A Boltzmann map for quantum oscillators

Abstract

We define a map τ on the space of quasifree states of the CCR or CAR of more than one harmonic oscillator which increases entropy except at fixed points ofx. The map τ is the composition of a doubly stochastic map T^* and the quasifree reductionQ. Under mixing conditions onT, iterates of τ take any initial state to the Gibbs state, provided that the oscillator frequencies are mutually rational. We give an example of a system with three degrees of freedom with energies ω₁, ω₂, and ω₃ mutually irrational, but obeying a relation n₁ω₁+n₂ω₂=n₃ω₃,n _i ∈ℤ. The iterated Boltzmann map converges from an initial statep to independent Gibbs states of the three oscillators at betas (inverse temperatures) β₁,β₂, β₃ obeying the equation n₁ω₁β₁+n₂ω₂β₂=n₃ω₃β₃. The equilibrium state can be rewritten as a grand canonical state. We show that for two, three, or four fermions we can get the usual rate equations as a special case.