Positive Commutators in Non-Equilibrium Statistical Mechanics

Abstract

The method of positive commutators, developed for zero temperature problems over the last twenty years, has been an essential tool in the spectral analysis of Hamiltonians in quantum mechanics. We extend this method to positive temperatures, i.e. to non-equilibrium quantum statistical mechanics. We use the positive commutator technique to give an alternative proof of a fundamental property of a certain class of large quantum systems, called {\it Return to Equilibrium}. This property says that equilibrium states are (asymptotically) stable: if a system is slightly perturbed from its equilibrium state, then it converges back to that equilibrium state as time goes to infinity.

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References in corpus (1)

• Ergodic Properties of the Spin - Boson System

Cited by in corpus (24)

• Strong Coupling Thermodynamics of Open Quantum Systems
• Open quantum system dynamics and the mean force Gibbs state
• Another Return of 'Return to Equilibrium'
• Theory of Non-Equilibrium Sationary States as a Theory of Resonances
• Thermal Ionization
• Dynamics of Open Quantum Systems II, Markovian Approximation
• Adaiabtic theorems and reversible isothermal processes
• On the irreversible dynamics emerging from quantum resonances
• Dynamics of Open Quantum Systems I, Oscillation and Decay
• Ergodicity of the Spin-Boson Model for arbitrary coupling strength
• Ionization of Atoms in a Thermal Field
• Stability of Equilibria with a Condensate
• Large deviation generating function for energy transport in the Pauli-Fierz model
• Open-system eigenstate thermalization in a noninteracting integrable model
• Overlapping Resonances in Open Quantum Systems
• Equivalence between the second order steady state for the spin-Boson model and its quantum mean force Gibbs state
• Return to Equilibrium for an Anharmonic Oscillator coupled to a Heat Bath
• Disjointness of inertial KMS states and the role of Lorentz symmetry in thermalization
• Quantum Diffusion with Drift and the Einstein Relation II
• Diffusion for a quantum particle coupled to phonons in $d\ge 3$
• Entropic Fluctuation Theorems for the Spin-Fermion Model
• Positive commutators, Fermi golden rule and the spectrum of zero temperature Pauli-Fierz Hamiltonians
• Level shift operators for open quantum systems
• Thermal Ionization for Short-Range Potentials