Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Entropic Fluctuation Theorems for the Spin-Fermion Model

Tristan Benoist, Laurent Bruneau, Vojkan Jakšić, Annalisa Panati, Claude-Alain Pillet

TL;DR

The paper proves quantum fluctuation theorems (Evans-Searles and Gallavotti-Cohen) for the Spin--Fermion open quantum system by developing a spectral resonance framework for quantum transfer operators. It builds a detailed α-Liouvillean formalism, connects entropic functionals to quasi-energy resonances, and shows PREF holds for small coupling, yielding real-analytic, strictly convex rate functions that exhibit TRI symmetry when temperatures are equal. The analysis relies on the glued Araki-Wyss representation, complex deformation techniques, and a tight link between level-sh Shift operators and deformed Davies generators. The results provide a rigorous bridge between non-equilibrium entropic fluctuations and the spectral data of quantum dynamical generators, with explicit results for the simplest spin-fermion instance. This work advances non-equilibrium quantum statistical mechanics by delivering a concrete, resonance-based pathway to quantify entropy production fluctuations in realistic open quantum systems.

Abstract

We study entropic fluctuations in the Spin-Fermion model describing an $N$-level quantum system coupled to several independent thermal free Fermi gas reservoirs. We establish the quantum Evans-Searles and Gallavotti-Cohen fluctuation theorems and identify their link with entropic ancilla state tomography and quantum phase space contraction of non-equilibrium steady state. The method of proof involves the spectral resonance theory of quantum transfer operators developed by the authors in previous works.

Entropic Fluctuation Theorems for the Spin-Fermion Model

TL;DR

The paper proves quantum fluctuation theorems (Evans-Searles and Gallavotti-Cohen) for the Spin--Fermion open quantum system by developing a spectral resonance framework for quantum transfer operators. It builds a detailed α-Liouvillean formalism, connects entropic functionals to quasi-energy resonances, and shows PREF holds for small coupling, yielding real-analytic, strictly convex rate functions that exhibit TRI symmetry when temperatures are equal. The analysis relies on the glued Araki-Wyss representation, complex deformation techniques, and a tight link between level-sh Shift operators and deformed Davies generators. The results provide a rigorous bridge between non-equilibrium entropic fluctuations and the spectral data of quantum dynamical generators, with explicit results for the simplest spin-fermion instance. This work advances non-equilibrium quantum statistical mechanics by delivering a concrete, resonance-based pathway to quantify entropy production fluctuations in realistic open quantum systems.

Abstract

We study entropic fluctuations in the Spin-Fermion model describing an -level quantum system coupled to several independent thermal free Fermi gas reservoirs. We establish the quantum Evans-Searles and Gallavotti-Cohen fluctuation theorems and identify their link with entropic ancilla state tomography and quantum phase space contraction of non-equilibrium steady state. The method of proof involves the spectral resonance theory of quantum transfer operators developed by the authors in previous works.

Paper Structure

This paper contains 32 sections, 17 theorems, 188 equations, 1 figure.

Table of Contents

  1. Introduction
  2. The Spin--Fermion Model
  3. Description of the model
  4. Entropy production and entropic functionals
  5. Fluctuation theorems and the principle of regular entropic fluctuations
  6. Main results
  7. Quantum transfer operators approach to the PREF
  8. The "Glued" Araki--Wyss representation
  9. The modular structure of the spin--fermion model
  10. Two families of Liouvilleans
  11. Liouvillieans: spectra, resonances and dynamics
  12. Complex deformation of $\mathcal{L}_{\lambda,\alpha}$
  13. (a)
  14. (b)
  15. (c)
  16. ...and 17 more sections

Key Result

Proposition 2.3

Suppose SFMtwo holds. Then, for all $t\in\mathbb{R}$, $([D\omega_t:D\omega]_{\alpha})_{\alpha\in\mathrm{i}\mathbb{R}}\subset\pi(\mathcal{O})$.

Figures (1)

  • Figure 1: Picture of the $z$-plane (it is assumed here that $\kappa r<r_\mathsf{S}/4$). The black dots are the eigenvalues of $\mathcal{L}_\mathsf{S}$. If $\theta$ is in the hashed area, the spectrum of the deformed Liouvillean $\mathcal{L}_{\lambda,\alpha}(\theta)$ is in the shaded areas (which extends to infinity on the top side).

Theorems & Definitions (32)

  • Remark 2.1
  • Remark 2.2
  • Proposition 2.3
  • Theorem 2.4
  • Definition 2.5
  • Remark 2.6
  • Remark 2.7
  • Theorem 2.8
  • Remark 2.9
  • Remark 3.1
  • ...and 22 more