Approach to equilbrium in nano-scale systems at finite temperatur

TL;DR

This study investigates how a small quantum spin system reaches equilibrium when interacting with an environment at finite temperature, showing that the reduced density matrix converges to a stationary state regardless of interaction strength or initial temperature.

Contribution

It demonstrates that the stationary state of a finite quantum spin system can be directly derived from its pure state evolution without relying on weak coupling or large system assumptions.

Findings

Eigenvalues of the reduced density matrix converge to stationary values.

System entropy relaxes to a stationary value.

Stationary state depends on initial environment temperature.

Abstract

We study the time evolution of the reduced density matrix of a system of spin-1/2 particles interacting with an environment of spin-1/2 particles. The initial state of the composite system is taken to be a product state of a pure state of the system and a pure state of the environment. The latter pure state is prepared such that it represents the environment at a given finite temperature in the canonical ensemble. The state of the composite system evolves according to the time-dependent Schr{\"{o}}dinger equation, the interaction creating entanglement between the system and the environment. It is shown that independent of the strength of the interaction and the initial temperature of the environment, all the eigenvalues of the reduced density matrix converge to their stationary values, implying that also the entropy of the system relaxes to a stationary value. We demonstrate that the difference between the canonical density matrix and the reduced density matrix in the stationary state increases as the initial temperature of the environment decreases. As our numerical simulations are necessarily restricted to a modest number of spin-1/2 particles ($<36$), but do not rely on time-averaging of observables nor on the assumption that the coupling between system and environment is weak, they suggest that the stationary state of the system directly follows from the time evolution of a pure state of the composite system, even if the size of the latter cannot be regarded as being close to the thermodynamic limit.