### 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 c^{2} exp(-2ρct @#@). For a finite system, the function ψ_{N}(t) is periodic with period 2 L/c. We also show that for more general velocity distribution functions (particles can have velocities ±c_{i}, i = 1,...), ψ_{N}(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 ψ_{N}(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) |
---|---|

Pages (from-to) | 157-171 |

Number of pages | 15 |

Journal | Journal of Statistical Physics |

Volume | 6 |

Issue number | 2-3 |

DOIs | |

State | Published - Nov 1 1972 |

Externally published | Yes |

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### All Science Journal Classification (ASJC) codes

- Statistical and Nonlinear Physics
- Mathematical Physics

### Cite this

*Journal of Statistical Physics*,

*6*(2-3), 157-171. https://doi.org/10.1007/BF01023684

}

*Journal of Statistical Physics*, vol. 6, no. 2-3, pp. 157-171. https://doi.org/10.1007/BF01023684

**The velocity autocorrelation function of a finite model system.** / Lebowitz, Joel; Sykes, J.

Research output: Contribution to journal › Article

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.

UR - http://www.scopus.com/inward/record.url?scp=0040312790&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0040312790&partnerID=8YFLogxK

U2 - https://doi.org/10.1007/BF01023684

DO - https://doi.org/10.1007/BF01023684

M3 - Article

VL - 6

SP - 157

EP - 171

JO - Journal of Statistical Physics

JF - Journal of Statistical Physics

SN - 0022-4715

IS - 2-3

ER -