Ergodic Properties of the Quantum Ideal Gas in the Maxwell-Boltzmann Statistics

It is proved that the quantization 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|→∞ lim m,L→∞ m/L→ρ ω β,L(e mAem B) = lim m,L→∞ m/L→ρ ω β,L(A)· lim m,L→∞ m/L→ρ ω β,L(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; ωm β,L(A) is the corresponding quantum Gibbs state. Moreover one has lim m,L→∞ m/L→ρ 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. On leave at: Mathematics Department, Princeton University, Princeton, NJ 08544, U.S.A. E-mail: [email protected]