On the asymptotic behaviour of the free gas and its fluctuations in the hydrodynamical limit

Zessin HN (1988)
Probability Theory and Related Fields 77(4): 605-622.

Zeitschriftenaufsatz | Veröffentlicht | Englisch
 
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Abstract / Bemerkung
We consider the time evolved statesP¯t=P¯∘θ−1t of the free motionθ t (q, v)=(q+tv,v),q,v∈ℝd, starting in some non-equilibrium stateP¯ and look at the associated processX ε t of fluctuations of the actual numberθ t/ε (μ)(1εA×B) of particles of the realization μ in1ε.A with velocities inB at timet/ε around its mean as ε→0 (i.e., in the hydrodynamic limit). It is shown that under natural conditions on the initial stateP¯, especially a mixing condition in the space variables, for eacht the laws of the fluctuations become Gaussian in the hydrodynamic limit in the following sense:P¯∘(Xεt)−1⇒Q¯t as ε→0, where ⇒ denotes weak convergence andQ¯t is a centered Gaussian state, which is translation invariant in the space variables. Furthermore the time evolution(Q¯t)t is also given by the free motion in the sense thatQ¯t=Q¯0∘θ−1t On the other hand we shall see thatP¯t⇒Pz⋅λ×σ ast→∞, whereP zηλ×τ is the Poisson process with intensity measurez·λ×τ, i.e., the equilibrium state for the free motion with particle densityz and velocity distribution τ. In the hydrodynamic limit this behaviour corresponds to the ergodic theorem for the fluctuation process:Q¯t⇒Q¯ ast→∞. HereQ¯ is a centered Gaussian state describing the equilibrium fluctuations, i.e., the fluctuations ofP zηλ×τ . Thus we prove the central limit theorem for the ideal gas: fluctuations are Gaussian even in non-equilibrium. The proofs rest on an adaption of the method of moments for sequences of generalized fields.
Erscheinungsjahr
1988
Zeitschriftentitel
Probability Theory and Related Fields
Band
77
Ausgabe
4
Seite(n)
605-622
ISSN
0178-8051
Page URI
https://pub.uni-bielefeld.de/record/1654361

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Zessin HN. On the asymptotic behaviour of the free gas and its fluctuations in the hydrodynamical limit. Probability Theory and Related Fields . 1988;77(4):605-622.
Zessin, H. N. (1988). On the asymptotic behaviour of the free gas and its fluctuations in the hydrodynamical limit. Probability Theory and Related Fields , 77(4), 605-622. doi:10.1007/BF00959620
Zessin, H. N. (1988). On the asymptotic behaviour of the free gas and its fluctuations in the hydrodynamical limit. Probability Theory and Related Fields 77, 605-622.
Zessin, H.N., 1988. On the asymptotic behaviour of the free gas and its fluctuations in the hydrodynamical limit. Probability Theory and Related Fields , 77(4), p 605-622.
H.N. Zessin, “On the asymptotic behaviour of the free gas and its fluctuations in the hydrodynamical limit”, Probability Theory and Related Fields , vol. 77, 1988, pp. 605-622.
Zessin, H.N.: On the asymptotic behaviour of the free gas and its fluctuations in the hydrodynamical limit. Probability Theory and Related Fields . 77, 605-622 (1988).
Zessin, Hans Norbert. “On the asymptotic behaviour of the free gas and its fluctuations in the hydrodynamical limit”. Probability Theory and Related Fields 77.4 (1988): 605-622.

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