Markov transitions and the propagation of chaos
- Univ. of California, Berkeley, CA (United States)
The propagation of chaos is a central concept of kinetic theory that serves to relate the equations of Boltzmann and Vlasov to the dynamics of many-particle systems. Propagation of chaos means that molecular chaos, i.e., the stochastic independence of two random particles in a many-particle system, persists in time, as the number of particles tends to infinity. We establish a necessary and sufficient condition for a family of general n-particle Markov processes to propagate chaos. This condition is expressed in terms of the Markov transition functions associated to the n-particle processes, and it amounts to saying that chaos of random initial states propagates if it propagates for pure initial states. Our proof of this result relies on the weak convergence approach to the study of chaos due to Sztitman and Tanaka. We assume that the space in which the particles live is homomorphic to a complete and separable metric space so that we may invoke Prohorov's theorem in our proof. We also s how that, if the particles can be in only finitely many states, then molecular chaos implies that the specific entropies in the n-particle distributions converge to the entropy of the limiting single-particle distribution.
- Research Organization:
- Lawrence Berkeley National Lab. (LBNL), Berkeley, CA (United States)
- Sponsoring Organization:
- USDOE Office of Science (SC)
- DOE Contract Number:
- AC03-76SF00098
- OSTI ID:
- 6538
- Report Number(s):
- LBNL-42839; TRN: US200305%%767
- Resource Relation:
- Other Information: TH: Thesis (Ph.D.); Supercedes report DE00006538; Submitted to the Univ. of California, Dept. of Mathematics, Berkeley, CA (US); PBD: 1 Dec 1998; PBD: 1 Dec 1998
- Country of Publication:
- United States
- Language:
- English
Similar Records
Convergence properties of polynomial chaos approximations for L2 random variables.
ONSET OF CHAOS IN A MODEL OF QUANTUM COMPUTATION