We study the nature of and approach to thermal equilibrium in isolated quantum systems. An individual isolated macroscopic quantum system in a pure or mixed state is regarded as being in thermal equilibrium if all macroscopic observables assume rather sharply the values obtained from thermodynamics. Of such a system (or state) we say that it is in macroscopic thermal equilibrium (MATE). A stronger requirement than MATE is that even microscopic observables (i.e., ones referring to a small subsystem) have a probability distribution in agreement with that obtained from the micro‐canonical, or equivalently the canonical, ensemble for the whole system. Of such a system we say that it is in microscopic thermal equilibrium (MITE). The distinction between MITE and MATE is particularly relevant for systems with many‐body localization (MBL) for which the energy eigenfuctions fail to be in MITE while necessarily most of them, but not all, are in MATE. However, if we consider superpositions of energy eigenfunctions (i.e., typical wave functions ψ) in an energy shell, then for generic macroscopic systems, including those with MBL, most ψ are in both MATE and MITE. We explore here the properties of MATE and MITE and compare the two notions, thereby elaborating on ideas introduced in . imageWe study the nature of and approach to thermal equilibrium in isolated quantum systems. An individual isolated macroscopic quantum system in a pure or mixed state is regarded as being in thermal equilibrium if all macroscopic observables assume rather sharply the values obtained from thermodynamics. Of such a system (or state) we say that it is in macroscopic thermal equilibrium (MATE). A stronger requirement than MATE is that even microscopic observables (i.e., ones referring to a small subsystem) have a probability distribution in agreement with that obtained from the micro‐canonical, or equivalently the canonical, ensemble for the whole system. Of such a system we say that it is in microscopic thermal equilibrium (MITE). The distinction between MITE and MATE is particularly relevant for systems with many‐body localization (MBL) for which the energy eigenfuctions fail to be in MITE while necessarily most of them, but not all, are in MATE. However, if we consider superpositions of energy eigenfunctions (i.e., typical wave functions ψ) in an energy shell, then for generic macroscopic systems, including those with MBL, most ψ are in both MATE and MITE. We explore here the properties of MATE and MITE and compare the two notions, thereby elaborating on ideas introduced in . image

arXiv:1610.02312Annalen der PhysikFeb 9, 201745 citations

Thermalization in Isolated Quantum Systems

Osamu Fukushima

Springer thesesJan 1, 20250 citations

Approach to thermal equilibrium of macroscopic quantum systems

Sheldon Goldstein, Joel L. Lebowitz, Christian Mastrodonato +2 more

We consider an isolated macroscopic quantum system. Let $\mathcal{H}$ be a microcanonical ``energy shell,'' i.e., a subspace of the system's Hilbert space spanned by the (finitely) many energy eigenstates with energies between $E$ and $E+\ensuremath{\delta}E$. The thermal equilibrium macrostate at energy $E$ corresponds to a subspace ${\mathcal{H}}_{eq}$ of $\mathcal{H}$ such that $\text{dim}\text{ }{\mathcal{H}}_{eq}/\text{dim}\text{ }\mathcal{H}$ is close to 1. We say that a system with state vector $\ensuremath{\psi}∊\mathcal{H}$ is in thermal equilibrium if $\ensuremath{\psi}$ is ``close'' to ${\mathcal{H}}_{eq}$ . We show that for ``typical'' Hamiltonians with given eigenvalues, all initial state vectors ${\ensuremath{\psi}}_{0}$ evolve in such a way that ${\ensuremath{\psi}}_{t}$ is in thermal equilibrium for most times $t$ . This result is closely related to von Neumann's quantum ergodic theorem of 1929.We consider an isolated macroscopic quantum system. Let $\mathcal{H}$ be a microcanonical ``energy shell,'' i.e., a subspace of the system's Hilbert space spanned by the (finitely) many energy eigenstates with energies between $E$ and $E+\ensuremath{\delta}E$. The thermal equilibrium macrostate at energy $E$ corresponds to a subspace ${\mathcal{H}}_{eq}$ of $\mathcal{H}$ such that $\text{dim}\text{ }{\mathcal{H}}_{eq}/\text{dim}\text{ }\mathcal{H}$ is close to 1. We say that a system with state vector $\ensuremath{\psi}∊\mathcal{H}$ is in thermal equilibrium if $\ensuremath{\psi}$ is ``close'' to ${\mathcal{H}}_{eq}$. We show that for ``typical'' Hamiltonians with given eigenvalues, all initial state vectors ${\ensuremath{\psi}}_{0}$ evolve in such a way that ${\ensuremath{\psi}}_{t}$ is in thermal equilibrium for most times $t$. This result is closely related to von Neumann's quantum ergodic theorem of 1929.

arXiv:0911.1724Physical Review EJan 7, 2010152 citations

Thermalization and its mechanism for generic isolated quantum systems

Marcos Rigol, Vanja Dunjko, Maxim Olshanii

arXiv:0708.1324NatureApr 16, 20083,065 citations

We consider an isolated macroscopic quantum system. Let

\mathcal{H}

be a microcanonical ``energy shell,'' i.e., a subspace of the system's Hilbert space spanned by the (finitely) many energy eigenstates with energies between

E

and $E+\ensuremath{\delta}E$. The thermal equilibrium macrostate at energy

E

corresponds to a subspace

{\mathcal{H}}_{eq}

\mathcal{H}

such that

\text{dim}\text{ }{\mathcal{H}}_{eq}/\text{dim}\text{ }\mathcal{H}

is close to 1. We say that a system with state vector $\ensuremath{\psi}∊\mathcal{H}$ is in thermal equilibrium if $\ensuremath{\psi}$ is ``close'' to

{\mathcal{H}}_{eq}

. We show that for ``typical'' Hamiltonians with given eigenvalues, all initial state vectors ${\ensuremath{\psi}}_{0}$ evolve in such a way that ${\ensuremath{\psi}}_{t}$ is in thermal equilibrium for most times

t

. This result is closely related to von Neumann's quantum ergodic theorem of 1929.