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

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.

A generalization of the Ehrenfest urn model

American Journal of Physics, 1989

This article presents a generalization of the Ehrenfest urn model in order to obtain the time evolution of the number of molecules n(t) in any subvolume V of a vessel V0 containing N molecules of a gas. The formalism makes use of the equilibrium (stationary) probability distribution PN(n,V,V0)≡P0(n) characterizing the gas. Two cases of pedagogical value, the ideal gas (binomial distribution) and the lattice gas (hypergeometric distribution) are considered.

Equilibrium in Boltzmannian Statistical Mechanics

2021

A gas that is confined to the left half of a container starts expanding as soon as the confining wall is removed and eventually spreads evenly over the entire available space. The state of being spread evenly is the equilibrium state and the process of expansion culminating in that state is the approach to equilibrium. It is one of the aims of statistical mechanics (SM) to give an exact characterization of equilibrium, and to explain why and how systems approach the state of equilibrium in terms of the dynamical laws that govern the individual molecules of which the gas is made up of. What is it about molecules and their motions that lead them to spread when the wall is removed? An important answer to these questions was suggested by Boltzmann (1909 [1877]), and variants of this answer are currently regarded by many as a promising option. As is customary in the current literature on SM, we refer to the approach that originates in Boltzmann’s 1877 paper as Boltzmannian SM (BSM). In t...

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.

Boltzmann’s Approach to Statistical Mechanics

Lecture Notes in Physics, 2001

In the last quarter of the nineteenth century, Ludwig Boltzmann explained how irreversible macroscopic laws, in particular the second law of thermodynamics, originate in the time-reversible laws of microscopic physics. Boltzmann's analysis, the essence of which I shall review here, is basically correct. The most famous criticisms of Boltzmann's later work on the subject have little merit. Most twentieth century innovations -such as the identification of the state of a physical system with a probability distribution on its phase space, of its thermodynamic entropy with the Gibbs entropy of , and the invocation of the notions of ergodicity and mixing for the justification of the foundations of statistical mechanics -are thoroughly misguided.

Statistical mechanics of colloids and Boltzmann’s definition of the entropy

American Journal of Physics, 2006

The Boltzmann entropy as traditionally presented in statistical mechanics textbooks is only a special case and not Boltzmann's fundamental definition. The difference becomes important when the traditional expression for the entropy is applied to colloids, for which it makes incorrect predictions. Boltzmann's original definition of the entropy in terms of the probabilities of states of composite systems leads to consistent and correct statistical mechanics and thermodynamics.

Ergodicity in hard-ball systems and Boltzmann’s entropy

Physical Review E, 1996

The ergodic property of the hard-ball systems is investigated numerically by comparing the microcanonical single-particle distributions and the corresponding numerical results in two-and three-dimensional cases. A Boltzmann-entropy-like quantity H(n c ) is defined and its evolution is discussed. The deviation of this entropy from the equilibrium value obeys a power decay law in the long time limit, and the origin of this tail is explored heuristically.

On the Statistics of Ideal Gas Particles

The usual exponential decay (with e, the Base of Natural Logarithm) of particle density with respect to energy levels proposed by Boltzmann does not hold up to scrutiny!. In fact the base of the exponential function depends on the ratio of Number of Particles to Number of energy levels.