On the Equilibrium State of a Small System with Random Matrix Coupling to Its Environment

TL;DR

This paper models a small quantum system interacting with a large environment using random matrices and shows that its equilibrium state approximates the canonical ensemble under certain conditions, regardless of interaction strength.

Contribution

It extends previous results to general small systems, demonstrating the robustness of the canonical state approximation in a random matrix interaction model.

Findings

The reduced density matrix converges to the canonical form for large environments.

The result holds for all interaction strengths, not just weak coupling.

Supports the use of canonical states for nano-systems in equilibrium with environments.

Abstract

We consider a random matrix model of interaction between a small $n$-level system, $S$, and its environment, a $N$-level heat reservoir, $R$. The interaction between $S$ and $R$ is modeled by a tensor product of a fixed $% n\times n$ matrix and a $N\times N$ hermitian Gaussian random matrix. We show that under certain "macroscopicity" conditions on $R$, the reduced density matrix of the system $\rho _{S}=\mathrm{Tr}_{R}\rho _{S\cup R}^{(eq)} $, is given by $\rho _{S}^{(c)}\sim \exp {\{-\beta H_{S}\}}$, where $H_{S}$ is the Hamiltonian of the isolated system. This holds for all strengths of the interaction and thus gives some justification for using $% \rho _{S}^{(c)}$ to describe some nano-systems, like biopolymers, in equilibrium with their environment \cite{Se:12}. Our results extend those obtained previously in \cite{Le-Pa:03,Le-Co:07} for a special two-level system.