Propagation of Chaos in Classical and Quantum Kinetics

The concept of molecular chaos dates back to Boltzmann [3], who derived the fundamental equation of the kinetic theory of gases under the hypothesis that the molecules of a nonequilibrium gas are in a state of “molecular disorder.” The concept of propagation of molecular chaos is due to Kac [8, 9], who called it “propagation of the Boltzmann property” and used it to derive the homogeneous Boltzmann equation in the infinite-particle limit of certain Markovian gas models (see also [5, 17]). McKean [12, 13] proved the propagation of chaos for systems of interacting diffusions that yield diffusive Vlasov equations in the mean-field limit. Spohn [16] used a quantum analog of the propagation of chaos to derive time-dependent Hartree equations for mean-field Hamiltonians, and his work was extended in [1] to open quantum mean-field systems. This article examines the relationship between classical and quantum propagation of chaos. The rest of this introduction reviews some ideas of quantum probability and dynamics. Section 1.2 discusses the classical and quantum concepts of propagation of chaos. In Section 1.3, classical propagation of chaos is shown to occur when quantum systems that propagate quantum molecular chaos are suitably prepared, allowed to evolve without interference, and then observed. Our main result is Corollary 1.3.7, which may be paraphrased as follows: Let O be a complete observable of a single particle, taking its values in a countable set J , and let Oi denote the observable O of particle i in a system of n distinguishable particles of the same species. Suppose we allow that quantum n-particle system to evolve freely, except that we periodically measure O1,O2, . . . ,On. The resulting time series of measurements is a Markov chain in J. If the sequence of n-particle dynamics propagates quantum molecular chaos, then these derived Markov chains propagate chaos in the classical sense.