A generalization of the Ehrenfest urn model (original) (raw)

A probabilistic model for the equilibration of an ideal gas

Physica A: Statistical Mechanics and its Applications, 2004

I present a generalization of the Ehrenfest urn model that is aimed at simulating the approach to equilibrium in a dilute gas. The present model differs from the original one in two respects: 1) the two boxes have different volumes and are divided into identical cells with either multiple or single occupancy; 2) particles, which carry also a velocity vector, are subjected to random, but elastic, collisions, both mutual and against the container walls. I show, both analytically and numerically, that the number and energy of particles in a given urn evolve eventually to an equilibrium probability density W which, depending on cell occupancy, is binomial or hypergeometric in the particle number and beta-like in the energy. Moreover, the Boltzmann entropy ln W takes precisely the same form as the thermodynamic entropy of an ideal gas. This exercise can be useful for pedagogical purposes in that it provides, although in an extremely simplified case, a probabilistic justification for the maximum-entropy principle.

The ideal gas as an urn model: derivation of the entropy formula

European Journal of Physics, 2005

The approach of an ideal gas to equilibrium is simulated through a generalization of the Ehrenfest ball-and-box model. In the present model, the interior of each box is discretized, {\it i.e.}, balls/particles live in cells whose occupation can be either multiple or single. Moreover, particles occasionally undergo random, but elastic, collisions between each other and against the container walls. I show, both analitically and numerically, that the number and energy of particles in a given box eventually evolve to an equilibrium distribution WWW which, depending on cell occupations, is binomial or hypergeometric in the particle number and beta-like in the energy. Furthermore, the long-run probability density of particle velocities is Maxwellian, whereas the Boltzmann entropy lnW\ln WlnW exactly reproduces the ideal-gas entropy. Besides its own interest, this exercise is also relevant for pedagogical purposes since it provides, although in a simple case, an explicit probabilistic foundation for the ergodic hypothesis and for the maximum-entropy principle of thermodynamics. For this reason, its discussion can profitably be included in a graduate course on statistical mechanics.

A Purely Probabilistic Representation for the Dynamics of a Gas of Particles

Foundations of Physics, 2000

The aim of the present paper is to give a purely probabilistic account for the approach to equilibrium of classical and quantum gas. The probability function used is classical. The probabilistic dynamics describes the evolution of the state of the gas due to unary and binary collisions. A state change amounts to a destruction in a state and the creation in another state. Transitions probabilities are splittled into destructions terms, denoting the random choice of the colliding particle(s), and creation terms, describing the allocation of the same particle(s) on the final state(s). While the destruction term is the same for all types of particles, the creation one depends upon a parameter bound to the interpraticle correlation. The transition probabilities give rise to a homogeneous Markov chain. The equilibrium distributions satisfy the principle of detailed balance. Relaxation times depend upon the interparticle correlation. Relationships with the Ehrenfest urn model, Brillouin unified method, ensemble interpretation, and quantum H-theorem are considered too.

Generalization of the Ehrenfest urn model to a complex network

Physical Review E, 2015

The Ehrenfest urn model is extended to a complex directed network, over which a conserved quantity is transported in a random fashion. The evolution of the conserved number of packets in each urn, or node of the network, is illustrated by means of a stochastic simulation. Using mean-field theory we were able to compute an approximation to the ensemble-average evolution of the number of packets in each node which, in the thermodynamic limit, agrees quite well with the results of the stochastic simulation. Using this analytic approximation we are able to find the asymptotic dynamical state of the system and the time scale to approach the equilibrium state, for different networks. The study is extended to large scale-free and small-world networks, in which the relevance of the connectivity distribution and the topology of the network for the distribution of time scales of the system is apparent. This analysis may contribute to the understanding of the transport properties in real networks subject to a perturbation, e.g., the asymptotic state and the time scale required to approach it.

Gas Model

2016

We propose a model for a two dimensional, associative water-like lattice gas with one single variable representing both long and short-range interactions. The corresponding hamiltonian was solved exactly, by state enumeration in a finite lattice, so to obtain an analytic expression for the partition function. The lattice dimensions were chosen based on geometric characteristics of the stable phases found in previous works using Monte Carlo simulations. An expression for the particle density in the finite lattice was then obtained, and coexistence curves with a region of anomaly in the density presented in a phase diagram. In the end, a phenomenological theory for the system density is proposed and compared to the previous results.

Condensation and equilibration in an urn model

Chaos, Solitons & Fractals, 2015

After reviewing the general scaling properties of aging systems, we present a numerical study of the slow evolution induced in the zeta urn model by a quench from a high temperature to a lower one where a condensed equilibrium phase exists. By considering both one-time and two-time quantities we show that the features of the model fit into the general framework of aging systems. In particular, its behavior can be interpreted in terms of the simultaneous existence of equilibrated and aging degrees with different scaling properties.

Statistical mechanics of granular gases in compartmentalized systems

PHYSICAL REVIEW E, 2003

We study the behavior of an assembly of N granular particles contained in two compartments within a simple kinetic approach. The particles belonging to each compartment collide inelastically with each other and are driven by a stochastic heat bath. In addition, the fastest particles can change compartment at a rate which depends on their kinetic energy. Via a Boltzmann velocity distribution approach we first study the dynamics of the model in terms of a coupled set of equations for the populations in the containers and their granular temperatures and find a crossover from a symmetric high-temperature phase to an asymmetric low temperature phase. Finally, in order to include statistical fluctuations, we solve the model within the Direct Simulation Monte Carlo approach. Comparison with previous studies are presented.

Ehrenfest urn revisited: Playing the game on a realistic fluid model

Physical Review E, 2007

A fundamental question in statistical mechanics is the reconciliation of the irreversibility of thermodynamics with the reversibility of the microscopic equations of motion governed by classical mechanics. In 1872 Boltzmann gave an answer with his H theorem 1, describing the increase in the entropy of an ideal gas as an irreversible process. However, the proof of this theorem contained the Stoßzahlansatz—ie, the assumption of molecular chaos. The result was subject to two main objections: Loschmidt's Umkehreinwand reversibility ...

On the ellipsoidal statistical model for polyatomic gases

Continuum Mechanics and Thermodynamics, 2009

The aim of this article is to construct a BGK-type model for polyatomic gases which gives in the hydrodynamic limit the proper transport coefficient. Its construction relies upon a systematic procedure: minimizing Boltzmann entropy under suitable moments constraints ([20, 9]). The obtained model corresponds to the ellipsoidal statistical model introduced in [2]. We also study the return to equilibrium of its solutions in the homogeneous case.

Distributions in the Ehrenfest process

Statistics & Probability Letters, 2006

The quest continues for cases of interest where the differential equations for the Po´lya process are amenable to an asymptotic solution. We introduce a tenable class of urns that generalize the classical Ehrenfest model, and analyze the Ehrenfest process obtained by embedding the discrete evolution in real time. We show that lurking under the Ehrenfest process is a limiting binomial distribution, whose number of trials is an integer invariant property of the process. Crown