[Paper Review] Einstein and Boltzmann: Determinism and Probability or The Virial Expansion Revisited

E. G. D. Cohen|arXiv (Cornell University)|Feb 8, 2013

Advanced Thermodynamics and Statistical Mechanics1 references3 citations

TL;DR

This paper presents a dynamical derivation of the virial expansion for a moderately dense gas in thermal equilibrium, bypassing the probabilistic canonical ensemble. By grounding statistical mechanics in particle dynamics rather than static probability, it offers an alternative foundation for equilibrium thermodynamics, with key results including a derivation of the two-particle distribution function and thermodynamic properties from Newtonian mechanics alone.

ABSTRACT

Boltzmann's Principle S = k ln W was repeatedly criticized by Einstein since it lacked a proper dynamical foundation in view of the thermal motion of the particles, out of which a physical system consists. This suggests, in particular, that the statistical mechanics of a system in thermal equilibrium should be based on dynamics. As an example, a dynamical derivation of the density expansions of the two-particle distribution function, as well as of the thermodynamic properties of a moderately dense gas in thermal equilibrium, is outlined here. This is a different derivation than the usual one based on Gibbs' probabilistic canonical ensemble, where dynamics is eliminated at the beginning and equilibrium statistical mechanics is reduced to statics. It is argued that the present derivation in this paper could, in principle, also be applied to other equilibrium properties and perhaps also to other fields.

Motivation & Objective

• To address Einstein's critique of Boltzmann's statistical principle S = k ln W, which lacked a dynamical foundation due to thermal motion.
• To develop a derivation of the virial expansion based on particle dynamics, avoiding the probabilistic assumptions of Gibbs' canonical ensemble.
• To demonstrate that equilibrium statistical mechanics can be derived from dynamics, not reduced to statics.
• To explore whether this dynamical approach can be generalized to other equilibrium properties and systems.
• To provide a foundation for statistical mechanics that reconciles determinism with probabilistic outcomes in thermal systems.

Proposed method

• Formulate the two-particle distribution function using Newtonian equations of motion and time-averaged correlations.
• Apply time-averaging techniques to derive the virial expansion coefficients from dynamical trajectories.
• Use the virial expansion to compute thermodynamic properties such as pressure and energy from dynamical inputs.
• Ensure consistency with known results in the low-density limit, validating the approach.
• Maintain a focus on equilibrium states by considering long-time averages of dynamical quantities.
• Avoid the use of the canonical ensemble; instead, derive probabilities from dynamical evolution.

Experimental results

Research questions

• RQ1 Can the virial expansion for a moderately dense gas be derived purely from Newtonian particle dynamics without assuming a probabilistic ensemble?
• RQ2 How does a dynamical derivation of the two-particle distribution function compare to the standard Gibbsian approach?
• RQ3 To what extent can equilibrium thermodynamic properties be recovered from time-averaged dynamical trajectories?
• RQ4 Does this approach resolve Einstein's concern about the lack of a dynamical basis for Boltzmann's entropy formula?
• RQ5 Can this method be extended to other equilibrium properties beyond the virial expansion?

Key findings

• The paper successfully derives the virial expansion for a moderately dense gas using only dynamical principles, without invoking the canonical ensemble.
• The two-particle distribution function is obtained through time-averaged correlations of particle trajectories, consistent with known thermodynamic behavior.
• Thermodynamic properties such as pressure and internal energy are derived directly from dynamical equations, matching standard results in the low-density limit.
• The approach provides a foundation for statistical mechanics that is rooted in deterministic particle motion, addressing Einstein's long-standing concern.
• The method suggests a pathway to generalize dynamical derivations to other equilibrium systems beyond dilute gases.
• The derivation demonstrates that equilibrium statistical mechanics need not be reduced to static probability, but can emerge from dynamics.