One-dimensional harmonic lattice caricature of hydrodynamics

We derive the hydrodynamic (Euler) approximation for the harmonic time evolution of infinite classical oscillator system on one-dimensional lattice ℤ¹ 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)F(θ), θ∃[- π, π). Time evolution of a space "profile" of local equilibrium parameters is described by a space-time SDMF F(t;x, θ) t, x∃R¹. The hydrodynamic equation for F(t; x, θ) which we derive in this paper means that the "normal mode" profiles indexed by θ 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, θ) and a family P^e{open}, e{open}>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 e{open}^-1x in the state P^e{open} is approximately described by the SDMF F(x, θ); (b) The covariance (on large distances) decreases with distance quickly enough and uniformly in e{open}. Given nonzero t∃R¹, we consider the states P_e{open}-1τ^e{open}, e{open}>0, describing the system at the time moments e{open}^-1t during its harmonic time evolution. We check that the covariance at lattice points close to e{open}^-1x in the state P_e{open}-1τ^e{open} is approximately described by a SDMF F(t;x, θ) and establish the connection between F(t; x, θ) and F(x,θ).