ISSN:
1089-7658
Source:
AIP Digital Archive
Topics:
Mathematics
,
Physics
Notes:
We prove that the quantum dynamics of a class of infinite harmonic crystals becomes ergodic and mixing in the following sense: if Hm is the m-particle Schrödinger operator, ωβ,m(A)=Tr(A exp−βHm)/Tr(exp−βHm) the corresponding quantum Gibbs distribution over the observables A,B,ψm,λ the coherent states in the mth particle Hilbert space, gm,λ=(exp−βHm)ψm,λ then limt→∞ limn→∞ limm→∞(1/T)∫T0〈eiHntAe−iHntψm,λ,ψ m,λ〉dt=limm→∞ ωβ,m(A) if the classical infinite dynamics is ergodic, and limt→∞ limn→∞ limm→∞ ωβ,m(e /iiHntAe−iHntB)=limm→∞ ωβ,m(A)limm→∞ωβ,m(B) if it is in addition mixing. The classical ergodicity and mixing properties are recovered as (h-dash-bar)→0, and limm→∞ ωβ,m(A) turns out to be the average over a classical Gibbs measure of the symbol generating A under Weyl quantization. © 1996 American Institute of Physics.
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1063/1.531741
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