Geometric structure of thermal cones

TL;DR

This paper explores the geometric structure of thermal cones in thermodynamics, revealing how states evolve and relate within the state space, with implications for resource theories and probabilistic transformations.

Contribution

It provides explicit constructions of past thermal cones and incomparable regions for classical systems, extending the understanding of thermodynamic state space geometry.

Findings

Explicit construction of past thermal cones and incomparable regions.

Analysis of thermal cone volumes as thermodynamic monotones.

Generalization to probabilistic state transformations.

Abstract

The second law of thermodynamics imposes a fundamental asymmetry in the flow of events. The so-called thermodynamic arrow of time introduces an ordering that divides the system's state space into past, future and incomparable regions. In this work, we analyse the structure of the resulting thermal cones, i.e., sets of states that a given state can thermodynamically evolve to (the future thermal cone) or evolve from (the past thermal cone). Specifically, for a $d$-dimensional classical state of a system interacting with a heat bath, we find explicit construction of the past thermal cone and the incomparable region. Moreover, we provide a detailed analysis of their behaviour based on thermodynamic monotones given by the volumes of thermal cones. Results obtained apply also to other majorisation-based resource theories (such as that of entanglement and coherence), since the partial ordering describing allowed state transformations is then the opposite of the thermodynamic order in the infinite temperature limit. Finally, we also generalise the construction of thermal cones to account for probabilistic transformations.