Definitions and Evolutions of Statistical Entropy for Hamiltonian Systems

TL;DR

This paper revisits foundational ideas of non-equilibrium statistical entropy, synthesizes them into a formal framework, and clarifies the conditions under which entropy evolves, shedding light on the second law of thermodynamics for Hamiltonian systems.

Contribution

It develops a coherent formalism combining Boltzmann and Gibbs entropy concepts and analyzes entropy evolution, clarifying the origin of the thermodynamic arrow of time.

Findings

Boltzmann entropy obeys a Stochastic H-Theorem.

Gibbs entropy increases monotonically under certain conditions.

Initial Boltzmann state determines the direction of entropy evolution.

Abstract

Regardless of studies and debates over a century, the statistical origin of the second law of thermodynamics still remains illusive. One essential obstacle is the lack of a proper theoretical formalism for non-equilibrium entropy. Here I revisit the seminal ideas about non-equilibrium statistical entropy due to Boltzmann and due to Gibbs, and synthesize them into a coherent and precise framework. Using this framework, I clarify the anthropomorphic principle of entropy, and analyze the evolution of entropy for classical Hamiltonian systems under different experimental setups. I find that evolution of Boltzmann entropy obeys a Stochastic H-Theorem, which relates probability of Boltzmann entropy increasing to that of decreasing. By contrast, the coarse-grained Gibbs entropy is monotonically increasing, if the microscopic dynamics is locally mixing, and the initial state is a Boltzmann state. These results clarify the precise meaning of the second law of thermodynamics for classical systems, and demonstrate that it is the initial condition as a Boltzmann state that is ultimately responsible for the arrow of time.