One-dimensional harmonic lattice caricature of hydrodynamics

R. L. Dobrushin, A. Pellegrinotti, Iouri M. Soukhov, L. Triolo

Research output: Contribution to journalArticle

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Abstract

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).

Original languageEnglish (US)
Pages (from-to)571-607
Number of pages37
JournalJournal of Statistical Physics
Volume43
Issue number3-4
DOIs
StatePublished - May 1 1986

Fingerprint

Spectral Density Matrix
Matrix Function
Density Function
Hydrodynamics
Harmonic
hydrodynamics
harmonics
Hydrodynamic Equations
Lattice Points
hydrodynamic equations
profiles
Local Equilibrium
Invariant
Linear Relation
Normal Modes
Equilibrium State
Quadratic form
Euler
fixing
Space-time

All Science Journal Classification (ASJC) codes

  • Statistical and Nonlinear Physics
  • Mathematical Physics

Cite this

Dobrushin, R. L. ; Pellegrinotti, A. ; Soukhov, Iouri M. ; Triolo, L. / One-dimensional harmonic lattice caricature of hydrodynamics. In: Journal of Statistical Physics. 1986 ; Vol. 43, No. 3-4. pp. 571-607.
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title = "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 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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One-dimensional harmonic lattice caricature of hydrodynamics. / Dobrushin, R. L.; Pellegrinotti, A.; Soukhov, Iouri M.; Triolo, L.

In: Journal of Statistical Physics, Vol. 43, No. 3-4, 01.05.1986, p. 571-607.

Research output: Contribution to journalArticle

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AU - Triolo, L.

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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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