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| title | chunk | source | category | tags | date_saved | instance |
|---|---|---|---|---|---|---|
| Eigenstate thermalization hypothesis | 3/5 | https://en.wikipedia.org/wiki/Eigenstate_thermalization_hypothesis | reference | science, encyclopedia | 2026-05-05T09:59:23.990681+00:00 | kb-cron |
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{\displaystyle {\overline {A}}=\sum _{\alpha =1}^{D}|c_{\alpha }|^{2}A_{\alpha \alpha }\approx A\sum _{\alpha =1}^{D}|c_{\alpha }|^{2}=A,}
where we have assumed that the initial state is normalized appropriately. Likewise, the prediction of the microcanonical ensemble becomes
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{\displaystyle \langle A\rangle _{\text{mc}}={\frac {1}{\mathcal {N}}}\sum _{\alpha '=1}^{\mathcal {N}}A_{\alpha '\alpha '}\approx {\frac {1}{\mathcal {N}}}\sum _{\alpha '=1}^{\mathcal {N}}A=A.}
The two ensembles are therefore in agreement. This constancy of the values of
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{\displaystyle A_{\alpha \alpha }}
over small energy windows is the primary idea underlying the eigenstate thermalization hypothesis. Notice that in particular, it states that the expectation value of
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{\displaystyle {\hat {A}}}
in a single energy eigenstate is equal to the value predicted by a microcanonical ensemble constructed at that energy scale. This constitutes a foundation for quantum statistical mechanics which is radically different from the one built upon the notions of dynamical ergodicity.
== Tests == Several numerical studies of small lattice systems appear to tentatively confirm the predictions of the eigenstate thermalization hypothesis in interacting systems which would be expected to thermalize. Likewise, systems which are integrable tend not to obey the eigenstate thermalization hypothesis. Some analytical results can also be obtained if one makes certain assumptions about the nature of highly excited energy eigenstates. The original 1994 paper on the ETH by Mark Srednicki studied, in particular, the example of a quantum hard sphere gas in an insulated box. This is a system which is known to exhibit chaos classically. For states of sufficiently high energy, Berry's conjecture states that energy eigenfunctions in this many-body system of hard sphere particles will appear to behave as superpositions of plane waves, with the plane waves entering the superposition with random phases and Gaussian-distributed amplitudes (the precise notion of this random superposition is clarified in the paper). Under this assumption, one can show that, up to corrections which are negligibly small in the thermodynamic limit, the momentum distribution function for each individual, distinguishable particle is equal to the Maxwell–Boltzmann distribution
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{\displaystyle f_{\rm {MB}}\left(\mathbf {p} ,T_{\alpha }\right)=\left(2\pi mkT\right)^{-3/2}e^{-\mathbf {p} ^{2}/2mkT_{\alpha }},}
where
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{\displaystyle \mathbf {p} }
is the particle's momentum, m is the mass of the particles, k is the Boltzmann constant, and the "temperature"
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{\displaystyle T_{\alpha }}
is related to the energy of the eigenstate according to the usual equation of state for an ideal gas,
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{\displaystyle E_{\alpha }={\frac {3}{2}}NkT_{\alpha },}
where N is the number of particles in the gas. This result is a specific manifestation of the ETH, in that it results in a prediction for the value of an observable in one energy eigenstate which is in agreement with the prediction derived from a microcanonical (or canonical) ensemble. Note that no averaging over initial states whatsoever has been performed, nor has anything resembling the H-theorem been invoked. Additionally, one can also derive the appropriate Bose–Einstein or Fermi–Dirac distributions, if one imposes the appropriate commutation relations for the particles comprising the gas. Currently, it is not well understood how high the energy of an eigenstate of the hard sphere gas must be in order for it to obey the ETH. A rough criterion is that the average thermal wavelength of each particle be sufficiently smaller than the radius of the hard sphere particles, so that the system can probe the features which result in chaos classically (namely, the fact that the particles have a finite size ). However, it is conceivable that this condition may be able to be relaxed, and perhaps in the thermodynamic limit, energy eigenstates of arbitrarily low energies will satisfy the ETH (aside from the ground state itself, which is required to have certain special properties, for example, the lack of any nodes ).
== Alternatives == Three alternative explanations for the thermalization of isolated quantum systems are often proposed: