# Process of equilibration in many-body isolated systems: Diagonal versus thermodynamic entropy

## TL;DR

This paper investigates how many-body isolated quantum systems equilibrate and thermalize, linking the width of the local density of states to diagonal and thermodynamic entropy, supported by analytical and numerical analysis.

## Contribution

It establishes a direct relation between the width of the local density of states and the thermodynamic entropy in isolated quantum systems, combining analytical derivations with numerical validation.

## Key findings

- The width of the LDoS is related to the decay rate of survival probability.
- Diagonal entropy can be linked to thermodynamic entropy after relaxation.
- Analytical expressions are confirmed in models of bosons with random and deterministic interactions.

## Abstract

As recently manifested , the quench dynamics of isolated quantum systems consisting of a finite number of particles, is characterized by an exponential spreading of wave packets in the many-body Hilbert space. This happens when the inter-particle interaction is strong enough, thus resulting in a chaotic structure of the many-body eigenstates considered in an unperturbed basis. The semi-analytical approach used here, allows one to estimate the rate of the exponential growth as well as the relaxation time, after which the equilibration (thermalization) emerges. The key ingredient parameter in the description of this process is the width $\Gamma$ of the Local Density of States (LDoS) defined by the initially excited state, the number of particles and the interaction strength. In this paper we show that apart from the meaning of $\Gamma$ as the decay rate of survival probability, the width of the LDoS is directly related to the diagonal entropy and the latter can be linked to the thermodynamic entropy of a system equilibrium state emerging after the complete relaxation. The analytical expression relating the two entropies is derived phenomenologically and numerically confirmed in a model of bosons with random two-body interaction, as well as in a deterministic model which becomes completely integrable in the continuous limit.