Quantum mechanical evolution towards thermal equilibrium

Abstract

The circumstances under which a system reaches thermal equilibrium, and how to derive this from basic dynamical laws, has been a major question from the very beginning of thermodynamics and statistical mechanics. Despite considerable progress, it remains an open problem. Motivated by this issue, we address the more general question of equilibration. We prove, with virtually full generality, that reaching equilibrium is a universal property of quantum systems: Almost any subsystem in interaction with a large enough bath will reach an equilibrium state and remain close to it for almost all times. We also prove several general results about other aspects of thermalisation besides equilibration, for example, that the equilibrium state does not depend on the detailed micro-state of the bath.Leave your hot cup of coffee or cold beer alone for a while and they soon lose their appeal -the coffee cools down and the beer warms up and they both reach room temperature. And it is not only coffee and beer -reaching thermal equilibrium is a ubiquitous phenomenon: everything does it. Thermalization is one of the most fundamental facts of nature.But how exactly does thermalization occur? How can one derive the existence of this phenomenon from the basic dynamical laws of nature (such as Newton's or Schrödinger's equations)? These have been open questions since the very beginning of statistical mechanics more than a century and a half ago.One -but by no means the only -stumbling block has been the fact that the basic postulates of statistical mechanics rely on subjective lack of knowledge and ensemble averages, which is very controversial as a physical principle. Recently however there has been significant progress: it was realized that ensemble averages and subjective ignorance are not needed, because individual quantum states of systems can exhibit statistical properties. This is a purely quantum phenomenon, and the key is entanglement, which leads to objective lack of knowledge. Namely, in quantum mechanics even when we have complete knowledge of the state of a system, i.e. it is in a pure state and has zero entropy, the state of a subsystem may be mixed and have non-zero entropy. In this situation we cannot describe the subsystem by any particular pure state, and for all purposes it behaves as if we have a lack of knowledge about what its pure state is (i.e. it behaves as a probability distribution over pure states). This is in stark contrast to classical physics where complete knowledge of the state of the whole system implies complete knowledge of the state of any subsystem, and hence probabilities can only arise as purely subjective lack of knowledge (i.e. the subsystem has a well-defined state only we don't know what that is).This approach has become a very fruitful direction of research in recent years [1, 2,3], see also important earlier work [4,5,6,7] and numerical studies [8]. Most notably, it was shown in [1, 2,3] that almost all (pure) states of a large system are such that any small subsystem is in a canonical stat...

One-Shot Decoupling

“…For example, the evolution of a thermodynamical system towards thermal equilibrium can be understood as a decoupling process, where the system under consideration decouples from the observer (somewhat analogous to the considerations in [LPSW09,Par89a,Par89b]). Recent work indeed shows that there is a close relation between smooth entropies and quantities that are relevant in thermodynamics [DRRV09,dRAR + 11,Hut11,FDOR12,Abe13,HO13].…”

Quench of a symmetry-broken ground state

“…quantities associated to operators with supports completely included in S, is indistinguishable by the time evolution of a system going towards a thermal equilibrium with a bath. In other words a closed quantum system may locally thermalize [6][7][8][9][10]. This implies that, in the steady state, local physical quantities will lose all the informations about the initial state with the exception of the effective temperature [6,11,12].…”

Yet another time about time … Part I: An essay on the phenomenology of physical time

“…It could also possibly give us the answer to the fundamental question about the source of the irreversible arrow of time, and hence about the nature of time itself. In their 2009 article, Linden and colleagues demonstrate that energy disperses and objects equilibrate, because of the way elementary particles become intertwined during their interactions (Linden et al, 2009). Thus, physical systems reach equilibrium, or a state of uniform energy distribution, within an infinite "amount of time" 60 by becoming quantum mechanically entangled with their surroundings.…”