### Abstract

We derive the hydrodynamic (Euler) approximation for the harmonic time evolution of infinite classical oscillator system on one-dimensional lattice ℤ^{1} 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∃R^{1}. 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}^{-1}x 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∃R^{1}, we consider the states P_{e{open}-1τ}^{e{open}}, e{open}>0, describing the system at the time moments e{open}^{-1}t during its harmonic time evolution. We check that the covariance at lattice points close to e{open}^{-1}x 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).

Original language | English (US) |
---|---|

Pages (from-to) | 571-607 |

Number of pages | 37 |

Journal | Journal of Statistical Physics |

Volume | 43 |

Issue number | 3-4 |

DOIs | |

State | Published - May 1 1986 |

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### All Science Journal Classification (ASJC) codes

- Statistical and Nonlinear Physics
- Mathematical Physics

### Cite this

*Journal of Statistical Physics*,

*43*(3-4), 571-607. https://doi.org/10.1007/BF01020654

}

*Journal of Statistical Physics*, vol. 43, no. 3-4, pp. 571-607. https://doi.org/10.1007/BF01020654

**One-dimensional harmonic lattice caricature of hydrodynamics.** / Dobrushin, R. L.; Pellegrinotti, A.; Soukhov, Iouri M.; Triolo, L.

Research output: Contribution to journal › Article

TY - JOUR

T1 - One-dimensional harmonic lattice caricature of hydrodynamics

AU - Dobrushin, R. L.

AU - Pellegrinotti, A.

AU - Soukhov, Iouri M.

AU - Triolo, L.

PY - 1986/5/1

Y1 - 1986/5/1

N2 - 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 Pe{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 Pe{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 Pe{open}-1τe{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 Pe{open}-1τe{open} is approximately described by a SDMF F(t;x, θ) and establish the connection between F(t; x, θ) and F(x,θ).

AB - 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 Pe{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 Pe{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 Pe{open}-1τe{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 Pe{open}-1τe{open} is approximately described by a SDMF F(t;x, θ) and establish the connection between F(t; x, θ) and F(x,θ).

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U2 - 10.1007/BF01020654

DO - 10.1007/BF01020654

M3 - Article

VL - 43

SP - 571

EP - 607

JO - Journal of Statistical Physics

JF - Journal of Statistical Physics

SN - 0022-4715

IS - 3-4

ER -