On the asymptotic behaviour of the free gas and its fluctuations in the hydrodynamical limit
Zessin, Hans Norbert
Zessin
Hans Norbert
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.
77
4
605-622
605-622
Springer
1988