Unification of Stochastic and Quantum Thermodynamics in Scalar Field Theory via a Model with Brownian Thermostat

TL;DR

This work proposes a field-theoretic unification of stochastic and quantum thermodynamics for a real scalar field by coupling it to a Brownian thermostat and applying canonical quantization to derive a quantum master equation. While the resulting dynamics are generally non-CPTP, a CPTP (GKSL) form can be recovered for a free-field Hamiltonian by tuning the bath parameters, and the classical limit recovers stochastic thermodynamics. The framework delivers clear definitions of heat, work, and entropy that satisfy the first and second laws in both classical and quantum regimes, and it clarifies the connection between the second law and relative entropy. Overall, the paper extends quantum thermodynamics into field theory, establishing a quantum-classical correspondence for open systems and providing tools to study entropy production via information-theoretic methods in quantum fields.

Abstract

We present a systematic procedure to derive a quantum master equation for thermal relaxation in real scalar field theory, expanding on the method proposed in [Koide and Nicacio, Phys. Lett. A494, 129277 (2024)]. We begin by introducing a generalized model for a classical scalar field interacting with a Brownian thermostat, consistent with stochastic thermodynamics. Applying canonical quantization to this model, we derive the corresponding quantum master equation, that is applicable to any form of the scalar field Hamiltonian. While its evolution is generally non-CPTP (Completely Positive and Trace-Preserving), it can be adjusted to describe a CPTP evolution, such as those found in the GKSL (Gorini-Kossakowski-Sudarshan-Lindblad) equation by appropriately tuning the parameters of the model. In this framework, we define heat, work, and entropy in a way that satisfies the first and second laws of quantum thermodynamics. This suggests that the quantum-classical correspondence extends beyond closed systems governed by unitary time evolution to open systems as well. We further investigate the relation between the second law in quantum thermodynamics and relative entropy, providing insights into the study of quantum fluctuations through information-theoretical techniques in quantum field theory.