Foundations of a Finite Non-Equilibrium Statistical Thermodynamics: Extrinsic Quantities

TL;DR

The paper tackles foundational questions in non‑equilibrium statistical thermodynamics for finite systems by reframing the microstate description as subjective uncertainty and by modifying Gibbs entropy to permit entropy production without altering phase space dynamics. It defines extrinsic quantities for the fundamental thermodynamic relation that reproduce classical equilibrium limits, and uses an ideal gas diffusion example to illustrate entropy densities and convergence to equilibrium while exposing finite‑size deviations. The work resolves long‑standing paradoxes through information‑theoretic interpretations and demonstrates how thermodynamic quantities can be distributed over arbitrary subsystems without ensembles. It argues that this framework is both consistent with established thermodynamics in the equilibrium limit and practically applicable to molecular dynamics, while signaling that intrinsic quantities remain an open area for future work.

Abstract

Statistical thermodynamics is valuable as a conceptual structure that shapes our thinking about equilibrium thermodynamic states. A cloud of unresolved questions surrounding the foundations of the theory could lead an impartial observer to conclude that statistical thermodynamics is in a state of crisis though. Indeed, the discussion about the microscopic origins of irreversibility has continued in the scientific community for more than a hundred years. This paper considers these questions while beginning to develop a statistical thermodynamics for finite non-equilibrium systems. Definitions are proposed for all of the extrinsic variables of the fundamental thermodynamic relation that are consistent with existing results in the equilibrium thermodynamic limit. The probability density function on the phase space is interpreted as a subjective uncertainty about the microstate, and the Gibbs entropy formula is modified to allow for entropy creation without introducing additional physics or modifying the phase space dynamics. Resolutions are proposed to the mixing paradox, Gibbs' paradox, Loschmidt's paradox, and Maxwell's demon thought experiment. Finally, the extrinsic variables of the fundamental thermodynamic relation are evaluated as functions of time and space for a diffusing ideal gas, and the initial and final values are shown to coincide with the expected equilibrium values when interpreted in a classical context.

Figures (10)

• Figure 1: A schematic showing the confinement of a microcanonical system to a small part of the configuration space. The microcanonical ensemble (left) stipulates that all microstates of a given total energy (blue, with the potential energy in black) be equally probable. For a physical system (right), the trajectory is confined to the path-connected region containing the initial condition.
• Figure 2: Representations of two partitions of the phase space containing a fixed number of elements, with the probability distribution in blue. The right partition maximizes the coarse-grained entropy.
• Figure 3: The thermodynamic system $A$ is assumed to be isolated, and the closed subsystem $B$ is separated from the surroundings $A - B$ by a heat-conductive barrier.
• Figure 4: The evolution of the marginalized probability distributions $\rho_{x}(x,t)$ (left) and $\rho_{p}(p,t)$ (right) for a single xenon atom that is initially confined to the left half of a one-dimensional box of length $l = 1.2 \times 10^{-6} \, \mathrm{m}$ at $T = 298 \, \mathrm{K}$.
• Figure 5: The evolution of the particle density of the single-particle solution as a function of position. The total number of particles is conserved at all times with a relative error on the order of $10^{-15}$.