We study a one-dimensional gas of $n$ charged particles confined by a potential and interacting through the Riesz potential or a more general potential. In equilibrium, and for symmetric potential the particles arrange themselves symmetrically around the origin within a finite region. Various models will be studied by modifying both the confining potential and the interaction potential. Focusing on the statistical properties of the system, we analyze the position of the rightmost particle, $x_{\text{max}}$, and show that its typical fluctuations are described by a limiting distribution different from the Tracy-Widom distribution found in the one-dimensional log-gas. We also derive the large deviation functions governing the atypical fluctuations of $x_{\text{max}}$ far from its mean.
Based on the calculation of the quasiequilibrium statistical sum by means of the functional integration method, we obtained a nonequilibrium statistical operator for the electron subsystem of a semibounded metal in the framework of the generalized jellium model in the Gaussian and higher approximations with respect to the dynamic electron correlations. This approach allows one to go beyond the linear approximation with respect to the gradient of the electrochemical potential corresponding to weakly nonequilibrium processes and to obtain generalized transport equations that describe nonlinear p…
Viscoelastic description of the electron subsystem of a semi-bounded metal on the basis of the generalized "jellium" model using the method of nonequilibrium statistical Zubarev operator is proposed. The nonequilibrium statistical operator and the quasi-equilibrium partition function calculated by means of the functional integration method are obtained. Transport equations for nonequilibrium mean values of electron density and momentum are received in the Gaussian approximation and in the following higher approximation that corresponds to the third-order cumulant averages in calculation of the…
We consider the jellium model of $N$ particles on a line confined in an external harmonic potential and with a pairwise one-dimensional Coulomb repulsion of strength $α> 0$. Using a Coulomb gas method, we study the statistics of $s = (1/N) \sum_{i=1}^N f(x_i)$ where $f(x)$, in principle, is an arbitrary smooth function. While the mean of $s$ is easy to compute, the variance is nontrivial due to the long-range Coulomb interactions. In this paper we demonstrate that the fluctuations around this mean are Gaussian with a variance ${\rm Var}(s) \approx b/N^3$ for large $N$. We provide an exact comp…
We consider the two-dimensional one-component plasma (jellium) of mobile pointlike particles with the same charge $e$, interacting pairwisely by the logarithmic Coulomb potential and immersed in a fixed neutralizing background charge density. Particles are in thermal equilibrium at the inverse temperature $β$, the only relevant dimensionless parameter is the coupling constant $Γ\equivβe^2$. In the bulk fluid regime and for any value of the coupling constant $Γ=2\times{\rm integer}$, Šamaj and Percus [J. Stat. Phys. {\bf 80}, 811--824 (1995)] have derived an infinite sequence of sum rules for t…
The surface energy, the one-particle distribution function of electrons, etc. of a semi-bounded metal within the framework of the semi-infinite jellium are calculated. The influence of the potential barrier height on these characteristics is studied. The barrier height is found from the condition of the minimum of the surface energy. The surface energy is positive in the entire domain of the Wigner-Seitz radius of metals, and it is in sufficiently good agreement with experimental data.
We find the first three (even) structure factor moments for a (non-quantum) one-component Jellium made of particles living in three dimensions and interacting with a Coulomb pair-potential plus a short-range term with either a finite range or decaying exponentially fast at large distances. Starting from the hierarchical Born-Green-Yvon equations we show that they are all form invariant respect to the addition of the short-range term. We discuss the relevance of the present study to interpret the failure of the moment sum-rules of ionic-liquids at criticality.
We adopt the fixed node restricted path integral Monte Carlo method within the "Worm algorithm" to simulate Wigner's Jellium model at finite, non zero, temperatures using free-particle nodes of the density matrix. The new element is that we incorporate the Worm algorithm paradigm of Prokof'ev and Svistunov in the grand canonical ensemble in order to more efficiently handle the fermionic exchanges. We present results for the structure and thermodynamic properties of the ideal Fermi gas and three points for the interacting electron gas. We treat explicitly the case of the partially polarized ele…
The model under consideration is the classical two-dimensional one-component plasma (jellium) of pointlike particles with charge $e$, interacting pairwisely via the logarithmic Coulomb potential and immersed in a uniform neutralizing background charge density. The system is in thermal equilibrium at the inverse temperature $β$, its thermodynamics depends only on the coupling constant $Γ=βe^2$. We put into an infinite (homogeneous and translationally invariant) plasma a guest particle of charge $Ze$ with $Z>-2/Γ$ in order to prevent from the collapse of the jellium charges onto it. The guest pa…
In the equilibrium statistical mechanics of classical Coulomb fluids, the long-range tail of the Coulomb potential gives rise to the Stillinger-Lovett sum rules for the charge correlation functions. For the jellium model of mobile particles of charge $q$ immersed in a neutralizing background, the fixing of one of the $q$-charges induces a screening cloud of the charge density whose zeroth and second moments are determined just by the Stillinger-Lovett sum rules. In this paper, we generalize these sum rules to the screening cloud induced around a pointlike guest charge $Z q$ immersed in the bul…
In this paper we define and study the classical Uniform Electron Gas (UEG), a system of infinitely many electrons whose density is constant everywhere in space. The UEG is defined differently from Jellium, which has a positive constant background but no constraint on the density. We prove that the UEG arises in Density Functional Theory in the limit of a slowly varying density, minimizing the indirect Coulomb energy. We also construct the quantum UEG and compare it to the classical UEG at low density.
Formation energies of charged point defects in semiconductors are calculated using periodic supercells, which entail a divergence arising from long-range Coulombic interactions. The divergence is typically removed by the so-called jellium approach. Recently, Wu, Zhang and Pantelides [WZP, Phys. Rev. Lett. 119, 105501 (2017)] traced the origin of the divergence to the assumption that charged defects are formed by physically removing electrons from or adding electrons to the crystal, violating charge neutrality, a key principle of statistical mechanics that determines the Fermi level. An alterna…
We study, through the diffusion Monte Carlo method, a spin one-half fermion fluid, in the three dimensional Euclidean space, at zero temperature. The point particles, immersed in a uniform "neutralizing" background, interact with a pair-potential which can be continuously changed from zero to the Coulomb potential depending on a parameter $μ$. We determine the radial distribution functions of the system for various values of density, $μ$, and polarization. We discuss about the importance, in a computer experiment, of the choice of suitable estimators to measure a physical quantity. The radial …
Quasi one-dimensional systems are systems of particles in domains which are of infinite extent in one direction and of uniformly bounded size in all other directions, e.g. on a cylinder of infinite length. The main result proven here is that for such particle systems with Coulomb interactions and neutralizing background, the so-called "jellium", at any temperature and at any finite-strip width there is translation symmetry breaking. This extends the previous result on Laughlin states in thin, two-dimensional strips by Jansen, Lieb and Seiler (2009). The structural argument which is used here b…
The model considered here is the `jellium' model in which there is a uniform, fixed background with charge density $-eρ$ in a large volume $V$ and in which $N=ρV$ particles of electric charge $+e$ and mass $m$ move --- the whole system being neutral. In 1961 Foldy used Bogolubov's 1947 method to investigate the ground state energy of this system for bosonic particles in the large $ρ$ limit. He found that the energy per particle is $-0.402 r_s^{-3/4} {me^4}/{\hbar^2}$ in this limit, where $r_s=(3/4πρ)^{1/3}e^2m/\hbar^2$. Here we prove that this formula is correct, thereby validating, for the fi…
We consider a classical system of n charged particles in an external confining potential, in any dimension d larger than 2. The particles interact via pairwise repulsive Coulomb forces and the coupling parameter scales like the inverse of n (mean-field scaling). By a suitable splitting of the Hamiltonian, we extract the next to leading order term in the ground state energy, beyond the mean-field limit. We show that this next order term, which characterizes the fluctuations of the system, is governed by a new "renormalized energy" functional providing a way to compute the total Coulomb energy o…
For inhomogeneous classical Coulomb fluids in thermal equilibrium, like the jellium or the two-component Coulomb gas, there exists a variety of exact sum rules which relate the particle one-body and two-body densities. The necessary condition for these sum rules is that the Coulomb fluid possesses good screening properties, i.e. the particle correlation functions or the averaged charge inhomogeneity, say close to a wall, exhibit a short-range (usually exponential) decay. In this work, we study equilibrium statistical mechanics of an electric double layer with counter-ions only, i.e. a globally…
We propose a unified framework to study the turbulent transport problem from the perspective of nonequilibrium statistical mechanics. By combining Krarichnan's turbulence thermalization assumption and Ruelle's recent work on nonequilibrium statistical mechanics settings for fluids, we show that the equation for viscous fluid can be viewed as the non-canonical Hamiltonian system perturbed by different thermostats. This allows an analogy between the viscous fluid and the nonequilibrium heat conduction model where the Fourier modes can be regarded as the ''particles''. With this framework, we ref…
A time-dependent statistical description of multiple particle breakage is presented. The approach combines the Tsallis non-extensive entropy with a fractal kinetic equation for the time variation of the number of fragments. The obtained fragment size distribution function is tested by fitting some experimental reports.
The exact equations of motion for microscopic density of classical particles number with account of inter-particle interactions and external field in closed form are derived. An integral equation for equilibrium distributions of the particles is deduced. No statistical or probabilistic hypotheses and assumption in these deductions have been used. Some well known results of equilibrium statistical mechanics are deduced from the obtained equations as simple limiting cases. The wave equation for almost homogeneous systems with inter-particle interactions are obtained. Connection between inter-par…
The s=1/2 Ising chain with uniform nearest-neighbor and next-nearest-neighbor coupling is used to construct a system of floating particles characterized by motifs of up to six consecutive local spins. The spin couplings cause the assembly of particles which, in turn, remain free of interaction energies even at high density. All microstates are configurations of particles from one of three different sets, excited from pseudo-vacua associated with ground states of periodicities one, two, and four. The motifs of particles and elements of pseudo-vacuum interlink in two shared site variables. The s…
Within the theory of statistical ensemble, the so-called $μPT$ ensemble describes equilibrium systems that exchange energy, particles, and volume with the surrounding. General, model-independent features of volume and particle number statistics are derived. Non-analytic points of the partition function are discussed in connection with divergent fluctuations and ensemble equivalence. Quantum and classical ideal gases, and a model of Bose gas with mean-field interactions are discussed as examples of the above considerations.
We introduce a fundamental theory for the kinetics of systems of classical particles. The theory represents a unification of kinetic theory, Brownian motion and field theory. It is self-consistent and is the dynamic generalization of the functional theory of static equilibrium fluids. This gives one a powerful tool for investigating the existence of ergodic-nonergodic transitions near the liquid-glass transition.
The equilibrium statistical mechanics of one-dimensional lattice gases with interactions of arbitrary range and shape between first-neighbor atoms is solved exactly on the basis of statistically interacting vacancy particles. Two sets of vacancy particles are considered. In one set all vacancies are of one-cell size. In the other set the sizes of vacancy particles match the separation between atoms. Explicit expressions are obtained for the Gibbs free energy and the distribution of spaces between atoms at thermal equilibrium. Applications to various types of interaction potentials are discusse…
We present predictions for the statistical error due to finite sampling in the presence of thermal fluctuations in molecular simulation algorithms. Specifically, we establish how these errors depend on Mach number, Knudsen number, number of particles, etc. Expressions for the common hydrodynamic variables of interest such as flow velocity, temperature, density, pressure, shear stress and heat flux are derived using equilibrium statistical mechanics. Both volume-averaged and surface-averaged quantities are considered. Comparisons between theory and computations using direct simulation Monte Car…
Appealing to the 1902 Gibbs' formalism for classical statistical mechanics (SM), the first SM axiomatic theory ever that successfully explained equilibrium thermodynamics, we will here show that already at the classical level there is a strong correlation between the Renyi's exponent $α$ and the number of particles for very simple systems. No reference to heat baths is needed for such a purpose.
Consistent statistical physical description is given for systems where the elementary excitations are composite objects. Explicit calculational scheme is constructed for the energy density and the total number of thermodynamical degrees of freedom, based on the spectral function of the system. One demonstrates through characteristic examples that single quasiparticle contributions combine non-linearly in these quantities. Relation to the Gibbs paradox is also discussed.
For open quantum systems, the Gaussian environmental dissipative effect can be represented by statistical quasi-particles, namely, dissipatons. We exploit this fact to establish the dissipaton thermofield theory. The resulting generalized Langevin dynamics of absorptive and emissive thermofield operators are effectively noise-resolved. The system-bath entanglement theorem is then readily followed between an important class of nonequilibrium steady-state correlation functions. All these relations are validated numerically. A simple corollary is the transport current expression, which exactly re…
We use the non-equilibrium statistical field theory for classical particles recently developed by Mazenko and Das and Mazenko, together with the free generating functional for particles initially correlated in phase space derived in Bartelmann et al. to study the impact of initial correlations on the equation of state of real gases. We first show that we can reproduce the well known van der Waals equation of state for uncorrelated initial conditions using this approach. We then impose correlated initial conditions and study their qualitative and quantitative effect on the equation of state of …
We argue that a particle language provides a conceptually simple framework for the description of anomalous equilibration in isolated quantum systems. We address this paradigm in the context of integrable models, which are those with particles that are stable against decay. In particular, we demonstrate that a complete description of equilibrium ensembles for interacting integrable models requires a formulation built from the mode occupation numbers of the underlying particle content, mirroring the case of non-interacting particles. This yields an intuitive physical interpretation of generaliz…
A classical particle system coupled with a thermostat driven by an external constant force reaches its steady state when the ensemble-averaged drift velocity does not vary with time. The statistical mechanics of such a system is derived merely based on the equal probability and ergodicity principles, free from any conclusions drawn on equilibrium statistical mechanics or local equilibrium hypothesis. The momentum space distribution is determined by a random walk argument, and the position space distribution is determined by employing the equal probability and ergodicity principles. The express…
Quantum statistical mechanics is formulated as an integral over classical phase space. Some details of the commutation function for averages are discussed, as is the factorization of the symmetrization function used for the grand potential and for the multi-particle density. Three binary choices (eight routes) for the average energy are shown to be mutually consistent. An expression for the phase space function that gives the average virial pressure is derived.
It was recently shown by Bartelmann et al. how correlated initial conditions can be introduced into the statistical field theory for classical particles pioneered by Das and Mazenko. In this paper we extend this development from the canonical to the grand canonical ensemble for a system satisfying statistical homogeneity and isotropy. We do this by translating the probability distribution for the initial phase space coordinates of the particles into an easy diagrammatic representation and then using a variant of the Mayer cluster expansion to sum over particle numbers. The grand canonical gene…
We present a formalism for obtaining the statistical properties of functionals and inverse functionals of the paths of a particle diffusing in a one-dimensional quenched random potential. We demonstrate the implementation of the formalism in two specific examples: (1) where the functional corresponds to the local time spent by the particle around the origin and (2) where the functional corresponds to the occupation time spent by the particle on the positive side of the origin, within an observation time window of size $t$. We compute the disorder average distributions of the local time, the in…
As a rough model for the collective motions of cells and organisms we develop here the statistical mechanics of swarms of self-propelled particles. Our approach is closely related to the recently developed theory of active Brownian motion and the theory of canonical-dissipative systems. Free motion and motion of a swarms confined in an external field is studied. Briefly the case of particles confined on a ring and interacting by repulsive forces is studied. In more detail we investigate self-confinement by Morse-type attracting forces. We begin with pairs N = 2; the attractors and distribution…
A chain of kinetic equations for non-equilibrium one-particle, two-particle and $ s $-particle distribution functions of particles which take into account nonlinear hydrodynamic fluctuations is proposed. The method of Zubarev non-equilibrium statistical operator with projection is used. Nonlinear hydrodynamic fluctuations are described with non-equilibrium distribution function of collective variables that satisfies generalized Fokker-Planck equation. On the basis of the method of collective variables, a scheme of calculation of non-equilibrium structural distribution function of collective va…
A statistical approach to a self-consistent description of kinetic and hydrodynamic processes in systems of interacting particles is formulated on the basis of the nonequilibrium statistical operator method by D.N.Zubarev. It is shown how to obtain the kinetic equation of the revised Enskog theory for a hard sphere model, the kinetic equations for multistep potentials of interaction and the Enskog-Landau kinetic equation for a system of charged hard spheres. The BBGKY hierarchy is analyzed on the basis of modified group expansions. Generalized transport equations are obtained in view of a self…
A fundamental pillar of quantum mechanics concerns indistinguishable quantum particles. In three dimensions they may be classified into fermions or bosons, having, respectively, antisymmetric or symmetric wave functions under particle exchange. One of numerous manifestations of this quantum statistics is the tendency of fermions (bosons) to anti-bunch (bunch). In a two-particle scattering experiment with two possible outgoing channels, the probability of the two particles to arrive each at a different terminal is enhanced (with respect to classical particles) for fermions, and reduced for boso…
We discuss recent results of the replica approach to statistical mechanics of a single classical particle placed in a random N(>>1)-dimensional Gaussian landscape. The particular attention is paid to the case of landscapes with logarithmically growing correlations and to its recent generalisations. Those landscapes give rise to a rich multifractal spatial structure of the associated Boltzmann-Gibbs measure. We also briefly mention related results on counting stationary points of random Gaussian surfaces, as well as ongoing research on statistical mechanics in a random landscape constructed loc…
We propose a method to study the transition to chaos in isolated quantum systems of interacting particles. It is based on the concept of delocalization of eigenstates in the energy shell, controlled by the Gaussian form of the strength function. We show that although the fluctuations of energy levels in integrable and non-integrable systems are principally different, global properties of the eigenstates may be quite similar, provided the interaction between particles exceeds some critical value. In this case the quench dynamics can be described analytically, demonstrating the universal statist…
We show that extremely simple systems of a not too large number of particles can be simultane- ously thermally stable and complex. To such an end, we extend the statistical complexity's notion to simple configurations of non-interacting particles, without appeal to probabilities, and discuss configurational properties.
We propose a new look at the heat bath for two Brownian particles, in which the heat bath as a `system' is both perturbed and sensed by the Brownian particles. Non-local thermal fluctuation give rise to bath-mediated static forces between the particles. Based on the general sum-rule of the linear response theory, we derive an explicit relation linking these forces to the friction kernel describing the particles' dynamics. The relation is analytically confirmed in the case of two solvable models and could be experimentally challenged. Our results point out that the inclusion of the environment …
Recently Mazenko and Das and Mazenko introduced a non-equilibrium field theoretical approach to describe the statistical properties of a classical particle ensemble starting from the microscopic equations of motion of each individual particle. We use this theory to investigate the transition from those microscopic degrees of freedom to the evolution equations of the macroscopic observables of the ensemble. For the free theory, we recover the continuity and Jeans equations of a collisionless gas. For a theory containing two-particle interactions in a canonical perturbation series, we find the m…
We study the transformation of the statistical mechanics of N particles to the statistical mechanics of fields, that are the collective coordinates, describing the system. We give an explicit expression for the functional Fourier transform of the Jacobian of the transformation from particle to collective coordinate and derive the Fokker-Planck equation in terms of the collective coordinates. Simple approximations, leading to Debye-Huckel theory and to the hard sphere Percus-Yevick equation are discussed.
Chern-Simons gauge field theory has provided a natural framework to gain deep insight about many novel phenomena in two-dimensional condensed matter. We investigate the nonequilibrium thermodynamics properties of a (two-dimensional) dissipative harmonic particle when the Abelian topological gauge action and the (linear) Brownian motion dynamics are treated on an equal footing. We find out that the particle exhibits remarkable magneticlike features in the quantum domain that are beyond the celebrated Landau diamagnetism: this could be viewed as the non-relativistic Brownian counterpart of the c…
Basing on main principles of statistical mechanics only, an exact virial expansion for path probability distribution of molecular Brownian particle in a fluid is derived which connects response of the distribution to perturbations of the fluid and statistical correlations of its molecules with Brownian particle. The expansion implies that (i) spatial spread of these correlations is finite, (ii) this is inconsistent with Gaussian distribution involved by the ``molecular chaos'' hypothesis, and (iii) real path distribution possesses power-law long tails. This means that actual Brownian path neve…
The Lorentz covariant classical and quantum statistical mechanics and thermodynamics of an ideal relativistic gas of bradyons (particles slower than light), luxons (particles moving with the speed of light) and tachyons (hypothetical particles faster than light) is discussed. The Lorentz covariant formulation is based on the preferred frame approach which among others enables consistent, free of paradoxes description of tachyons. The thermodynamic functions within the covariant approach are obtained both in classical and quantum case.
We model a quantum walk of identical particles that can change their exchange statistics by hopping across a domain wall in a 1D lattice. Such a "statistical boundary" is transparent to single particles and affects the dynamics only by swapping multiple particles arriving together. We find that the two-particle interference is dramatically altered by reflections of these bunched waves at the interface, producing strong measurable asymmetries. Depending on the phases on the two sides, a bunched wavepacket can get completely reflected or split into a superposition of a reflected wave and an anti…
A new class of identical particles which may exhibit both Bose and Fermi statistics with respective probabilities $p_b$ and $p_f$ is introduced. Such an uncertainity may be either an intrinsic property of a particle or can be viewed as an ``experimental uncertainity''. Statistical equivalence of such particles and particles obeying parastatistics of infinite order is shown. Generalized statistical distributions are derived and statistical and thermodynamical properties of an ideal gas of the particles are investigated. The physical nature of such particles and the implications of this investig…
