Properties of Fractional Exclusion Statistics in Interacting Particle Systems
Dragoş-Victor Anghel
备注：
8 pages, 3 figures, to appear in Rom. J. Phys. Is a continuation of J. Phys. A: Math. Theor. 40 F1013-F1019 (2007); arXiv:0710.0724, and arXiv:0710.0728
We show that fractional exclusion statistics is manifested in general in interacting systems and we discuss the conjecture recently introduced (J. Phys. A: Math. Theor. 40, F1013, 2007), according to which if in a thermodynamic system the mutual exclusion statistics parameters are not zero, then they have to be proportional to the dimension of the Hilbert space on which they act. By using simple, intuitive arguments, but also concrete calculations in interacting systems models, we show that this conjecture is not some abstract consequence of unphysical modeling, but is a natural--and for a long time overlooked--property of fractional exclusion statistics. We show also that the fractional exclusion statistics is the consequence of interaction between the particles of the system and it is due to the change from the description of the system in terms of free-particle energies, to the description in terms of the quasi-particle energies. From this result, the thermodynamic equivalence of systems of the same, constant density of states, but any exclusion statistics follows immediately.
We introduce the hypothesis of incomplete information into the fractional exclusion statistics in order to apply the latter to some correlated heavy fermion systems. It is shown that the actual inexplicit distribution function of FES may be replaced by an equivalent explicit one of which the form does not change with $α$ value.
We show that it is possible to replace the actual implicit distribution function of the fractional exclusion statistics by an explicit one whose form does not change with the parameter $α$. This alternative simpler distribution function given by a generalization of Pauli exclusion principle from the level of the maximal occupation number is not completely equivalent to the distributions obtained from the level of state number counting of the fractional exclusion particles. Our result shows that the two distributions are equivalent for weakly bosonized fermions ($α>>0$) at not very high tempera…
The analytical expressions of the Joule-Thomson coefficient for homogeneous and harmonically trapped three-dimensional ideal anyons which obey Haldane fractional exclusion statistics are derived. For an ideal Fermi gas, the Joule-Thomson coefficient is negative, which means that there is no maximum Joule-Thomson inversion temperature. With careful study, it is found that there exists a Joule-Thomson inversion temperature in the fractional exclusion statistics model. Furthermore, the relations between the Joule-Thomson inversion temperature and the statistical parameter $g$ are investigated.
A fractional Fick's law and fractional hydrostatics for the one dimensional exclusion process with long jumps in contact with infinite reservoirs at different densities on the left and on the right are derived.
I discuss the concept of fractional exclusion statistics (FES) and I show that in order to preserve the thermodynamic consistency of the formalism, the exclusion statistics parameters should change if the species of particles in the system are divided into subspecies. Using a simple and intuitive model I deduce the general equations that have to be obeyed by the exlcusion statistics parameters in any FES system.
I introduce an ansatz for the exclusion statistics parameters of fractional exclusion statistics (FES) systems and I apply it to calculate the statistical distribution of particles from both, bosonic and fermionic perspectives. Then, to check the applicability of the ansatz, I calculate the FES parameters in three well-known models: in a Fermi liquid type of system, a one-dimensional quantum systems described in the thermodynamic Bethe ansatz and quasiparticle excitations in the fractional quantum Hall (FQH) systems. The FES parameters of the first two models satisfy the ansatz, whereas those …
The quantum statistical mechanics of an ideal gas with a general free-particle energy obeying fractional exclusion statistics are systematically investigated in arbitrary dimensions. The pressure relations, the relation between pressure and internal energy, the equation of state, as well as the thermodynamic properties are thoroughly discussed. Some novel results are obtained.
I discuss Haldane's concept of generalised exclusion statistics (Phys. Rev. Lett. {\bf 67}, 937, 1991) and I show that it leads to inconsistencies in the calculation of the particle distribution that maximizes the partition function. These inconsistencies appear when mutual exclusion statistics is manifested between different subspecies of particles in the system. In order to eliminate these inconsistencies, I introduce new mutual exclusion statistics parameters, which are proportional to the dimension of the Hilbert sub-space on which they act. These new definitions lead to properly defined p…
I show that fractional exclusion statistics (FES) is manifested in general interacting systems and I calculate the exclusion statistics parameters. Most importantly, I show that the mutual exclusion statistics parameters--when the presence of particles in one Hilbert space influences the dimension of another Hilbert space--are proportional to the dimension of the Hilbert space on which they act. This result, although surprising and different from the usual way of understanding the FES, renders this statistics consistent and valid in the thermodynamic limit, in accordance with the conjucture in…
Fractional exclusion statistics (FES) is a generalization of the Bose and Fermi statistics. Typically, systems of interacting particles are described as ideal FES systems and the properties of the FES systems are calculated from the properties of the interacting systems. In this paper I reverse the process and I show that a FES system may be described in general as a gas of quasiparticles which obey Bose or Fermi distributions; the energies of the newly defined quasiparticles are calculated starting from the FES equations for the equilibrium particle distribution. In the end I use a system in …
The k-body Gaussian Embedded Ensemble of Random Matrices is considered for N bosons distributed on two single-particle levels. When k = N, the ensemble is equivalent to the Gaussian Orthogonal Ensemble (GOE), and when k = 2 it corresponds to the Two-body Random Ensemble (TBRE) for bosons. It is shown that the energy spectrum leads to a rank function which is of the form of a discrete generalized beta distribution. The same distribution is obtained assuming N non-interacting quasiparticles that obey the fractional exclusion statistics introduced by Haldane two decades ago.
We present a stochastic method for the simulation of the time evolution in systems which obey generalized statistics, namely fractional exclusion statistics and Gentile's statistics. The transition rates are derived in the framework of canonical ensembles. This approach introduces a tool for describing interacting fermionic and bosonic systems in non-equilibrium as ideal FES systems, in a computationally efficient manner. The two types of statistics are analyzed comparatively, indicating their intrinsic thermodynamic differences and revealing key aspects related to the species size.
The violation of the Pauli principle has been surmised in several models of the Fractional Exclusion Statistics and successfully applied to several quantum systems. In this paper, a classical alternative of the exclusion statistics is studied using the maximum entropy methods. The difference between the Bose-Einstein statistics and the Maxwell-Boltzmann statistics is understood in terms of a separable quantity, namely the degree of indistinguishability. Starting from the usual Maxwell-Boltzmann microstate counting formula, a special restriction related to the degree of indistinguishability is …
The low energy properties of the one-dimensional anyon gas with $δ$-function interaction are discussed in the context of its Bethe ansatz solution. It is found that the anyonic statistical parameter and the dynamical coupling constant induce Haldane exclusion statistics interpolating between bosons and fermions. Moreover, the anyonic parameter may trigger statistics beyond Fermi statistics for which the exclusion parameter $α$ is greater than one. The Tonks-Girardeau and the weak coupling limits are discussed in detail. The results support the universal role of $α$ in the dispersion relations.
We show that below the degenerate temperature the distribution profiles of strongly interacting anyons in one dimension coincide with the most probable distributions of ideal particles obeying generalized exclusion statistics (GES). In the strongly interacting regime the thermodynamics and the local two-particle correlation function derived from the GES are seen to agree for low temperatures with the results derived for the anyon model using the thermodynamic Bethe Ansatz. The anyonic and dynamical interactions implement a continuous range of GES, providing a signature of strongly interacting …
It is mentioned that anyon thermodynamic potential $Q(α, N)$ could not be factorized in terms characteristic of the ideal boson $α=0$ and fermion $α=1$ gases by the relation $Q(α, N) = (1-α) Q(0, N_b)+ αQ(1, N_f)$ in which $N=N_f +N_b$, that claimed in Phys. Rev. Lett. 78, 3233 (1997). Our analyses indicate that the thermodynamic quantities of anyon gas may be factorized as $Q(α) = αQ(1) + (1-α) Q(0)$ only in the two-dimension system.
We calculate the partition function of a gas of particles obeying Haldane exclusion statistics, using a definition of a Hilbert space having a `fractional dimension' and constructing appropriate coherent states. The fractional dimension is expressed though the form of the identity operator in the Hilbert space. We find that there many possible generalisations of the Pauli exclusion principle, with particular choices of the scalar product leading to consistency either with Haldane's original definition of the effective dimensionality of the Hilbert space or with the combinatorial procedure invo…
We show the possibility of describing fractional exclusion statistics (FES) as an occupancy process with global and \textit{local} exclusion constraints. More specifically, using combinatorial identities, we show that FES can be viewed as "ball-in-box" models with appropriate weighting on the set of occupancy configurations (merely represented by a partition of the total number of particles). As a consequence, the following exact statement of the generalized Pauli principle is derived: for an $N$-particles system exhibiting FES of extended parameter \mbox{$g=q/r$} ($q$ and $r$ are co-prime int…
We describe a mean field interacting particle system in any number of dimensions and in a generic external potential as an ideal gas with fractional exclusion statistics (FES). We define the FES quasiparticle energies, we calculate the FES parameters of the system and we deduce the equations for the equilibrium particle populations. The FES gas is "ideal", in the sense that the quasiparticle energies do not depend on the other quasiparticle levels populations and the sum of the quasiparticle energies is equal to the total energy of the system. We prove that the FES formalism is equivalent to t…
In this paper, we study shocks and related transitions in asymmetric simple exclusion processes of particles with nearest neighbor interactions. We consider two kinds of inter-particle interactions. In one case, the particle-hole symmetry is broken due to the interaction. In the other case, particles have an effective repulsion due to which the particle-current-density drops down near the half filling. These interacting particles move on a one dimensional lattice which is open at both the ends with injection of particles at one end and withdrawal of particles at the other. In addition to this,…
We propose a new method for investigating the exclusion statistics of quasi-particles in Conformal Field Theory (CFT) spectra. The method leads to one-particle distribution functions, which generalize the Fermi-Dirac distribution. For the simplest $su(n)$ invariant CFTs we find a generalization of Gentile parafermions, and we obtain new distributions for the simplest $Z_N$-invariant CFTs. In special examples, our approach reproduces distributions based on `fractional exclusion statistics' in the sense of Haldane. We comment on applications to fractional quantum Hall effect edge theories.
We show that the $N$-particle Sutherland model with inverse-square and harmonic interactions exhibits orthogonality catastrophe. For a fixed value of the harmonic coupling, the overlap of the $N$-body ground state wave functions with two different values of the inverse-square interaction term goes to zero in the thermodynamic limit. When the two values of the inverse-square coupling differ by an infinitesimal amount, the wave function overlap shows an exponential suppression. This is qualitatively different from the usual power law suppression observed in the Anderson's orthogonality catastrop…
For a large class of inhomogeneous interacting particle systems (IPS) on a lattice we develop a rigorous method for mapping them onto homogeneous IPS. Our novel approach provides a direct way of obtaining the statistical properties of such inhomogeneous systems by studying the far simpler homogeneous systems. In the cases when the latter can be solved exactly our method yields an exact solution for the statistical properties of an inhomogeneous IPS. This approach is illustrated by studies of three of IPS, namely those with particles of different sizes, or with varying (between particles) maxim…
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 (s+1)-dimensional space, formed by the direct product between the s-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 analytica…
The Fermi liquid theory may provide a good description of the thermodynamic properties of an interacting particle system when the interaction between the particles contributes to the total energy of the system with a quantity which may depend on the total particle number, but does not depend on the temperature. In such a situation, the ideal part of the Hamiltonian, i.e. the energy of the system without the interaction energy, also provides a good description of the system's thermodynamics. If the total interaction energy of the system, being a complicated function of the particle populations,…
In Phys. Rev. Lett. 67, 937 (1991) [1], Haldane introduced the fruitful concept of fractional exclusion statistics (FES). One of the most influential papers in which the thermodynamics of FES systems was deduced is Y.-S. Wu, Phys. Rev. Lett. 73, 922 (1994). Unfortunately, some important, but eventually subtle, properties of the exclusion statistics parameters were overlooked in the original paper [1] and in all the papers after that, including [2]. This omission makes the thermodynamics of FES systems inconsistent when mutual exclusion statistics is manifesting in the system. By this Comment I…
We develop a bosonization approach to study the low temperature properties of one-dimensional gas of particles obeying fractional exclusion statistics (FES). It is shown that such ideal gas reproduces the low-energy excitations and asymptotic exponents of a one-component Luttinger liquid (with no internal degrees of freedom). The bosonized effective theory at low energy (or temperature) is identified to a $c=1$ conformal field theory (CFT) with compactified radius determined by the statistics parameter $λ$. Moreover, this CFT can be put into a form of the harmonic fluid description for Lutting…
This paper is a revision of the combinatorics of fractional exclusion statistics (FES). More specifically, the following exact statement of the generalized Pauli principle is derived: for an $N$-particles system exhibiting FES of extended parameter $g=q/r$ ($q$ and $r$ are co-prime integers such that $0 < q \leq r$), we found that the allowed occupation number of a state is smaller than or equal to $r-q+1$ and \emph{not} to $1/g$ whenever $q\neq 1$ and, moreover, the global occupancy shape (merely represented by a partition of $N$) is admissible if the number of states occupied by at least two…
We briefly explain the notion of exclusion statistics and in particular discuss the concept of an ideal exclusion statistics gas. We then review a recent work where it is demonstrated that a {\em two-dimensional} Bose gas with repulsive delta function interactions obeys ideal exclusion statistics, with a fractional parameter related to the interaction strength.
We extend our earlier study about the fractional exclusion statistics to higher dimensions in full physical range and in the non-relativistic and ultra-relativistic limits. Also, two other fractional statistics, namely Gentile and Polychronakos fractional statistics, will be considered and similarities and differences between these statistics will be explored. Thermodynamic geometry suggests that a two dimensional Haldane fractional exclusion gas is more stable than higher dimensional gases. Also, a complete picture of attractive and repulsive statistical interaction of fractional statistics i…
Complex biological and physical transport processes are often described through systems of interacting particles. Excluded-volume effects on these transport processes are well studied, however the interplay between volume exclusion and reactions between heterogenous particles is less well known. In this paper we develop a novel framework for modeling reaction-diffusion processes which directly incorporates volume exclusion. From an off-lattice microscopic individual based model we use the Fokker--Planck equation and the method of matched asymptotic expansions to derive a low-dimensional macros…
In this paper, the particles of quantum gases, that is, bosons and fermions are regarded as g-ons which obey fractional exclusion statistics. With this point of departure the thermostatistical relations concerning the Bose and Fermi systems are unified under the g-on formulation where a fractal approach is adopted. The fractal inspired entropy, the partition function, distribution function, the thermodynamics potential and the total number of g-ons have been found for a grand canonical g-on system. It is shown that from the g-on formulation; by a suitable choice of the parameters of the nonext…
By constructing the super-particle representation of the free boson gas, we propose a new statistics in which the particles are non-exclusive. This statistics can be considered as a generalization of Bose-Einstein's. The possible condensation of this statistical system is studied. It is found that the chemical potential below the condensation temperature is linearly proportional to the temperature rather than a constant. With an proper choice of the exclusion factors $γ_l$, Hadane-Wu's fractional statistics is retrieved in this representation.
We study a continuous-space version of the totally asymmetric simple exclusion process (TASEP), consisting of interacting Brownian particles subject to a driving force in a periodic external potential. Particles are inserted at the leftmost site at rate $α$, hop to the right at unit rate, and are removed at the rightmost site at rate $β$. Our study is motivated by recent experiments on colloidal particles in optical tweezer arrays. The external potential is of the form generated by such an array. Particles spend most of the time near potential minima, approximating the situation in the lattice…
We propose a quasi-particle formulation of effective edge theories for the fractional quantum Hall effect. For the edge of a Laughlin state with filling fraction ν=1/m, our fundamental quasi-particles are edge electrons of charge -e and edge quasi-holes of charge +e/m. These quasi-particles satisfy exclusion statistics in the sense of Haldane. We exploit algebraic properties of edge electrons to derive a kinetic equation for charge transport between a ν=1/m fractional quantum Hall edge and a normal metal. We also analyze alternative `Boltzmann' equations that are directly based on the exclusio…
We determine the exclusion statistics properties of the fundamental edge quasi-particles over a specific $ν=\half$ non-abelian quantum Hall state known as the pfaffian. The fundamental excitations are the edge electrons of charge $-e$ and the edge quasi-holes of charge $+{e \over 4}$. We explicitly determine thermodynamic distribution functions and establish a duality which generalizes the duality for fractional exclusion statistics in the sense of Haldane.
We prove a law of large numbers and a central limit theorem for a tagged particle in a symmetric simple exclusion process in the one-dimensional lattice with variable diffusion coefficient. The scaling limits are obtained from a similar result for the current through -1/2 for a zero-range process with bond disorder. For the CLT, we prove convergence to a fractional Brownian motion of Hurst exponent 1/4.
Here we review a method for constructing exact eigenvalues and eigenfunctions of a many-particle quantum system, which is obtained by adding some nonhermitian but PT invariant (i.e., combined parity and time reversal invariant) interaction to the Calogero model. It is shown that such extended Calogero model leads to a real spectrum obeying generalised exclusion statistics. It is also found that the corresponding exchange statistics parameter differs from the exclusion statistics parameter and exhibits a `reflection symmetry' provided the strength of the PT invariant interaction exceeds a criti…
We show that the TASEP of a driven system of particles of arbitrary size, with nearest neighbor repulsive interaction, on an open lattice is equivalent to the TASEP of interacting monomers on an open lattice whose size fluctuates in response to the entry and exit of particles. We have presented the maximal current profile as a function of the interaction strength for dimers and tetramers, obtained in Monte Carlo simulation; the results agree well with the ones computed by applying a specific rod-to-monomer mapping to the steady state current and density predicted by a mean-field theory of inte…
The thermodynamic potential of ideal gases described by the simplest non-abelian statistics is investigated. I show that the potential is the linear function of the element of the abelian-part statistics matrix. Thus, the factorizable property in the Haldane (abelian) fractional exclusion shown by the author [W. H. Huang, Phys. Rev. Lett. 81, 2392 (1998)] is now extended to the non-abelian case. The complete expansion of the thermodynamic potential is also given.
I discuss ideal and interacting quantum gases obeying general fractional exclusion statistics. For systems with constant density of single-particle states, described in the mean field approximation, the entropy depends neither on the microscopic exclusion statistics, nor on the interaction. Such systems are called {\em thermodynamically equivalent} and I show that the microscopic reason for this equivalence is a one-to-one correspondence between the excited states of these systems. This provides a method, different from the bosonisation technique, to transform between systems of different excl…
We study, using Monte Carlo simulations, the steady state properties of a system of particles interacting via hard core exclusion and moving in a discrete flashing disordered ratchet potential. Quenched disorder is introduced by breaking the periodicity of the ratchet potential through changing shape of the potential across randomly chosen but fixed periods. We show that the effects of quenched disorder can be broadly classified as strong or weak with qualitatively different behaviour of the steady state particle flux as a function of overall particle density. We further show that most of the …
We review the latest advances in the analytical modelling of single file diffusion. We focus first on the derivation of the fractional Langevin equation that describes the motion of a tagged file particle. We then propose an alternative derivation of the very same stochastic equation by starting from the diffusion-noise formalism for the time evolution of the file density.
Universität Innsbruck（因斯布鲁克大学）；CEREMADE, CNRS, Université Paris-Dauphine, Université PSL（法国国家科学研究中心巴黎第九大学PSL大学数学与决策研究实验室）；State Key Laboratory of Quantum Optics Technologies and Devices, Institute of Opto-Electronics, Shanxi University（山西大学光电研究所量子光学技术与器件国家重点实验室）；Universitat Politècnica de Catalunya（加泰罗尼亚理工大学）；School of Physics, Renmin University of China（中国人民大学物理学院）
In this paper we further elaborate on the notion of fractional exclusion statistics, as introduced by Haldane, in two-dimensional conformal field theory, and its connection to the Universal Chiral Partition Function as defined by McCoy and collaborators. We will argue that in general, besides the pseudo-particles introduced recently by Guruswamy and Schoutens, one needs additional `null quasi-particles' to account for the null-states in the quasi-particle Fock space. We illustrate this in several examples of WZW-models.
The exclusion process in which particles may jump any distance l>=1 with the probability that decays as l^-(1+sigma) is studied from coarse-grained equation for density profile in the limit when the lattice spacing goes to zero. For 1<sigma<2, the usual diffusion term of this equation is replaced by the fractional one, which affects dynamical-scaling properties of the late-time approach to the stationary state. When applied to an open system with totally asymmetric hopping, this approach gives two results: first, it accounts for the sigma-dependent exponent that characterizes the algebraic dec…
We propose that the phases of all vicinal surfaces can be characterized by four fixed lines, in the renormalization group sense, in a three-dimensional space of coupling constants. The observed configurations of several Si surfaces are consistent with this picture. One of these fixed lines also describes one-dimensional quantum particles with fractional exclusion statistics. The featureless steps of a vicinal surface can therefore be thought of as a realization of fractional-statistics particles, possibly with additional short-range interactions.
Let a lattice gas of constant density, described by the symmetric simple exclusion process, be brought in contact with a "target": a spherical absorber of radius $R$. Employing the macroscopic fluctuation theory (MFT), we evaluate the probability ${\mathcal P}(T)$ that no gas particle hits the target until a long but finite time $T$. We also find the most likely gas density history conditional on the non-hitting. The results depend on the dimension of space $d$ and on the rescaled parameter $\ell=R/\sqrt{D_0T}$, where $D_0$ is the gas diffusivity. For small $\ell$ and $d>2$, ${\mathcal P}(T)$ …
Closed form, analytical results for the finite-temperature one-body density matrix, and Wigner function of a $d$-dimensional, harmonically trapped gas of particles obeying exclusion statistics are presented. As an application of our general expressions, we consider the intermediate particle statistics arising from the Gentile statistics, and compare its thermodynamic properties to the Haldane fractional exclusion statistics. At low temperatures, the thermodynamic quantities derived from both distributions are shown to be in excellent agreement. As the temperature is increased, the Gentile dist…
We study current fluctuations in lattice gases in the hydrodynamic scaling limit. More precisely, we prove a large deviation principle for the empirical current in the symmetric simple exclusion process with rate functional I. We then estimate the asymptotic probability of a fluctuation of the average current over a large time interval and show that the corresponding rate function can be obtained by solving a variational problem for the functional I. For the symmetric simple exclusion process the minimizer is time independent so that this variational problem can be reduced to a time independen…
Many ribosomes simultaneously move on the same messenger RNA (mRNA), each separately synthesizing the protein coded by the mRNA. Earlier models of ribosome traffic represent each ribosome by a ``self-propelled particle'' and capture the dynamics by an extension of the totally asymmetric simple exclusion process (TASEP). In contrast, here we develope a theoretical model that not only incorporates the mutual exclusions of the interacting ribosomes, but also describes explicitly the mechano-chemistry of each of these individual cyclic machines during protein synthesis. Using analytical and numeri…
We verify the validity of the Cohen-Gallavotti fluctuation theorem for the strongly correlated problem of charge transfer through an impurity in a chiral Luttinger liquid, which is realizable experimentally as a quantum point contact in a fractional quantum Hall edge state device. This is accomplished via the development of an analytical method to calculate the full counting statistics (FCS) of the problem in all the parameter regimes involving the temperature, the Hall voltage, and the gate voltage.
Totally asymmetric simple exclusion processes, consisting of two coupled parallel lattice chains with particles interacting with hard-core exclusion and moving along the channels and between them, are considered. In the limit of strong coupling between the channels, the particle currents, density profiles and a phase diagram are calculated exactly by mapping the system into an effective one-channel totally asymmetric exclusion model. For intermediate couplings, a simple approximate theory, that describes the particle dynamics in vertical clusters of two corresponding parallel sites exactly and…
We study a problem of interacting fractional charges with $J_1$-$J_2$-$J_3$ Ising model on a checkerboard lattice under magnetic field. As a result of the interplay between repulsive interactions and particle density tuning by magnetic field, the fractional charges form a classical spin liquid (CSL) phase. The CSL phase is composed of degenerate spin configurations, which can be mapped to the trimer covering of dual square lattice. The CSL state shows macroscopic ground-state entropy, implying the emergence of novel quantum spin liquid phase when quantum fluctuations are turned on. In addition…
This paper presents the exact expressions of the transition probabilities of some non-determinantal Bethe ansatz solvable interacting particle systems: the two-sided PushASEP, the asymmetric avalanche process and the asymmetric zero range process. The time integrated currents of the asymmetric avalanche process and the asymmetric zero range process are immediate from the results of the asymmtric simple exclusion process.
