9.0 KiB
| title | chunk | source | category | tags | date_saved | instance |
|---|---|---|---|---|---|---|
| Eigenstate thermalization hypothesis | 4/5 | https://en.wikipedia.org/wiki/Eigenstate_thermalization_hypothesis | reference | science, encyclopedia | 2026-05-05T09:59:23.990681+00:00 | kb-cron |
For initial states of physical interest, the coefficients
c
α
{\displaystyle c_{\alpha }}
exhibit large fluctuations from eigenstate to eigenstate, in a fashion which is completely uncorrelated with the fluctuations of
A
α
α
{\displaystyle A_{\alpha \alpha }}
from eigenstate to eigenstate. Because the coefficients and matrix elements are uncorrelated, the summation in the diagonal ensemble is effectively performing an unbiased sampling of the values of
A
α
α
{\displaystyle A_{\alpha \alpha }}
over the appropriate energy window. For a sufficiently large system, this unbiased sampling should result in a value which is close to the true mean of the values of
A
α
α
{\displaystyle A_{\alpha \alpha }}
over this window, and will effectively reproduce the prediction of the microcanonical ensemble. However, this mechanism may be disfavored for the following heuristic reason. Typically, one is interested in physical situations in which the initial expectation value of
A
^
{\displaystyle {\hat {A}}}
is far from its equilibrium value. For this to be true, the initial state must contain some sort of specific information about
A
^
{\displaystyle {\hat {A}}}
, and so it becomes suspect whether or not the initial state truly represents an unbiased sampling of the values of
A
α
α
{\displaystyle A_{\alpha \alpha }}
over the appropriate energy window. Furthermore, whether or not this were to be true, it still does not provide an answer to the question of when arbitrary initial states will come to equilibrium, if they ever do. For initial states of physical interest, the coefficients
c
α
{\displaystyle c_{\alpha }}
are effectively constant, and do not fluctuate at all. In this case, the diagonal ensemble is precisely the same as the microcanonical ensemble, and there is no mystery as to why their predictions are identical. However, this explanation is disfavored for much the same reasons as the first. Integrable quantum systems are proved to thermalize under condition of simple regular time-dependence of parameters, suggesting that cosmological expansion of the Universe and integrability of the most fundamental equations of motion are ultimately responsible for thermalization.
== Temporal fluctuations of expectation values == The condition that the ETH imposes on the diagonal elements of an observable is responsible for the equality of the predictions of the diagonal and microcanonical ensembles. However, the equality of these long-time averages does not guarantee that the fluctuations in time around this average will be small. That is, the equality of the long-time averages does not ensure that the expectation value of
A
^
{\displaystyle {\hat {A}}}
will settle down to this long-time average value, and then stay there for most times. In order to deduce the conditions necessary for the observable's expectation value to exhibit small temporal fluctuations around its time-average, we study the mean squared amplitude of the temporal fluctuations, defined as
(
A
t
−
A
¯
)
2
¯
≡
lim
τ
→
∞
1
τ
∫
0
τ
(
A
t
−
A
¯
)
2
d
t
,
{\displaystyle {\overline {\left(A_{t}-{\overline {A}}\right)^{2}}}\equiv \lim _{\tau \to \infty }{\frac {1}{\tau }}\int _{0}^{\tau }\left(A_{t}-{\overline {A}}\right)^{2}dt,}
where
A
t
{\displaystyle A_{t}}
is a shorthand notation for the expectation value of
A
^
{\displaystyle {\hat {A}}}
at time t. This expression can be computed explicitly, and one finds that
(
A
t
−
A
¯
)
2
¯
=
∑
α
≠
β
|
c
α
|
2
|
c
β
|
2
|
A
α
β
|
2
.
{\displaystyle {\overline {\left(A_{t}-{\overline {A}}\right)^{2}}}=\sum _{\alpha \neq \beta }|c_{\alpha }|^{2}|c_{\beta }|^{2}|A_{\alpha \beta }|^{2}.}
Temporal fluctuations about the long-time average will be small so long as the off-diagonal elements satisfy the conditions imposed on them by the ETH, namely that they become exponentially small in the system size. Notice that this condition allows for the possibility of isolated resurgence times, in which the phases align coherently in order to produce large fluctuations away from the long-time average. The amount of time the system spends far away from the long-time average is guaranteed to be small so long as the above mean squared amplitude is sufficiently small. If a system poses a dynamical symmetry, however, it will periodically oscillate around the long-time average.
== Quantum fluctuations and thermal fluctuations == The expectation value of a quantum mechanical observable represents the average value which would be measured after performing repeated measurements on an ensemble of identically prepared quantum states. Therefore, while we have been examining this expectation value as the principal object of interest, it is not clear to what extent this represents physically relevant quantities. As a result of quantum fluctuations, the expectation value of an observable is not typically what will be measured during one experiment on an isolated system. However, it has been shown that for an observable satisfying the ETH, quantum fluctuations in its expectation value will typically be of the same order of magnitude as the thermal fluctuations which would be predicted in a traditional microcanonical ensemble. This lends further credence to the idea that the ETH is the underlying mechanism responsible for the thermalization of isolated quantum systems.