[Paper Review] Thermalisation of a quantum system from first principles

Grégoire Ithier, Florent Benaych-Georges|arXiv (Cornell University)|Oct 14, 2015

Advanced Thermodynamics and Statistical Mechanics23 references3 citations

TL;DR

This paper demonstrates that thermalisation in quantum systems emerges universally from unitary time evolution in a large closed composite system, without assuming initial thermalisation of the environment. Using measure concentration and overlap statistics between bare and dressed Hamiltonian eigenstates, it derives a generalized canonical partition function and shows that the reduced density matrix of the subsystem converges to the canonical state in the long-time limit, recovering the Boltzmann distribution under typical macroscopic conditions.

ABSTRACT

Why is thermalisation a universal phenomenon? How does a quantum system reach thermodynamical equilibrium? These questions are not new, dating even from the very birth of quantum theory and have been the subject of a renewed interest over the two last decades (see for instance the review in Eisert et al, Nature Physics 2015). In this work, we propose a universal model demonstrating that thermalisation of a small quantum system is an emergent property of the unitary evolution under a Schrödinger equation of a larger composite system, whose initial state can be arbitrary. We show that the origin of universality lies in the phenomenon of 'measure concentration', which provides self-averaging properties for the reduced density matrix characterizing the state of the small subsystem. Using our framework, we focus on the asymptotic state at long times and consider its stationary properties. In typical macroscopic conditions, we recover the canonical state and the Boltzmann distribution well known from statistical thermodynamics. This findings lead us to propose an alternative and more general definition of the canonical partition function which also allow us to describe non thermal stationary states.

Motivation & Objective

• To establish a first-principles derivation of quantum thermalisation without assuming initial thermalisation of the environment.
• To resolve foundational issues in quantum statistical mechanics by replacing ergodicity and coarse-graining with measure concentration.
• To derive a generalized canonical partition function that describes both thermal and non-thermal stationary states.
• To show that the reduced density matrix of a subsystem self-averages to the canonical state due to typicality in high-dimensional Hilbert spaces.

Proposed method

• Model a small quantum system S coupled to an environment E as a closed composite system governed by a total Hamiltonian H = H_s + H_e + W.
• Use unitary time evolution U_t = exp(-iHt/ħ) to describe the time evolution of the total density matrix ϱ(t).
• Compute the reduced density matrix ϱ_s(t) = Tr_E[ϱ(t)] to describe the subsystem's state.
• Apply measure concentration to show that ϱ_s(t) concentrates around a typical behavior independent of initial conditions.
• Analyze the statistics of overlaps between eigenvectors of the bare Hamiltonian (H_s + H_e) and the dressed Hamiltonian (H_s + H_e + W) to derive the self-averaging property.
• Derive a generalized canonical partition function from the ratio of densities of states of the environment and the composite system.

Experimental results

Research questions

• RQ1 Can thermalisation in a quantum system be derived from unitary evolution without assuming initial thermalisation of the environment?
• RQ2 What is the role of measure concentration in the emergence of typicality in the reduced density matrix of a subsystem?
• RQ3 How can the canonical ensemble be derived from first principles in a closed quantum system with arbitrary initial conditions?
• RQ4 What is the generalised form of the canonical partition function that includes non-thermal stationary states?
• RQ5 How does the ratio of densities of states of the environment and the composite system determine the stationary state of the subsystem?

Key findings

• The reduced density matrix of the subsystem ϱ_s(t) concentrates around a typical value due to measure concentration, ensuring self-averaging in high-dimensional Hilbert spaces.
• The norm of the gradient of ϱ_s with respect to the interaction W is bounded by √(2τ² dim H_s), proving stability and concentration.
• A generalized canonical partition function is derived from the ratio of the density of states of the environment and the composite system.
• In typical macroscopic conditions, the stationary state of the subsystem converges to the canonical ensemble, recovering the Boltzmann distribution.
• The derivation holds for arbitrary initial states of the composite system, without assuming thermalisation of the environment.
• The framework allows description of both thermal and non-thermal stationary states through the generalized partition function.