Open quantum systems and the grand canonical ensemble

TL;DR

This paper explores deriving the grand canonical ensemble directly from the Lindblad equation for open quantum systems, proposing a modified Hamiltonian approach that naturally yields the equilibrium state without external assumptions.

Contribution

It introduces a modified Lindblad equation with a Hamiltonian including the chemical potential term, deriving the grand canonical state from first principles.

Findings

The modified Lindblad equation naturally produces the grand canonical state.

Including the $b N$ term in the Hamiltonian aligns the dynamics with statistical mechanics.

The approach removes the need for external assumptions in deriving equilibrium states.

Abstract

The celebrated Lindblad equation governs the non-unitary time evolution of density operators used in the description of open quantum systems. It is usually derived from the von Neumann equation for a large system, at given physical conditions, when a small subsystem is explicitly singled out and the rest of the system acts as an environment whose degrees of freedom are traced out. In the specific case of a subsystem with variable particle number, the equilibrium density operator is given by the well-known grand canonical Gibbs state. Consequently, solving the Lindblad equation in this case should automatically yield, without any additional assumptions, the corresponding density operator in the limiting case of statistical equilibrium. Current studies of the Lindblad equation with varying particle number assume, however, the grand canonical Gibbs state a priori: the chemical potential is externally imposed rather than derived from first principles, and hence the corresponding density operator is not obtained as a natural solution of the equation. In this work, we investigate the compatibility of grand canonical statistical mechanics with the derivation of the Lindblad equation. We propose an alternative and complementary approach to the current literature that consists in using a generalized system Hamiltonian which includes a term $\mu N$. In a previous paper, this empirically well-known term has been formally derived from the von Neumann equation for the specific case of equilibrium. Including $\mu N$ in the system Hamiltonian leads to a modified Lindblad equation which yields the grand canonical state as a natural solution, meaning that all the quantities involved are obtained from the physics of the system without any external assumptions.