Kinetic and Hydrodynamic Scalings in an Exactly-Soluble Model for the Brownian Motion

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Abstract
The macroscopic system exhibits hierarchical dynamic processes with different time scales, each level of which has a characteristic scaling. This important feature is rigorously investigated for the Brownian motion of a heavy impurity in a harmonic linear chain in thermal equilibrium which can treated analytically in the full context. The model considered is a slightly-extended one-dimensional Rubin's model with an arbitrary force constant K′ between the impurity and its neighboring particles. It is shown that the characteristic scaling of the hydrodynamic process is x →Lx and t →L2t (L ≫1), x being the position of the Brownian particle, and leads to the diffusion equation. When the Brownian particle is an isotope with mass M, the kinetic process is characterized by the scaling M →L2M and t →L2t, and the asymptotic form of the probability density in µ space agrees with that of the phenomenological theory of the Brownian motion of free particles. When the Brownian particle is an impurity, the scaling K′ →K′/L2, M →L2M and t →L2t leads to a new non-Markov kinetic process. Thus it is shown for this model that each of the kinetic and the hydrodynamic process has a characteristic scaling, and the macroscopic scale invariance rigorously holds.

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