Fractional exclusion statistics in non-homogeneous interacting particle systems

# Fractional exclusion statistics in non-homogeneous interacting particle systems

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###### Abstract

We develop a model based on the fractional exclusion statistics (FES) applicable to non-homogeneous interacting particle systems. Here the species represent elementary volumes in an -dimensional space, formed by the direct product between the -dimensional space of positions and the quasiparticle energy axis. The model is particularly suitable for systems with localized states. We prove the feasibility of our method by applying it to systems of different degrees of complexities. We first apply the formalism on simpler systems, formed of two sub-systems, and present numerical and analytical thermodynamic calculations, pointing out the quasiparticle population inversion and maxima in the heat capacity, in contrast to systems with only diagonal (direct) FES parameters. Further we investigate larger, non-homogeneous systems with repulsive screened Coulomb interactions, indicating accumulation and depletion effects at the interfaces. Finally, we consider systems with several degrees of disorder, which are prototypical for models with glassy behavior. We find that the disorder produces a spatial segregation of quasiparticles at low energies which significantly affects the heat capacity and the entropy of the system.

fractional exclusion statistics non-homogeneous system screened Coulomb interaction heat capacity

1,2]G. A. Nemnes 2]D. V. Anghel 1]University of Bucharest, Faculty of Physics, “Materials and Devices for Electronics and Optoelectronics” Research Center, P.O. Box MG-11, 077125 Măgurele-Ilfov, Romania. 2]Horia Hulubei National Institute for Physics and Nuclear Engineering, P.O. Box MG-6, 077126 Măgurele-Ilfov, Romania.

## 1 Introduction

The concept of fractional exclusion statistics (FES), which is a generalization of the Pauli exclusion principle, was introduced by Haldane in Ref. [1] and the statistical mechanics of FES systems was formulated by several authors, employing different methods [2, 3, 4, 5, 6, 7]. Other generalizations of the Bose and Fermi statistics include the Gentile’s statistics [8, 9], obtained by fixing the maximum occupation number of a single-particle state, anyonic statistics [10, 11, 12, 13, 14], which is connected to the braid group, and q-deformed statistics, concerning systems of particles in arbitrary dimensions which satisfy quon algebras [15].

The FES was applied to quasiparticle excitations at the lowest Landau level in the fractional quantum Hall effect, spinon excitations in a spin- quantum antiferomagnet [1, 4, 16], Bose and Fermi systems described in the thermodynamic Bethe ansatz [5, 17, 18, 19], excitations [1] or motifs of spins [20, 21] in spin chains, elementary volumes obtained by coarse-graining in the phase-space of a system [5, 22, 23, 24, 25, 26, 27], interacting particles described in the mean-field approximation [28, 29, 30, 31], etc. Generic FES systems in different numbers of dimensions or interacting with external fields have also been studied for example in Refs. [32, 33, 34, 35, 36, 37, 38, 39, 40]. Correlations in FES systems have been calculated in [41]. In Refs. [28, 42] it was shown that systems of constant density of states with equal and diagonal FES parameters have the same thermodynamic properties under canonical conditions. This property extends the Bose-Fermi thermodynamic equivalence in 2D systems discovered long time ago [43, 44, 45], to the FES systems [46].

A stochastic method for the simulation of the time evolution of FES systems was introduced in Ref. [47] as a generalization of a similar method used for Bose and Fermi systems [48] whereas the relatively recent experimental realization of the Fermi degeneracy in cold atomic gases have renewed the interest in the theoretical investigation of non-ideal Fermi systems at low temperatures and their interpretation as ideal FES systems [49, 37, 38, 50, 51].

The FES formalism was amended to include the change of the FES parameters at the change of the particle species [24, 52] and this allows the implementation of FES as a general method to describe interacting particle systems as ideal gases of quasiparticles [25, 26, 53, 31, 54, 20, 21]. Moreover, FES was applied also to systems of classical particles in Refs. [55, 56].

In this paper we present the procedure of describing systems of particles with long-range interaction as ideal FES systems. The method was introduced briefly in Refs. [53, 31] and we apply it here to systems consisting of randomly and (in general) non-uniformly distributed localized states which can be occupied by particles. The particle-particle interaction potential depends only on the distance between the particles, like for example the screened Coulomb interaction, . We define the quasiparticle energies and the fractional exclusion statistics parameters. Using these we write the FES equations for equilibrium particle distribution. In Ref. [31] it was shown that the solution provided by the FES equations is equivalent to the standard solution by Landau’s Fermi liquid theory (FLT). We show here the feasibility of the FES method by solving numerically the FES equations for a variety of one-dimensional systems, with homogeneous and non-homogeneous particle distributions.

This method may find applications in glassy systems such as the Coulomb glasses, systems of bosons trapped in optical lattices, but also to mesoscopic transport in an already implemented Monte Carlo framework [47].

Another FES method for the description of one-dimensional systems of particles which occupy localized states, was proposed in [57]. It is not our purpose to compare the two methods here, but they may be complementary to each other. Our method may describe systems of particles with long range interaction in any number of dimensions, with non-homogeneous sites distributions and different energies per site, but assumes a large number of particles and sites within the range of the particle-particle interaction (see also Refs. [53, 31]). On the other hand, the method of Ref. [57] describes one-dimensional homogeneous lattice gases of the same energy per site, on the basis of statistically interacting vacancy particles. The method is accurate for any range of the particle-particle interaction and is especially suitable for extracting the distributions of spaces between individual particles.

The structure of the paper is as follows. In the following subsection we introduce briefly the notations and the basic concepts of FES. In Section 2 we introduce our model, in which species are elementary volumes in the -dimensional space formed by the direct product between the -dimensional space of positions, , and the energy axis, or . These species are related by FES parameters, which we calculate. We prove the feasibility of the method by applying it to a few physically relevant systems of different complexities. First we apply it to homogeneous systems of particles interacting by repulsive (screened) Coulomb potentials. Then, in Section 3, we apply our formalism on a few test cases with reduced number of species, which are analytically tractable in order to capture essential features of interacting inhomogeneous systems. Next, the accumulation and depletion of particles subject to screened Coulomb interactions is investigated in larger, non-uniform systems, and interface phenomena are emphasized. Finally, systems with several degrees of disorder are considered and the spatial segregation of quasiparticles is pointed out together with its consequences in the thermodynamic behavior.

### 1.1 Basic definitions

A FES system consists of a countable set of species, indexed by . Each species contains a finite number of single-particle states and particles, denoted by and , respectively. The number of states in the species depend on the number of particles. For small variations of the number of particles around some reference distribution, , the number of states changes by

 δGi=−∑jαijδNj, (1)

where by we denote the particle variations and ’s are called the FES parameters [1].

The FES parameters must satisfy certain rules [24, 52, 21, 55], namely if we split an arbitrary species, , into a number of sub-species, , then all the parameters , with both, and different from , remain unchanged, whereas the rest of the parameters must satisfy the relations:

 αij = αij0=αij1=…, for any i, i≠j (2a) αji = αj0i+αj1i+…, for any i, i≠j (2b) αjj = αj0j0+αj1j0+…=αj0j1+αj1j1+…=… (2c)

These rules are satisfied by the ansatz [24, 27],

 αij=α(e)ij+α(s)iδij, (3a) where the parameters α(e)ij, called the “extensive” parameters, are proportional to Gi, α(e)ij≡aijGi. (3b)

The parameters refer to only one species and do not depend on .

The number of microscopic configurations compatible to a given distribution of particles on species, , is

 W(−)({Ni})=∏i(G(−)i+Ni−1)!Ni!(G(−)i−1)!andW(+)({Ni})=∏iG(+)i!Ni!(G(+)i−Ni)!, (4)

if the particles are bosons and fermions, respectively.

If for each species of particles, say species , we associate an energy, , and a chemical potential , then the equilibrium particle distribution, , is obtained by maximizing the partition function,

 Z(±)≡∑{Ni}{W(±)({Ni})exp[β∑j(ϵj−μj)Nj]}, (5)

with respect to the distribution , taking into account that the ’s vary with according to (1)–in Eq. (5) is the inverse temperature.

The maximization of with the conditions (1) gives

 β(μi−ϵi)+ln1∓n(±)ini=∓∑jαjiln[1∓n(±)j]. (6a) where the all the upper signs are for fermions and all the lower signs are for bosons. If one uses the ansatz (3) which is relevant for the systems analyzed below, Eq. (6a) becomes [27] β(μi−ϵi)+ln[1∓n(±)i]1−α(s)in(±)i=∓∑jGjajiln[1∓n(±)j]. (6b)

In FES, a system of fermions with a set of parameters, , may be interpreted as a system of bosons with the parameters and vice-versa, a system of bosons of parameters may be interpreted as a system of fermions with parameters . Therefore it is more natural to refer to Bose and Fermi formulations, rather than to bosons and fermions.

The most used formulation of FES is the one employed by Wu in Ref. [4], which will be denoted here by “W”. To see how this is related to the Bose formulation we define the number of states in the absence of particles in the system, , and a new particle population, . Then the ’s are determined in two steps. First one solves the system

 (1+wi)∏j(wj1+wj)αji=e(ϵi−μ)/kBT, (7a) to determine the wi’s, and then the nWi’s are calculated from ∑j(δijwj+αijGWj/GWi)nj=1. (7b)

Comparing Eqs. (6a) and (7a) we observe that .

Equations (6) admit solutions of the Fermi liquid form [58]

 n(±)i=1eβ(~ϵi−μ)±1, (8)

where are Landau type of quasiparticle energies that satisfy the general relations

 ~ϵk = ϵk∓kBT∑iα(±)ikln[1∓n(±)i]. (9) = ϵk±kBT∑iα(±)ikln[1±e−β(~ϵi−μ)].

## 2 Model and formalism

The particles are localized on random sites in a solid -dimensional matrix. The positions of the sites are denoted by , , where is the total number of sites. We assume that the wavefunctions of the particles do not overlap and the total energy of the system is

 E=∑IϵrInrI+12∑I,JVrIrJnrInrJ. (10)

where is the energy and is the occupation number of the site ; the total particle number is .

We shall work in the continuous limit, so we define the density of sites, , and the particle density, . The average particle population in an arbitrary D volume, , is , where we assume that the volume is large enough, so that we have . In these notations the total energy of the system (10) becomes

 E = ∫Ωdsr∫ϵmaxϵminϵρ(r,ϵ)dϵ (11) +12∫Ωdsr∫Ωdsr′∫ϵmaxϵmindϵ∫ϵmaxϵmindϵ′ρ(r,ϵ)ρ(r′,ϵ′)Vrr′

where is total the volume of the system and is the interval in which takes values, with . We shall assume that , (where is the Boltzmann constant and is the temperature) and the interaction energy depends only on the distance between the sites, i.e. . Because , we shall take . For concreteness, we analyze only Fermi system, but the formalism can be easily extended to bosons.

To apply FES, we have to divide the system into species. We do this by coarse-graining the parameters space of the system, , into the elementary volumes, . By the lower case Greek letters, e.g. , we identify the spatial volume and by the lower case Latin letters, e.g. , we identify the energy intervals, . We take .

We identify a species either directly, by , or by the subscripts, . We assume that each elementary volume, , is centered at and contains a large enough number of sites and particles to justify the application of the statistical methods and, in particular the Stirling approximation for the logarithms of factorial numbers. In each of the volumes, say , we have a distribution of sites, , of energies . Under the assumption that the number of sites is large enough, the set of energies form a (quasi)continuous distribution along the axis, with a density . The number of states in the species is then and the number of particles is . We shall use the notations for the particle density and for the volume particle density.

We define the quasiparticle energies, , in a similar way as in Ref. [25, 26, 31], by

 ~ϵrI=ϵrI+∑~ϵrJ<~ϵrIV(|rI−rJ|)nrJ. (12)

Because of the identity , the thermodynamics of the quasiparticle gas follows identically the thermodynamics of the original system. In the continuous limit, Eq. (12) becomes

 ~ϵrI=ϵrI+∫Ωdsr∫ϵrI0dϵV(|rI−r|)σ(r,ϵ)n(r,ϵ). (13)

We have a new parameters space, , in which, by construction, and . Using Eq. (13) we obtain the new DOS,

 ~σ[r,~ϵ(ϵ)]=σ(r,ϵ)∣∣∣d~ϵdϵ∣∣∣−1=σ(r,ϵ)∣∣1+∫Ωdsr′V(|r−r′|)σ(r′,ϵ)n(r′,ϵ)∣∣. (14)

Assuming that Eq. (13) defines a one-to-one function – which may also be inverted to – we split the space into species, , as we did with , in such a way that for any and each species contains states and particles, as before. By the application of the procedure of Refs. [25, 26] to this interaction potential and particle species we obtain the FES parameters [53, 31]

 αξi;ηj = [δijσξ(ϵi)+θ(i−j)δϵidσξ(ϵi)dϵi]V(|rξ−rη|) (15) ≡ [δijδΩξσ(rξ,ϵi)+θ(i−j)δϵiδΩξ∂σ(rξ,ϵi)∂ϵi]V(|rξ−rη|)

where the first doublet, , specifies the species in which the number of states changes, whereas the second doublet, , specifies the species in which the number of particle changes; is the step function, and . The manifestation of the FES parameters given by Eq. (15) for a system with , i.e. independent of , between species of the same quasiparticle energies is represented in Fig. 1.

We observe that the FES parameters (15) obey the rules (2) [52] and define a new ansatz, which is a generalization of (3) [27].

The fact that the definition species, quasiparticle energies and FES parameters is self-consistent and feasible was checked in Ref. [27] where it was shown that the FES formalism is equivalent to the standard FLT for a general class of mean-field systems which includes also a non-constant external potential.

### 2.1 Equilibrium thermodynamics

Since we have fermions in the systems, we employ the Fermi formulation [27] to calculate the equilibrium thermodynamics. Plugging the parameters (15) into the equations (6) we get

 0 = β(μ−~ϵi)+ln1−n(+)(rξ,~ϵi)n(+)(rξ,~ϵi)+∑ηjαηj;ξiln[1−n(+)(rη,~ϵj)]. (16)

In the continuous limit, Eq. (16) becomes

 0 = β(μ−~ϵ)+ln1−n(+)(r,~ϵ)n(+)(r,~ϵ)+∫Ωdsr′V(|r′−r|)σ[r′,ϵ(~ϵ)]ln[1−n(+)(r′,~ϵ)] (17) +∫Ωdsr′V(|r′−r|)∫∞~ϵd~ϵ′∂σ(r′,ϵ)∂ϵ∣∣∣ϵ(~ϵ′)ln[1−n(+)(r′,~ϵ′)].

In the bosonic formulation, as we mentioned above, , and the dimension of the species is the number of available states, (we used the notation to avoid confusion with the number of states in the species, ). This changes also the definition of the population to . Plugging the new quantities into Eqs. (6) we obtain the system of equations for :

 0 = β(μ−~ϵi)+ln1+n(−)(rξ,~ϵi)n(−)(rξ,~ϵi)−∑ηjα(−)ηj;ξiln[1+n(−)(rη,~ϵj)] (18)

and in the continuous limit,

 0 = β(μ−~ϵ)+ln1+n(−)(r,~ϵ)n(−)(r,~ϵ)−∫Ωdsr′V(|r′−r|)σ[r′,ϵ(~ϵ)]ln[1+n(−)(r′,~ϵ)] (19) −∫Ωdsr′V(|r′−r|)∫∞~ϵd~ϵ′∂σ(r′,ϵ)∂ϵ∣∣∣ϵ(~ϵ′)ln[1+n(−)(r′,~ϵ′)].

In Wu’s formulation [4] the dimension of the species is and . The equilibrium population is calculated from

 (1+wξi)∏η,j(wηj1+wηj)α(−)ηj;ξi=e(ϵξ,i−μ)/kT, (20a) and ∑η,j(δξηδijwηj+βξi,ηj)nWηj=1, (20b)

where and .

Having the quasiparticle populations, we can calculate any thermodynamical quantity. The internal energy of the system is

 U(T,μ)=∑InrI~ϵrI=∑ξinP(rξ,~ϵi)GPξi~ϵi (21)

or, in the continuous limit,

 U(T,μ) = ∫Ωdsr∫∞0d~ϵ~σP(r,~ϵ)nP(r,~ϵ)~ϵ≡∫Ωdsr∫∞0d~ϵ~ρ(r,~ϵ)~ϵ, (22)

where or and is given by Eq. (14). The other two densities of states are and and we have the relation . We shall use the notation for the quasi-particle density in , at quasiparticle energy , and for the volume quasi-particle density. One should note that by definition.

In order to define the DOS we have to define the densities of the FES parameters [24, 27], , by

 αξi;ηj = arξ~ϵi;rη~ϵjδϵiδΩξ (23a) where arξ~ϵi;rη~ϵj = [δ(~ϵi−~ϵj)σ(rξ,ϵi)+θ(~ϵi−~ϵj)∂σ(rξ,ϵi)∂ϵi]V(|rξ−rη|). (23b)

Using Eqs. (23) we write .

Similarly to Eq. (22), the total particle number is

 N(T,μ) = ∫Ωdsr∫∞0d~ϵ~σP(r,~ϵ)nP(r,~ϵ)≡∫Ωdsr∫∞0d~ϵ~ρ(r,~ϵ). (24)

The heat capacity and the entropy of the system are

 CV = (∂U∂T)N=∂U(T,μ)∂T−∂U(T,μ)∂μ∂N(T,μ)∂T(∂N(T,μ)∂μ)−1 (25)

and

 S=kBln(W(±)), (26)

respectively, and they satisfy the equation

 CV=T(∂S∂T)N. (27)

### 2.2 Homogeneous system

If is independent of and if we impose periodic boundary conditions or ignore the effect of the surfaces (deep inside the solid), then both, and , are independent of and in the Eqs. (17) and (19) we can perform the integrals over to obtain an equation only in :

 0 = β(μ−~ϵ)+ln1−n(+)(~ϵ)n(+)(~ϵ)+IV{σ[ϵ(~ϵ)]ln[1−n(+)(~ϵ)] (28) +∫∞~ϵd~ϵ′∂σ(ϵ)∂ϵ∣∣∣ϵ(~ϵ′)ln[1−n(+)(~ϵ′)]},

for fermions or

 0 = β(μ−~ϵ)+ln1+n(−)(~ϵ)n(−)(~ϵ)−IV{σ[ϵ(~ϵ)]ln[1+n(−)(~ϵ)] (29) −∫∞~ϵd~ϵ′∂σ(ϵ)∂ϵ∣∣∣ϵ(~ϵ′)ln[1+n(−)(~ϵ′)]},

for bosons, where

 IV=∫Ωdsr′V(|r′−r|). (30)

Moreover, if , i.e. it does not depend on energy, then one may define an effective FES parameter, , and Eqs. (28) and (29) simplify to

 β(~ϵ−μ)=ln[1−n(+)(r,~ϵ)]1+αeffn(+)(r,~ϵ) (31)

and

 β(~ϵ−μ)=ln[1+n(−)(r,~ϵ)]1−αeffn(−)(r,~ϵ), (32)

respectively.

## 3 Applications

### 3.1 Two sub-volumes system

First let’s consider the simple example of a system formed of two sub-volumes, and . Such a test-case has the advantage of being transparent enough for a detailed discussion while still providing significant insights into the thermodynamic of more general interacting inhomogeneous systems. Moreover, for properly chosen parameters the populations may be calculated analytically.

We define the energy independent densities of states for the non-interacting particles in the two sub-volumes, and , and if we take the interaction energies between the particles in the same volume to be and in different volumes , then the FES parameters are , where and denote the energy intervals, as explained above. The total number of particles is defined as , where is the average DOS and is the Fermi energy for the non-interacting particle system. In this example we take .

We investigate two types of systems: type 1 (figure 2 a), with (since the particles in the same volume, being closer together, interact stronger than the particles in different volumes), and type 2 (figure 2 b), with the particles in the same volume behaving like ideal fermions, but interact repulsively with the particles in the other volume: .

In figure 2 (a) we observe that the particle densities in both volumes decrease with the quasiparticle energy, as one would expect, whereas, due to the repulsive interaction between the atoms, the fermionic densities of states increase monotonically. At high energies the fermionic densities of states converge to the non-interacting densities, and . These curves are calculated for , but they are typical for this system. In the right inset of figure 2 (a) we plot the heat capacity, which is not much different from the heat capacity of an ideal Fermi system.

The second example (figure 2 b) is equivalent to two subsystems of ideal fermions, with mutual interaction and which can exchange particles between them. This choice of parameters lead to and . Choosing and denoting the fugacity by , we obtain from the system (16) a third order equation for ,

 g[n(+)1(g)]3+(1−g−g2)[n(+)1(g)]2−2n(+)1(g)+1=0, (33a) and an equation for n(+)0(ϵ), n(+)0(g)=1−g2[n(+)1(g)]2[1−n(+)1(g)]2 (33b)

The third degree equation from (33a) has three real, distinct solutions for . However, by imposing the conditions and , we remain with only one solution, which is represented in Fig. 2(b).

In contrast to the previous system, a maximum occurs in at , indicating a population inversion with respect to the quasiparticle energy in the volume with lower DOS. This is due to the mutual exclusion statistics (1), which reduces the density of states significantly at low energies, especially in the species 0, where the density of states is lower.

At larger quasiparticle energies, , the densities of particles are lower and therefore the statistical interaction effects are also diminished. It is worth mentioning that for the total particle density, , no population inversion occurs, i.e. is a monotonically decreasing function of the quasiparticle energy. The maximum observed in induces a minimum in , whereas increases monotonically with .

Systems of particles with the same constant DOS and direct FES parameters, , have the same heat capacity, which is independent of and which is monotonically increasing with [18, 43, 44, 45, 46]. In our case the existence of mutual FES parameters, for , not only changes the heat capacity, but also leads to the appearance of a maximum in . For , converges to 1, which is the Boltzmann limit, for any FES parameters.

### 3.2 Screened Coulomb interactions. Homogeneous systems.

We further assume a two-particle screened potential of the general form

 V(r;γ,λ)=κexp(−r/λ)rγ, (34)

where . In particular, if , we have the usual screened Coulomb and Yukawa type potentials. In the absence of screening (), such systems exhibit standard thermostatistical behavior if [59] and the interactions are classified as short ranged. If the systems obey non-extensive thermodynamics, i.e. quantities like total energy are not extensive due to the long range interactions. However if the screening is present, the interactions become short-ranged and the usual thermodynamics applies.

To remove the singularity at the origin that appear in the integrals over in Eqs. (17), (19) and (30), we introduce a cut-off at radius , below which the potential remains constant – for . With this assumption, for a homogeneous system,

 IV=2πs2Γ(s2)∫∞0drrs−1V(r)=IV1+IV2, (35)

where

 IV1 ≡ 2πs2Γ(s2)∫R00drrs−1V(r)=πs2Γ(s2+1)Rs0V0 IV2 ≡ IV−IV1=2πs2Γ(s2)κσλs−γΓ(s−γ,R0λ). (36)

Here is the upper incomplete gamma function.

For a screened Coulomb-type interaction in a -dimensional system (), the term can be expressed as:

 I1DV2 = 2κσE1(R0/λ) I2DV2 = 2πκσλexp(−R0λ) I3DV2 = 4πκσλ(R0+λ)exp(−R0λ), (37)

where is the exponential integral,

 E1(z)=∫∞ze−ttdt=∫∞1e−zttdt. (38)

Using the integrals (3.2) calculated analitically one can apply the formalism presented in Subsection 2.2 and therefore the complete thermodynamical behavior of the homogeneous system may be calculated.

### 3.3 Accumulation and depletion effects in one-dimensional systems

Using the FES formalism we next describe the accumulation and depletion effects for one dimensional systems of particles with screened Coulomb interactions and non-uniform DOS. Periodic boundary conditions are imposed using the minimum image convention, i.e. the interaction drops at a distance equal to half the length of the repetitive unit [60].

For simplicity we shall assume in the following that [and ] are independent of , so we shall simplify the notation of the DOS to (and ). In this case the FES parameters (15) can be written as