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

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.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