Ergodic properties of the quantum ideal gas in the Maxwell–Boltzmann statistics

Abstract

It is proved that the q uantization of the Volkovyski–Sinai model of ideal gas (in the Maxwell–Boltzmann statistics) enjoys at the thermodynamical limit the property of quantum mixing in the following sense: lim|t|→∞ limm/L→ρm,L→∞ ωβ ,Lm(eiHmt/ℏ×Ae−iHmt/ mCB)=limm/L→ρm,L→∞ ωβ,Lm( A)⋅limm/L→ρm,L→∞ ωβ,Lm(B ). Here Hm is the Schrödinger operator of m free particles moving on a circle of length L; A and B are the Weyl quantization of two classical observables a and b; ωmβ,L(A) is the corresponding quantum Gibbs state. Moreover, one has limm/L→ρm,L→∞ ωβ,m(A)=Pρ,β (a), where Pρ,β(a) is the classical Gibbs measure. The consequent notion of quantum ergodicity is also independently proven.