1996/02/02 by S. Graffi, Sandro Graffi, A. Martinez +1 · 8 citations
Mathematics · Physics and Astronomy · #Ergodic theory #Ergodicity #Harmonic measure #Hilbert space #Markov Chains and Monte Carlo Methods #Measure (data warehouse) #Mixing (physics) #Observable #Quantum #Random Matrices and Applications #Spectral Theory in Mathematical Physics #Stationary ergodic process #chao-dyn #cond-mat #nlin.CD #quant-ph
paper · pdf · doi:10.1063/1.531741
published in Journal of Mathematical Physics 37(10), 5111-5135 (American Institute of Physics) · 30 pages, plain LaTex
arxiv created 1996/02/02 · openalex publication_date 1996/10/01 · arxiv updated 2009/11/30 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/05
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,ψm,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 ℏ→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.