One-dimensional harmonic lattice caricature of hydrodynamics

Abstract

We derive the hydrodynamic (Euler) approximation for the harmonic time evolution of infinite classical oscillator system on one-dimensional lattice Z. It is known that equilibrium (i.e., time-invariant attractive) states for this model are translationally invariant Gaussian ones, with the mean 0, which satisfy some linear relations involving the interaction quadratic form. The natural ''parameter'' characterizing equilibrium states is the spectral density matrix function (SDMF) Ftheta, theta EPSILON (-..pi.., ..pi..). Time evolution of a space ''profile'' of local equilibrium parameters is described by a space-time SDMF F(t; x, theta) t, x EPSILON R/sup 1/. The hydrodynamic equation for F(t; x, theta) which we derive in this paper means that the ''normal mode'' profiles indexed by theta are moving according to linear laws and are mutually independent. The procedure of deriving the hydrodynamic equation is the following: We fix an initial SDMF profile F(x, theta) and a family (P /sup ue/, EPSILON>0) of mean 0 states which satisfy the two conditions imposed on the covariance of spins at various lattice points: (a) the covariance at points ''close'' to the value EPSILON/sup -1/ x in the state PEPSILON is approximately described by the SDMF F(x, theta); (b) The covariance (on large distances) decreasesmore » with distance quickly enough and uniformly in EPSILON. Given nonzero t EPSILON R/sup 1/ we consider the states P /sup ue/ /sub ue/ -1 /sub t/, e>0, describing the system at the time moments EPSILON/sup -1/ /sub t/ during its harmonic time evolution. We check that the covariance at lattice points close to EPSILON/sup -1/x in the state P /sup ue/ /sub ue/ -1 /sub t/ is approximately described by a SDMF F(t;x,theta) and establish the connection between F(t;,x,theta) and F(x, theta).« less