The velocity autocorrelation function of a finite model system

Joel Lebowitz, J. Sykes

Research output: Contribution to journalArticle

13 Citations (Scopus)

Abstract

We investigate in detail the dependence of the velocity autocorrelation function of a one-dimensional system of hard, point particles with a simple velocity distribution function (all particles have velocities ±c) on the size of the system. In the thermodynamic limit, when both the number of particles N and the length of the "box"L approach infinity and N/L → ρ, the velocity autocorrelation function ψ(t) is given simply by c2 exp(-2ρct @#@). For a finite system, the function ψN(t) is periodic with period 2 L/c. We also show that for more general velocity distribution functions (particles can have velocities ±ci, i = 1,...), ψN(t) is an almost periodic function of t. These examples illustrate the role of the thermodynamic limit in nonequilibrium phenomena: We must keep t fixed while letting the size of the system become infinite to obtain an auto-correlation function, such as ψ(t), which decays for all times and can be integrated to obtain transport coefficients. For any finite system, our ψN(t) will be "very close" to ψ(t) as long as t is small compared to the effective "size" of the system, which is 2 L/c for the first model.

Original languageEnglish (US)
Pages (from-to)157-171
Number of pages15
JournalJournal of Statistical Physics
Volume6
Issue number2-3
DOIs
StatePublished - Nov 1 1972
Externally publishedYes

Fingerprint

Finite Models
Autocorrelation Function
autocorrelation
Thermodynamic Limit
Velocity Distribution
Distribution Function
velocity distribution
distribution functions
Almost Periodic Functions
periodic functions
thermodynamics
Transport Coefficients
One-dimensional System
Infinite Systems
infinity
Non-equilibrium
boxes
transport properties
Infinity
Decay

All Science Journal Classification (ASJC) codes

  • Statistical and Nonlinear Physics
  • Mathematical Physics

Cite this

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title = "The velocity autocorrelation function of a finite model system",
abstract = "We investigate in detail the dependence of the velocity autocorrelation function of a one-dimensional system of hard, point particles with a simple velocity distribution function (all particles have velocities ±c) on the size of the system. In the thermodynamic limit, when both the number of particles N and the length of the {"}box{"}L approach infinity and N/L → ρ, the velocity autocorrelation function ψ(t) is given simply by c2 exp(-2ρct @#@). For a finite system, the function ψN(t) is periodic with period 2 L/c. We also show that for more general velocity distribution functions (particles can have velocities ±ci, i = 1,...), ψN(t) is an almost periodic function of t. These examples illustrate the role of the thermodynamic limit in nonequilibrium phenomena: We must keep t fixed while letting the size of the system become infinite to obtain an auto-correlation function, such as ψ(t), which decays for all times and can be integrated to obtain transport coefficients. For any finite system, our ψN(t) will be {"}very close{"} to ψ(t) as long as t is small compared to the effective {"}size{"} of the system, which is 2 L/c for the first model.",
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The velocity autocorrelation function of a finite model system. / Lebowitz, Joel; Sykes, J.

In: Journal of Statistical Physics, Vol. 6, No. 2-3, 01.11.1972, p. 157-171.

Research output: Contribution to journalArticle

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