# Necessity of eigenstate thermalization for equilibration towards unique expectation values when starting from generic initial states

## TL;DR

This paper demonstrates that eigenstate thermalization is necessary for initial state independent equilibration in closed quantum systems, supported by analytical derivations and numerical simulations showing deviations when it is violated.

## Contribution

It provides an analytical expression for deviations from equilibrium when eigenstate thermalization fails and confirms its necessity through numerical examples.

## Key findings

- Long time expectation values depend on initial states if eigenstate thermalization is violated.
- Analytical predictions match numerical results beyond theoretical limits.
- Eigenstate thermalization is necessary for initial state independent equilibration.

## Abstract

We investigate dynamical equilibration of expectation values in closed quantum systems for realistic non-equilibrium initial states. Thereby we find that the corresponding long time expectation values depend on the initial expectation values if eigenstate thermalization is violated. An analytical expression for the deviation from the expected ensemble value is derived for small displacements from equilibrium. Additional numerics for magnetization and energy equilibration in an asymmetric anisotropic spin-1/2-ladder demonstrate that the analytical predictions persist beyond the limits of the theory. The results suggest eigenstate thermalization as physically necessary condition for initial state independent equilibration.