Hamiltonian curl-forces in systems coupled to multiple thermal reservoirs
Abstract
We investigate the statistical mechanics of a system coupled to multiple heat reservoirs at different temperatures, a setting which typically sustains nontrivial circulation. By applying a canonical transformation, we map the system onto that of a particle coupled to a single reservoir yet driven by a deterministic curl-force---a reversible, velocity-independent non-conservative force with non-vanishing curl---that can be described by a Hamiltonian with an anisotropic kinetic energy term. Despite the Hamiltonian structure, the system does not relax to the corresponding Boltzmann distribution, and is genuinely out of equilibrium. The mapping provides a mechanical interpretation of the currents in multi-temperature systems, as we demonstrate by means of two examples. In particular, the direction of circulation is qualitatively predictable from the deterministic force field. For the exactly solvable case of an N-dimensional underdamped Ornstein-Uhlenbeck process, we derive explicit expressions for the steady-state distribution and the entropy production rate, highlighting the role of the (generalized) curl of the force field in driving the system out of thermal equilibrium. Finally, we show how a non-conservative force can be used to exactly equilibrate any system coupled to multiple reservoirs, and obtain the corresponding equilibrium distribution.
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BibTeX
@misc{chor2026hamiltonian,
title = {{Hamiltonian curl-forces in systems coupled to multiple thermal reservoirs}},
author = {Omer Chor and Aljaž Godec and Oren Raz},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.08423},
eprint = {2610.08423},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.08423},
}
RIS
TY - UNPB
TI - Hamiltonian curl-forces in systems coupled to multiple thermal reservoirs
AU - Omer Chor
AU - Aljaž Godec
AU - Oren Raz
PY - 2026
DA - 2026/10/06
PB - arXiv
DO - 10.48550/arxiv.2610.08423
UR - https://arxiv.org/abs/2610.08423
AB - We investigate the statistical mechanics of a system coupled to multiple heat reservoirs at different temperatures, a setting which typically sustains nontrivial circulation. By applying a canonical transformation, we map the system onto that of a particle coupled to a single reservoir yet driven by a deterministic curl-force---a reversible, velocity-independent non-conservative force with non-vanishing curl---that can be described by a Hamiltonian with an anisotropic kinetic energy term. Despite the Hamiltonian structure, the system does not relax to the corresponding Boltzmann distribution, and is genuinely out of equilibrium. The mapping provides a mechanical interpretation of the currents in multi-temperature systems, as we demonstrate by means of two examples. In particular, the direction of circulation is qualitatively predictable from the deterministic force field. For the exactly solvable case of an N-dimensional underdamped Ornstein-Uhlenbeck process, we derive explicit expressions for the steady-state distribution and the entropy production rate, highlighting the role of the (generalized) curl of the force field in driving the system out of thermal equilibrium. Finally, we show how a non-conservative force can be used to exactly equilibrate any system coupled to multiple reservoirs, and obtain the corresponding equilibrium distribution.
ER -