Abstract
We consider the time evolved states\(\bar P_t = \bar P \circ \theta _t^{ - 1}\) of the free motionθ t (q, v)=(q+tv,v),q,v∈ℝd, starting in some non-equilibrium state\(\bar P\) and look at the associated processX ε t of fluctuations of the actual numberθ t/ε (μ)\(\left( {\frac{1}{\varepsilon }A \times B} \right)\) of particles of the realization μ in\(\frac{1}{\varepsilon }.A\) 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 state\(\bar P\), 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:\(\bar P \circ \left( {X_t^\varepsilon } \right)^{ - 1} \Rightarrow \bar Q_t\) as ε→0, where ⇒ denotes weak convergence and\(\bar Q_t\) is a centered Gaussian state, which is translation invariant in the space variables. Furthermore the time evolution\((\bar Q_t )_t\) is also given by the free motion in the sense that\(\bar Q_t = \bar Q_0 \circ \theta _t^{ - 1}\) On the other hand we shall see that\(\bar P_t \Rightarrow P_{z \cdot \lambda \times \sigma }\) 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:\(\bar Q_t \Rightarrow \bar Q\) ast→∞. Here\(\bar Q\) 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.
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Zessin, H. On the asymptotic behaviour of the free gas and its fluctuations in the hydrodynamical limit. Probab. Th. Rel. Fields 77, 605–622 (1988). https://doi.org/10.1007/BF00959620
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DOI: https://doi.org/10.1007/BF00959620