# Paper Introduction: Eigenstate Thermalization Hypothesis.

In this article, I will talk about the Eigenstate Thermalization Hypothesis (ETH), a key concept in quantum information thermodynamics. It states that the eigenstate trace of a physical quantity in a quantum system coincides with a small canonical population after infinite time. This is an empirical law that has been shown to hold for non-integrable systems in general and for some integrable systems in particular. It is also consistent with quantum ergodicity and the fluctuation theorem. There are some non-integrable systems that become quantum integrable or localized by quantum quenching, and numerical experiments have been carried out using these systems. The single-particle next-nearest-neighbor hopping term in a chain of hard-core bosons and fermions with final Hamiltonians of length 21 and 24, and the wavenumber moments in a system with exchange and superexchange interactions clearly show a decrease in fluctuations with increasing size. The relationship between the coefficients of the superexchange interaction and the next-nearlest-neighbor hopping term, $t'=V'$, and the expectation value of the wavenumber for a discrete frequency k in a small canonical population, ﻿$\langle n_k \rangle_{mc}$﻿ , shows the same minima for bosons and fermions at different sizes, and the temperature dependence is also confirmed. Since these terms break the integrability of the particle difference, which is an integrable system, it is consistent with the theory that the average expectation value in the small canonical population deviates from the quantum mechanical expectation value as they approach zero.

This is an empirical rule imported from classical thermostatistics, but it plays an important role in connecting thermodynamics and quantum mechanics. From here, it evolves into quantum information thermodynamics.

https://arxiv.org/abs/1108.0928