Meaning of temperature in different thermostatistical ensembles

Abstract

Depending on the exact experimental conditions, the thermodynamic properties of physical systems can be related to one or more thermostatistical ensembles. Here, we survey the notion of thermodynamic temperature in different statistical ensembles, focusing in particular on subtleties that arise when ensembles become non-equivalent. The 'mother' of all ensembles, the microcanonical ensemble, uses entropy and internal energy (the most fundamental, dynamically conserved quantity) to derive temperature as a secondary thermodynamic variable. Over the past century, some confusion has been caused by the fact that several competing microcanonical entropy definitions are used in the literature, most commonly the volume and surface entropies introduced by Gibbs. It can be proved, however, that only the volume entropy satisfies exactly the traditional form of the laws of thermodynamics for a broad class of physical systems, including all standard classical Hamiltonian systems, regardless of their size. This mathematically rigorous fact implies that negative 'absolute' temperatures and Carnot efficiencies $>1$ are not achievable within a standard thermodynamical framework. As an important offspring of microcanonical thermostatistics, we shall briefly consider the canonical ensemble and comment on the validity of the Boltzmann weight factor. We conclude by addressing open mathematical problems that arise for systems with discrete energy spectrum.

11 pages, 1 figure

References in corpus (9)

• Finite quantum dissipation: the challenge of obtaining specific heat
• Specific heat anomalies of open quantum systems
• Construction of microcanonical entropy on thermodynamic pillars
• Thermodynamics of Peptide Aggregation Processes. An Analysis from Perspectives of Three Statistical Ensembles
• Thermodynamics and Fluctuation Theorems for a Strongly Coupled Open Quantum System: An Exactly Solvable Case
• Finite Bath Fluctuation Theorem
• The Gibbs "volume" entropy is incorrect
• Kinetic energy and microcanonical nonanalyticities in finite and infinite systems
• Thermodynamic laws in isolated systems

Cited by in corpus (20)

• {\it Colloquium:} Statistical Mechanics and Thermodynamics at Strong Coupling: Quantum and Classical
• Statistical Mechanics of Systems with Negative Temperature
• On the dispute between Boltzmann and Gibbs entropy
• Operational definition of the temperature of a quantum state
• Isotropic-Nematic Phase Transitions in Gravitational Systems
• Phase transitions at high energy vindicate negative microcanonical temperature
• Rényi entropy, abundance distribution and the equivalence of ensembles
• Entropy and Temperature in finite isolated quantum systems
• Merits and Qualms of Work Fluctuations in Classical Fluctuation Theorems
• Temperature as a quantum observable
• Non-Clausius heat transfer: The method of the nonstationary Langevin equation
• Deformed Jarzynski Equality
• Strong Coupling and non-Markovian Effects in the Statistical Notion of Temperature
• Entropy non-conservation and boundary conditions for Hamiltonian dynamical systems
• A microcanonical entropy correcting finite-size effects in small systems
• Non-Clausius heat transfer: the example of harmonic chain with an impurity
• Increase of quantum volume entropy in presence of degenerate eigenenergies
• Coarse-graining and symplectic non-squeezing
• Taking the temperature of quantum many-body scars
• Riemannian submersions for q-entropies