# 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 language English (US) 157-171 15 Journal of Statistical Physics 6 2-3 https://doi.org/10.1007/BF01023684 Published - Nov 1 1972 Yes

### 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.",
author = "Joel Lebowitz and J. Sykes",
year = "1972",
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language = "English (US)",
volume = "6",
pages = "157--171",
journal = "Journal of Statistical Physics",
issn = "0022-4715",
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In: Journal of Statistical Physics, Vol. 6, No. 2-3, 01.11.1972, p. 157-171.

Research output: Contribution to journalArticle

TY - JOUR

T1 - The velocity autocorrelation function of a finite model system

AU - Lebowitz, Joel

AU - Sykes, J.

PY - 1972/11/1

Y1 - 1972/11/1

N2 - 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.

AB - 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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