{\it Colloquium:} Statistical Mechanics and Thermodynamics at Strong Coupling: Quantum and Classical

TL;DR

This paper reviews the statistical mechanics and thermodynamics of small systems strongly coupled to environments, emphasizing the Hamiltonian of mean force and discussing ambiguities in defining heat and internal energy.

Contribution

It provides a comprehensive review of the thermodynamics of strongly coupled open systems using the Hamiltonian of mean force, highlighting conceptual ambiguities.

Findings

Hamiltonian of mean force characterizes reduced states and thermodynamics.

Thermodynamic potentials are differences between total and environment potentials.

Ambiguities in defining heat and internal energy in strong coupling regimes.

Abstract

The statistical mechanical description of small systems staying in thermal equilibrium with an environment can be achieved by means of the Hamiltonian of mean force. In contrast to the reduced density matrix of an open quantum system, or the reduced phase-space probability density function of a classical open system, the Hamiltonian of mean force not only characterizes the reduced state but also contains full information about the thermodynamics of the considered open system. The resulting thermodynamic potentials all assume the form as the difference of the potentials for the total system and the bare environment in the absence of the system. In contrast to work as a mechanical notion, one faces several problems with the definition of heat which turns out to be largely ambiguous in the case of strong coupling between system and environment. The general theory of the thermodynamics of open systems, in particular, in view of strong coupling, is reviewed and illustrated with several examples. The vagueness of heat is discussed in the context of the ambiguities in the definitions of a fluctuating internal energy and other fluctuating thermodynamic potentials.