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| title | chunk | source | category | tags | date_saved | instance |
|---|---|---|---|---|---|---|
| Eigenstate thermalization hypothesis | 1/5 | https://en.wikipedia.org/wiki/Eigenstate_thermalization_hypothesis | reference | science, encyclopedia | 2026-05-05T09:59:23.990681+00:00 | kb-cron |
The eigenstate thermalization hypothesis (ETH) is a set of ideas which purports to explain when and why an isolated quantum mechanical system can be accurately described using equilibrium statistical mechanics. In particular, it is devoted to understanding how systems which are initially prepared in far-from-equilibrium states can evolve in time to a state which appears to be in thermal equilibrium. The phrase "eigenstate thermalization" was first coined by Mark Srednicki in 1994, after similar ideas had been introduced by Josh Deutsch in 1991. The principal philosophy underlying the eigenstate thermalization hypothesis is that instead of explaining the ergodicity of a thermodynamic system through the mechanism of dynamical chaos, as is done in classical mechanics, one should instead examine the properties of matrix elements of observable quantities in individual energy eigenstates of the system.
== Motivation == In statistical mechanics, the microcanonical ensemble is a particular statistical ensemble which is used to make predictions about the outcomes of experiments performed on isolated systems that are believed to be in equilibrium with an exactly known energy. The microcanonical ensemble is based upon the assumption that, when such an equilibrated system is probed, the probability for it to be found in any of the microscopic states with the same total energy have equal probability. With this assumption, the ensemble average of an observable quantity is found by averaging the value of that observable
A
i
{\displaystyle A_{i}}
over all microstates
i
{\displaystyle i}
with the correct total energy:
A
¯
c
l
a
s
s
i
c
a
l
=
1
N
∑
i
=
1
N
A
i
{\displaystyle {\bar {A}}_{\mathrm {classical} }={\frac {1}{N}}\sum _{i=1}^{N}A_{i}}
Importantly, this quantity is independent of everything about the initial state except for its energy. The assumptions of ergodicity are well-motivated in classical mechanics as a result of dynamical chaos, since a chaotic system will in general spend equal time in equal areas of its phase space. If we prepare an isolated, chaotic, classical system in some region of its phase space, then as the system is allowed to evolve in time, it will sample its entire phase space, subject only to a small number of conservation laws (such as conservation of total energy). If one can justify the claim that a given physical system is ergodic, then this mechanism will provide an explanation for why statistical mechanics is successful in making accurate predictions. For example, the hard sphere gas has been rigorously proven to be ergodic. This argument cannot be straightforwardly extended to quantum systems, even ones that are analogous to chaotic classical systems, because time evolution of a quantum system does not uniformly sample all vectors in Hilbert space with a given energy. Given the state at time zero in a basis of energy eigenstates
|
Ψ
(
0
)
⟩
=
∑
α
c
α
|
E
α
⟩
,
{\displaystyle |\Psi (0)\rangle =\sum _{\alpha }c_{\alpha }|E_{\alpha }\rangle ,}
the expectation value of any observable
A
^
{\displaystyle {\hat {A}}}
is
⟨
A
^
⟩
t
≡
⟨
Ψ
(
t
)
|
A
^
|
Ψ
(
t
)
⟩
=
∑
α
,
β
c
α
∗
c
β
A
α
β
e
−
i
(
E
β
−
E
α
)
t
/
ℏ
.
{\displaystyle \langle {\hat {A}}\rangle _{t}\equiv \langle \Psi (t)|{\hat {A}}|\Psi (t)\rangle =\sum _{\alpha ,\beta }c_{\alpha }^{*}c_{\beta }A_{\alpha \beta }e^{-i\left(E_{\beta }-E_{\alpha }\right)t/\hbar }.}
Even if the
E
α
{\displaystyle E_{\alpha }}
are incommensurate, so that this expectation value is given for long times by
⟨
A
^
⟩
t
≈
t
→
∞
∑
α
|
c
α
|
2
A
α
α
,
{\displaystyle \langle {\hat {A}}\rangle _{t}{\overset {t\to \infty }{\approx }}\sum _{\alpha }\vert c_{\alpha }\vert ^{2}A_{\alpha \alpha },}
the expectation value permanently retains knowledge of the initial state in the form of the coefficients
c
α
{\displaystyle c_{\alpha }}
. In principle it is thus an open question as to whether an isolated quantum mechanical system, prepared in an arbitrary initial state, will approach a state which resembles thermal equilibrium, in which a handful of observables are adequate to make successful predictions about the system. However, a variety of experiments in cold atomic gases have indeed observed thermal relaxation in systems which are, to a very good approximation, completely isolated from their environment, and for a wide class of initial states. The task of explaining this experimentally observed applicability of equilibrium statistical mechanics to isolated quantum systems is the primary goal of the eigenstate thermalization hypothesis.
== Statement == Suppose that we are studying an isolated, quantum mechanical many-body system. In this context, "isolated" refers to the fact that the system has no (or at least negligible) interactions with the environment external to it. If the Hamiltonian of the system is denoted
H
^
{\displaystyle {\hat {H}}}
, then a complete set of basis states for the system is given in terms of the eigenstates of the Hamiltonian,
H
^
|
E
α
⟩
=
E
α
|
E
α
⟩
,
{\displaystyle {\hat {H}}|E_{\alpha }\rangle =E_{\alpha }|E_{\alpha }\rangle ,}
where
|
E
α
⟩
{\displaystyle |E_{\alpha }\rangle }
is the eigenstate of the Hamiltonian with eigenvalue
E
α
{\displaystyle E_{\alpha }}
. We will refer to these states simply as "energy eigenstates." For simplicity, we will assume that the system has no degeneracy in its energy eigenvalues, and that it is finite in extent, so that the energy eigenvalues form a discrete, non-degenerate spectrum (this is not an unreasonable assumption, since any "real" laboratory system will tend to have sufficient disorder and strong enough interactions as to eliminate almost all degeneracy from the system, and of course will be finite in size). This allows us to label the energy eigenstates in order of increasing energy eigenvalue. Additionally, consider some other quantum-mechanical observable
A
^
{\displaystyle {\hat {A}}}
, which we wish to make thermal predictions about. The matrix elements of this operator, as expressed in a basis of energy eigenstates, will be denoted by