Geometric structure of thermal cones

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 analyze 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 behavior based on thermodynamic monotones given by the volumes of thermal cones. Results obtained apply also to other majorization-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 generalize the construction of thermal cones to account for probabilistic transformations.

Automated summary

The second law of thermodynamics introduces an asymmetry in the flow of events, known as the thermodynamic arrow of time. This leads to an ordering of the system's state space into past, future, and incomparable regions. In this study, we analyze the structure of thermal cones, which represent the states that a given state can evolve to (future thermal cone) or evolve from (past thermal cone) thermodynamically. We provide explicit constructions of the past thermal cone and the incomparable region for a classical system interacting with a heat bath, and analyze their behavior based on thermodynamic monotones.