A thermodynamic analysis of nonsecular master equations, beyond strict energy conservation and Gibbs equilibrium state
Abstract
We present a systematic thermodynamic analysis of nonsecular GKLS master equations. When some energy transitions in the system are comparable with the dissipative rate, the secular approximation is not valid, and the positivity of the dynamics must be ensured via other approximations. The dynamics then presents features reminiscent from strong coupling to the environment: e.g. the equilibrium state with an environment at thermal equilibrium is not a Gibbs state of the local system Hamiltonian and the dynamics couples energy coherences and populations. Nonetheless, we build a unified, consistent thermodynamic framework for those dynamics. We focus on two approximations beyond the full secular approximation (the local and the more accurate geometric-arithmetic approximations), and then employ systematic perturbation theory on microscopic expressions for the first and second law of thermodynamics which relate irreversibility to the increase of system-environment correlations. We show that, in spite of the weak system-bath coupling, the system-bath interaction energy participates to the energy balance, and is directly linked to the Lamb-shift Hamiltonian. We compare this microscopic entropy production with the definition based on the contractivity of the reduced system dynamics (Spohn inequality). We show that, unlike for secular master equations, the two entropy production rates differ because of the presence of non-vanishing stationary coherences in the energy eigenbasis. However, in the case of a single thermal bath, the difference is purely transient, and no work can be cyclically extracted from the steady-state despite its non-Gibbs form. Finally, we illustrate our results with simple examples, clarifying and completing the thermodynamic picture of Markovian dynamics in the quantum regime.
Related
- Hyperuniformity and conservation laws in non-equilibrium systems
- Approximating thermodynamic equilibrium observables of quasi one-dimensional magnetic molecules and chains by thermal DMRG - a user perspective
- Global equilibrium at order 𝒪(ℏ2): pseudo-gauge ambiguity, Maxwell relations, and thermodynamic consistency
- qdmag: A Python package for simulating nonequilibrium magnetization of molecules using quantum master equations
- Quantum-Classical Neural Networks for Solving Lindblad Master Equations
BibTeX
@misc{boubakour2026thermodynamic,
title = {{A thermodynamic analysis of nonsecular master equations, beyond strict energy conservation and Gibbs equilibrium state}},
author = {Mohamed Boubakour and Talia Szikman and Cyril Elouard},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2606.13504},
eprint = {2606.13504},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2606.13504},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - A thermodynamic analysis of nonsecular master equations, beyond strict energy conservation and Gibbs equilibrium state
AU - Mohamed Boubakour
AU - Talia Szikman
AU - Cyril Elouard
PY - 2026
DA - 2026/10/13
PB - arXiv
DO - 10.48550/arxiv.2606.13504
UR - https://arxiv.org/abs/2606.13504
AB - We present a systematic thermodynamic analysis of nonsecular GKLS master equations. When some energy transitions in the system are comparable with the dissipative rate, the secular approximation is not valid, and the positivity of the dynamics must be ensured via other approximations. The dynamics then presents features reminiscent from strong coupling to the environment: e.g. the equilibrium state with an environment at thermal equilibrium is not a Gibbs state of the local system Hamiltonian and the dynamics couples energy coherences and populations. Nonetheless, we build a unified, consistent thermodynamic framework for those dynamics. We focus on two approximations beyond the full secular approximation (the local and the more accurate geometric-arithmetic approximations), and then employ systematic perturbation theory on microscopic expressions for the first and second law of thermodynamics which relate irreversibility to the increase of system-environment correlations. We show that, in spite of the weak system-bath coupling, the system-bath interaction energy participates to the energy balance, and is directly linked to the Lamb-shift Hamiltonian. We compare this microscopic entropy production with the definition based on the contractivity of the reduced system dynamics (Spohn inequality). We show that, unlike for secular master equations, the two entropy production rates differ because of the presence of non-vanishing stationary coherences in the energy eigenbasis. However, in the case of a single thermal bath, the difference is purely transient, and no work can be cyclically extracted from the steady-state despite its non-Gibbs form. Finally, we illustrate our results with simple examples, clarifying and completing the thermodynamic picture of Markovian dynamics in the quantum regime.
ER -