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A connection between mixing and Kac’s chaos. (English) Zbl 1412.37010

Summary: The Boltzmann equation is an integro-differential equation which describes the density function of the distribution of the velocities of the molecules of dilute monoatomic gases under the assumption that the energy is only transferred via collisions between the molecules. In 1956 {M. Kac} [in: Proc. 3rd Berkeley Sympos. Math. Statist. Probability 3, 171–197 (1956; Zbl 0072.42802)] studied the Boltzmann equation and defined a property of the density function that he called the ‘Boltzmann property’ which describes the behaviour of the density function at a given fixed time as the number of particles tends to infinity. The Boltzmann property has been studied extensively since then, and now it is simply called chaos, or Kac’s chaos. On the other hand, in ergodic theory, chaos usually refers to the mixing properties of a dynamical system as time tends to infinity. A relationship is derived between Kac’s chaos and the notion of mixing.

MSC:

37A60 Dynamical aspects of statistical mechanics
37A25 Ergodicity, mixing, rates of mixing
81Q50 Quantum chaos
28D05 Measure-preserving transformations
35Q20 Boltzmann equations
82B40 Kinetic theory of gases in equilibrium statistical mechanics
82C40 Kinetic theory of gases in time-dependent statistical mechanics
82C31 Stochastic methods (Fokker-Planck, Langevin, etc.) applied to problems in time-dependent statistical mechanics

Citations:

Zbl 0072.42802
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References:

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