact that we cannot antisymmetrize completely all the nucleons and that

heavy nuclei. However IBM contains fundamental problems as for instance the bosonization

(IBM) describes the main properties (i.e. energy levels, electric quadrupole moments) of

free neutral hydrocarbon aromatic radicals. In nuclear physics the independent boson model

wave function studied by Brovetto and Bussetti [8] to analyze the hyperfine structure of the

functions it is possible to describe the physical properties of systems. One example is the trial

are used being characterized by a partial antisymmetrisation: only with this kind of wave

symmetrisation. In some physical systems of many—idcntical fermions trial wave functions

wave functions. Let us introduce the concept of degree of antisymmetrisation or degree of

Indistinguishability is strictly related to antisymmetrisation and to symmetrisation of

nuclear, particle and condensed matter physics and in astrophysics and cosmology.

deviations from the Pauli principle. Implications of such deviations are reported in atomic,

fermions [5-7]. Experimental high precision methods have been developed to verify small

and annihilation operators obey to commutation relations interpolating between bosons and

and BE statistics has developed many studies on the intermediate statistics, where creation

early work by Gentile in 1940 Very recently the search for small violations of the FD

allowing that N fermions, in place of one alone, occupy a single state, started with the

The evolution equation can be written in the form: OCR Output

the transition probability rr(t,v —> u) from the point v to the point u in the phase space.

particle distribution at any point of the phase space. The particle kinetics is described by

We call n(t, v) the occupational number, it is a scalar Held which describes univocally the

Let us consider the kinetics of particles in a continuous space of arbitrary dimension D.

II. PARTICLE KINETICS IN THE PHASE SPACE

Conclusions are reported in Sec.VI.

interval between two close states. The distribution functions are also derived.

velocity (or discrete energy) space. We may deduce the quantum of energy which is the

In Sec.V, the kinetics of the previous section is studied on a lattice i.e. on a discrete

rc == :1:1, to the BE and FD distribution and when zz = 0 to the MB one.

to innnite we may derive the expression of the distribution function which coincides, when

tional to the particle velocity and the diffusion coeflicient is a constant. As the time goes

In Sec.IV, we develop the kinetics for Brownian particles: the drift coefficient is propor

is derived whose expression is valid for any value of the parameter re, between -1 and +1.

second order momentum of the transition rate different from zero. A stationary distribution

In Sec.III, we limit ourselves to consider nearest neighbors interactions and only first and

equation.

dimensional space that, in the case of rc = O, becomes the well—known, linear Fokker-Planck

In Sec.II, we introduce a general, non—linear kinetic equation in an arbitrary D

of intermediate statistics.

through inhibition or enhancement factors. We send to ref.[9] for a classical kinetic model

within the framework of classical, non—linear equations of motion containing quantum effects

be possible to develop interesting models for atomic, nuclear and particle quantum dynamics

of particles under consideration. The result obtained in this paper holds promise that it may

of rt we may choose the degree of indistinguishability or the degree of classicity of the system

t — : —M »V UN ml 5 <>OCR Output(—1)"‘ @'"¢l¤(i»V)l ——-——-————;-—·— Q O, Om »~ » M

The function 1/¤[n(t,v — u)] also, in the second integral of Eq.(3), can be written as:

u(,,uc,,...uO,m . (4)r(t, v + u, u)ep[n(t, v + u)] : go E,1 am r ¢ v)]}

We obtain:

expanded in a Taylor series as a function of v + u, in an interval around v, being u < 11.

first integral of the right hand side of Eq.(3) the quantity r(t, v + u, u) ¢p[n(t, V+ u)] can be

In this paper we consider physical systems evolving very slowly. This means that in the

<p[n(t,v)] [ r(t,v,u) zf»[n(t,v — u)] duu

,= 1b[n(t,v)] /r(t,v + u,u) ¤p[n(t,v + u)] dun8 r, ngV)

assumes the form:

The evolution equation, taking into account the expression of the transition probability,

degree, the transition rates, from site to site.

arrival site u. These two functions can inhibit or enhance, fully or partially, at a certain

bution at the initial state v while 1,b[n(t, u)] depends on the occupational distribution of the

number to be fully expressed: cp[n(t, v)] is a function depending on the occupational distri—

v — u of the particle state during the transition, <,0 and rb are functions of the occupational

where r(t, v, v-—u) is the transition rate, depending on the initial state v and on the variation

(2)1r(t,v —> u) = r(t,v,v — u) gp[n(t,v)] 1/¤[n(t, u)] ,

the following way:

Let us define the transition probability of a particle from the state v to the state u in

point of the space.

arriving in v from any point of the space and the particles leaving v and going towards any

The variation in time of n(t,v) is due exclusively to the difference between the particles

(1)= [¢r(t,u —> v) — 1r(t,v —> u)] duu6 ¢ T,g-?-2 /

equation given by Eq.(7) becomes: OCR Output

with the occupation number of the starting site. Taking into account Eq.s (9), the evolution

high number of particles,the probability that one of them leaves the state increases linearly

the arrival site increases. The relation ¤p(n) = n means that if the initial state contains a

xc is equal to 1 and the probability transition is ever larger as the occupational number n in

if n = O, we have = 1 and transitions shall take place. In the case particles are bosons

site n = 1, the inhibition factor is zero and transitions do not take place; on the other hand,

case are bosons. In the Hrst case, must represent an inhibition factor: if in the arrival

where it is a parameter which has the value -1 in case the particles are fermions and +1 in

cp(n) = n , : 1 + rm , (9)

¤p(n) and are linear functions of n, we have:

impose that its population affects the transition probability. In the more simple case where

site if, in it, particles are absent. If, on the contrary, the arrival site is populated, we can

the second condition imposes that the transition probability does not depend on the arrival

The hrst condition means we do not have transitions when the initial site is empty, while

@(0) = 0 , 1/»(0) = I (8)

initial or arrival site is empty:

We observe that the functions <p(n) and must obey the two conditions, in the case the

(7)m + (-1) +1C¤,¤2...¤.,.(% V)<Pl"(#»V)l )@"‘¤/¤l¤(( V)l

Bt 7,,:0 l 30,,, 3110,2 . . .0vOm¤(¢»V) Z i (1/][n(, v]@”‘{C¤m...¤m(¢»V)<Pl¤('i»V)l}

equation:

the integro-differential Eq.(3) can be transformed into the following infinite order differential

(6)u §O,,,,,___,,m(t,v) : r(t,v,u)u,,,uc,,...u,,md u ,1 RT/

After the introduction of the mm order momentum of the transition rate

10 OCR Output

which is the continuity equation and takes into account the local balance in the phase space.

Ga+ VJUN) = 0 » (15)

t 6n( iv)

Eq_.(13) has the expression:

the exclusion—inclusion principle. After introducing the particle current given by Eq.(14),

is a drift current depending on §¤(t,v), O§(,6(t,v)/Ovp and on the parameter rt related to

The last term on the right hand side of Eq.(l4) is the Fick current, while the first term

(14). J¤(*»V) = —lC¤(i»V) + ····g·—l¤(¢»V)l1 + ~¤(f,V)l — C¤¤(tV)··y Un U:BQ, (t,v) L 3n(t,v)

We define the particle current j0,(t,v) by means of the relation:

8% 6%t, l»tt, Gt, -—-————. 13 l¤( V)l 4 r¤( V)l +C ¤( V) } ( )——-——=—— qt, Gt 3Un{lC( V) +

3(_"¤p(t,v) 3n(t,v)0n(t,v) 3

Eq.(l0) can be written as it follows:

(12)g`O,p(t,v) : r(t,v,u)uO,up duué/

(Il)§O,(t,v)= /r(t,v,u}u(, duu ,

momentum of the transition rate are different from zero; we have:

In this section we consider the Eq.(10) in the case only the first and the second order

III. NEAREST NEIGHBOURS INTERACTION

In the case rc = 0 we obtain the well known, linear Fokker—Planck equation.

of how much the particle kinetics is affected by the particle population of the arrival site.

which describes a non—linear kinetics. We note that the parameter rc gives a quantification

u

1 ’ ( 0)t +""( "’) g O"‘[§., (t v)n(t v)] 8"‘n(t v) 102 crm 7 7 _1m+] Ot O OI fi 7 i av¤1ar,2...0t¤m fl l C 1 m( "’)avmat.,2...av,,mlGt ,:0 314,1 5210,2 . . . Hvdm

8n(t,v) @2 8"‘[§m,,2___L,m(t, v)n(t, v)]

11 OCR Output

variation dv of the position of a particle along the direction oz; we have:

local balance of the particles along the generic direction cx. Let us consider an infinitesimal

that is exactly the Eq.(15). This equation can be obtained also directly, by means of a

6v,,Gt Ovc,A _ ¤]¤(f»v)l1 + ~¤(¢»v)l + D.<t,v>¤i . uv= V{[J(¢.v) +

OD., t, ( v) 6 t, i’@3 t, "( V)

current into Eq.(10) we obtain:

In Eq. (18) repeated indices are not summed over. Substituting this expression of the

. ],;,(t,v) = —[JO,(t,v) + -—-———]n(t,v)[1 + zm(t,v)] — Dc,(i,v)——dv,. Ov, (18)GDC, t, LY2 3 t, ?iv>

mix with each other; we have:

consequence that during the transitions the various components of the current vector do not

Let us remark that the second order momentum is a diagonal matrix. This fact has the

Eq.(17) represents the conditions to be satisfied only for the nearest neighbors interaction.

Coq¤;...r.vm(t;v) : 0 m > 2

(c,p(t,v) : D,,,(t,v)6a,6 , (17)

g`O,(t,v) : JO,(t,v) ,

mornenta:

the transition rate and using Eq.(6) we have, for Av ——> 0 the following expressions for the

where D¤(t,v) and J,,(t,v) are the diffusion and the drift coefficients. With this choice of

(16)+lD¤(i»V) —§J¤(i,V)Avl<$(¤¤ + A10} »

(pT(¢,v,u) Z -5%; g]! 6(u,,){[n(,,(t,v) + gJC,(r,v)At]a(u(, - Av)

rate assumes the following form:

only one component of the position vector u can change of a quantity Av, the transition

tions among the nearest neighbors. If we make the hypothesis that during a single transition

We demonstrate that the kinetics described by Eq.(15) takes into account only interac

12 OCR Output

equation j(t, v) = O, when J,,(v) and DQ(v) are related by means of the following expression:

t —> oo the particle current is equal to zero. It is easy to integrate the first order differential

The stationary solutions n(v) can be obtained directly by observing that at the limit of

tn(v) = lim n(i, v) (28)

tD(,,(v) = lim D,(t,v) , (27)

tJO,(v) = liin J(,,(i,v) , (26)

We pose now our attention to the steady states of Eq.(19). We define:

consequently Eq.(19).

If we expand this expression, up to the first order in dv, we obtain immediately Eq.(18) and

(25)

jQ(t,v)dv = r(t,v,dv)n(t,v)[1+ :m(t,v + dv)] — r(t,v + dv, -—dv)n(t,v + dv)[1+ »m(t,v)]

and explicitly we have:

j,,(t,v)dv: 7r(t,v——> v+dv)—1r(t,v+dv -—>v) , (24)

The particle current along the direction oz is consequently given by:

1r(t, v + dv —> v) : r(t, v + dv, —dv)n(t,v + dv)[1 + rm(t,v)] (23)

(22)¢r(t,v —+ v + dv) : r(t,v,dv)n(t,v)[1+ ;m(t, v + dv)] ,

following expressions:

the transition probabilities of the two transitions v —> v + dv and v + dv —> v assume the

(21)r(t,v, ;l;dv) = D,,,(t,v) fc §.]¤(t,v)dv ,

Defining the transition rate by the relation:

(20)(dv),6 : 6,8,,,dv

13 OCR Output

diffusion coefficient is a constant:

this type of particles, the drift coefficient is proportional to the particle velocity while the

In this Section we consider the brownian particle kinetics in a 3—dimensional space; for

IV. BOSONS AND FERMIONS IN A CONTINUOUS SPACE

framework of validity of the nearest neighbors interaction hypothesis.

rc can vary continuously between -1 and +1: all kind of statistics can be derived, in the

and D,,,(v,,) the integration can be analytically or numerically accomplished. The parameter

motion along the various directions is decoupled. Depending on the expressions of J,,,(vO,)

Eq.(33) is the most general expression for the stationary distribution function n(v) when the

BD —·“1 Xplg, f `5QT·75lJ<~("¤) + J.,L,ld”¤— W) · Kn(v) : ———N (33)

In this case the steady state given by Eq.(30) assumes the expression:

D¤(V) = D¤(v¤) (32)

J¤(V) Z J¤(v¤) » (31)

the phase space is completely decoupled:

The condition given by Eq.(29) is satisfied if the motion along the various directions of

that expresses the conservation of the particle number.

where Hp is the integration constant and can be calculated from the normalization condition

a1exp[;f h,,,,(vO,)dvO, — Bp] —- rc

71(V) = (30)

where h,,(vO,) is an arbitrary function. We obtain with easy calculations:

011,,,J0,(v) : h0,(v(,)Dc,(v) (29)

OD, (V)

14

Gt+ <I>(t,v) = O (41) OCR Output

6N z, ( U)

the equation describing the evolution of the particles inside the sphere E., is:

*I>(i,v) =i(i»v)¤d¤» (40)/ EI}

If we indicate with <I>(t, v) the flux of particles outcoming the sphere Ev at time t we have:

N (t, oo) = N (39)

and therefore we have the condition:

Let us call N the total number of particles. This number must be conserved at any time

(38)N(t,v) = n(t,v)d3v[3

having the center at the origin and the radius equal to v; we have:

velocity §_ v. Of course these are the particles lying, in the velocity space, in a sphere E.,

We now indicate with N(t, 11) the number of particles having at time t the modulus of

E == §m(v$ + 11;+ vg)

The kinetic energy of a particle is given by:

we will show that Eq.(36) can be written as a continuity equation in the energy variable.

Let us remark that Eq.(36) is a continuity equation, written in the velocity space. Later on

t(36)1 V{cvn(t,v)[1+ Ȣn(t,v)] + c-/;5Vn(t, v)}3 t, ngV)

The kinetics equation assumes the form:

(35)j : —cvn(t,v)[1 + rm(t,v)] —— %—Vn(i,v)

where c is an arbitrary constant. The current is given by:

(34)JO,(t,v) := cv, , D(,(t,v) = , B = ff ,

15 OCR Output

course, the particle current in the energy axis and assumes the following form:

The flux of particles <I>(t, E) outcoming the sphere EU of radius v = (/2E/m represents, of

m

(48)a = 4c7r(;)3/2

We introduce the constant:

(47)pn, E) : 2¤(;)3/A/1T§n(t, E)

From Eq.(46) it follows that the function p(t, E) is given by:

(46)p(i,E)dE : 4rrv2n(t,E)dv

interval [11,21 + dv]; we therefore have:

shell delimited by the two spheres E,,+d,, and EU, having the modulus of the velocity in the

particles that lie in the interval [E, E + dE] is equal to the number of particles lying in the

space: we introduce the occupation probability p(t,E) at energy E, then the number of

We now want to interpret Eq.(45) as a continuity equation in the one-dimensional energy

8n(t,E) _ 2c G 3/2 l 3n(t,E)

and, after the introduction of the energy as the variable, we have:

at U20v -—--—-——- Hm gv }) »2 = ———— (v {v¤(i»V)l1+~¤(i»V)l+On(t,v) c 8 1 3n(t,v)

Then Eq.(42) assumes the following form:

<I>(t,v) = 4vrvvj(t,v) (43)

The values of <l>(t, v) can easily be calculated by Eq.(40) which assumes the expression:

Ot 41rv2 61242 ( )6‘n(t,v) 1 3<I>(t,v) I

Eq.(41) in the form:

We differentiate Eq.(4l) with respect to v and take into account Eq.(38); we can write

16 OCR Output

all the other values of rc a family of different statistical distributions.

Eq.(52) gives exactly the FD (rc :: -1), MB (rc = 0) and BE (rc = +1) statistics and for

(55)/°° .—[}y, CXP Z _ K`,.>”“ mdz Z N

the condition given by Eq.(53) assumes the form:

542W12 N = i —-— 3/

constant. After the introduction of the constant N, defined by:

V and h is a constant with the right dimensions which can be identified by the Planck

dn = d(mv;)dmi/h. N is the total number of particles of the system occupying the volume

where dr is the elementary volume in the phase space and is given by dr = dn dT2d·r;», with

n(E)d*r = N , (53)

The integration constant ;i can be determined from the normalization condition:

52 ( )E : —-————¥——————— fl <=><pB(E—#)—*<

After integration we have:

+ + : 0 (51)

bution at the limit t —> oo must satisfy by the first order differential equation:

If we set the particle current <I>(t, E) equal to zero we obtain that the stationary distri

Ot ' OE(50)

3p(t, E) I 3<I>(t,E)

space:

Eq.(45) assumes finally the form of a continuity equation in the one—dimensional energy

H OE(49)<I>(t, E) = —aE3/2{n(t, E)[1 + »m(t,E)] +

E

‘ T F _r·r··r

17 OCR Output

can write the expression for the particle current :

and are given by Eq.(9) and remember that now the velocity is a discrete variable, we

We consider the definition of the transition probability of Eq.(2), where the functions g0(n)

2D,(t) : r(t, 1);+1, ——Av) + r(t,v,+1, —Av) (61)

Ji(t) Av = r(t,v,, —l-Av) — r(t,v;, —Av) , (60)

transition rates:

We now introduce the drift J,(t) and the diffusion Di(t) coefficients by means of the

dt Av

d , t A Q t

nuity equation in a discrete space:

The Pauli master equation given by Eq.(56) assumes an expression analogous to the conti

Ajit) =J;(t)-J;-1(i) (58)

and call the currentvariation between two adjacent sites:

Mi) = lr(i,ve —> var) ·¢r(i,v¢+r —> v¢)lAv » (57)

occupation probability of the site i. We define the particle current as:

where rr(t, v, ——·> vj) is the transition probability from the site i to the site j and n;(t) is the

d= 7K'(t,'U;-1 —> 1),)+ rr(t,v,+1 ——> vi) — 7l'(t,Ug —> 11,+1) —— 7r(t,v; —> 11,-1) , (56)n; )

in the following way:

not change during the transition. The Pauli master equation for this process can be written

transitions to the nearest neighbors are allowed and that the direction of the particles does

interval between close levels is represented by a quantum of energy. Let us assume that only

particles. The results for this case can be useful when energies are discretized and the

In this section we wish to discuss, in a discrete velocity space, the kinetics of brownian

V. BOSONS AND FERMIONS IN A LATTICE

18

e = ——·—— ; Q +1 1 _ BAE? 69 ( ) OCR Output

Eq.(67) become:

fh ·· *6TL; =- —-—— , (68)

Introducing the variable qi:

1 + BAEK11;; = n +1 67 ( )

(I —

To deduce ni, let us write the recurrence relation of Eq.(66) in the form:

substitution if we remember the definition of the transition probability given by Eq.(2).

tute + with the quantity n,(1 + mm,1). It is straightforward to justify this

equation given by Eq.(51). On the other hand, when we discretize Eq.(5l) we must substi

In the limit to the continuum we can transform this recurrence relation into the differential

m(1 + mai) + (66)gkgmm

current equal to zero:

To obtain the stationary distribution of the system, it is enough to pose the particle

Mi) = -iAE{¤;(i)l1+ #<m+1(f)l + (65)Z,··§é}Qthe expression of the current assumes the form:

Av ·· "mimr » (64)

Considering the energy as the variable instead of the velocity and taking into account that:

(63)J;(t) = cv; , D;(t) :

have the following expressions:

We are considering brownian motion by imposing that the drift and the diffusion coefficients

Av Av(62). Je(i) = —{lJi(i) + —·l¤¢(¢)[1 + ~m+1(¢)l + De(*)·—}A6AD; i l2 AH; I Q

19

€Xp¤r»»r>—~ll i Z: ( 1 OCR Output

Eq.(59):

the stationary distribution given by Eq.(70) becomes the continuous distribution given by

AE—•Olim iAE = E , (75)

Let us remark that if we calculate the limit to the continuum and consider that

which express the detailed balance principle and is valid for any value of is E [-1, +1].

MU + mar) = €><p[—f’(E¢+i - Ei)l¤e+1(1+ me) » (74)

Once we consider relation (72), the Eq.(66) can be written in the following form:

where k is the Boltzmann constant.

may derive that the relation between w and T is given by eu : (kT/h) log[1/(1 —— AE/kT)],

lf we assume that E is proportional to w through the relation S = hw, from Eq.(72) we

ln conclusion the quantum E coincides with the lattice interval AE only when ,8 —-—> O (T ~—>

(73)2 e AE + gmaiz)2 + §52(AE)3 +.

of the system. Let us expand the energy E in power series of AE, we have:

two close states, is different from the lattice interval AE. This is due to the finite temperature

Let us remark that the quantum of energy E, which represents the energy interval between

H 1 _ ME8 = —l 0g 72 ( )1 1

and E has the expression:

Ei = ii , (71)

which is the discretized distribution. ln Eq.(70) the energy E, is given by:

€><1:>5(E¢—u) —·<TL; = )

We can easily calculate q, and then finally ni:

20 OCR Output

we must recall that in recent works [10] it has been shown that the quantum average fluc

tions. We have limited ourselves in considering explicitly only systems of brownian particles;

quantum dynamics is equivalent to classical dynamics with the inclusion of quantum fluctua

The results obtained in this work should not come as a surprise; in fact we know that

a small violation of the exclusion or inclusion principle.

whose wave functions must not be completely antisymmetrised or to physical processes with

sation or symmetrisation, therefore these intermediate statistics can be applied to systems

We give to rc the meaning of degree of indistinguishability or of degree of antisymmetri—

of different statistics.

classical MB (rc = 0, no quantum effects are considered). When A gé 0, :k1 we have a family

When the particles are brownian we find exactly the FD (rc ·: -1), the BE (rs : 1) and the

of distribution functions of many-particle systems which is the main interest in this work.

of transitions caused by a small violation of the Pauli principle rather than to the study

Green and by other authors in the past Their interest was mainly devoted to the study

The statistics which differ in a discrete way from the usual ones were introduced by

of the Pauli exclusion principle and of BE statistics.

Greenberg and collaborators [6,7] in view of considering the possibility of small violations

The exact validity of BE and FD statistics was questioned recently in many papers by

account, i.e. the particle wave functions are only partially antisymmetrised or symmetrised.

are derived in which the exclusion or the inclusion principle is only partially taken into

quantum effects are not considered, in all the other cases different statistical distributions

ple when rc = +1, while the case rc = 0 means that the distribution is a classical one because

and 1. We take fully into account the exclusion principle when rc = -1, the inclusion princi

ment factor. The factor contains a parameter KZ whose value ranges continuously from -1

effects introduced into the transition probabilities by means of an inhibition or an enhance

In this work we introduce a non—linear classical kinetic equation, containing quantum

21 OCR Output

We thank A. Erdas for critical reading of the manuscript.

ACKNOWLEDGMENTS

a dynamics very close to the quantum dynamics of a system of identical particles.

conclude that the non—linear kinetic equation we have introduced in this paper can describe

tuations of the quantum dynamics coincides with the brownian fluctuations. Therefore we

22

sitaf di Pavia, (1992) and references therein.

10 M.RoncadeI1i, A.Defendi, I cammini di Fei nman, uaderni di Fisica. Teorica, Univer- J

together with the present work.

[9] G.Kaniadakis, Classical intermediate statistics, submitted for publication to Phys.Rev.E

[8] P.Brovetto and G.Bussetti, Nuovo Cimento 64 B, 284 (1969).

[7] O.W.Greenberg, Phys.Rev.D 43, 4111 (1991).

[6] O.W.Greenberg, R.N.Mohapatra., Phys.Rev.D 39, 2032 (1989).

H.S.Grcon, Phys.Rev. 90, 270 (1953).[5]

G.Gonti1o, Nuovo Cimonto 17,493 (1940).[4]

J.M.Leinaas,J.Myrheim, Nuovo Cimonto 37 B, 1 (1977).[3]

G.Kaniadakis and P.Quara.ti, Phys.Rev.E 48,4263 (1993).[2]

C.Dorso, S.Duarte, J.Ramdrup, Phys.Lett.B 287,1988 (1987).[1]

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