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Lagrangian statistical mechanics applied to non-linear stochastic field equations

Author

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  • Edwards, Sam F.
  • Schwartz, Moshe

Abstract

We consider non-linear stochastic field equations such as the KPZ equation for deposition and the noise driven Navier–Stokes equation for hydrodynamics. We focus on the Fourier transform of the time dependent two-point field correlation, Φk(t). We employ a Lagrangian method aimed at obtaining the distribution function of the possible histories of the system in a way that fits naturally with our previous work on the static distribution. Our main result is a non-linear integro-differential equation for Φk(t), which is derived from a Peierls–Boltzmann type transport equation for its Fourier transform in time Φk,ω. That transport equation is a natural extension of the steady state transport equation, we previously derived for Φk(0). We find a new and remarkable result which applies to all the non-linear systems studied here. The long time decay of Φk(t) is described by Φk(t)∼exp(−a|k|tγ), where a is a constant and γ is system dependent.

Suggested Citation

  • Edwards, Sam F. & Schwartz, Moshe, 2002. "Lagrangian statistical mechanics applied to non-linear stochastic field equations," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 303(3), pages 357-386.
    • Handle: RePEc:eee:phsmap:v:303:y:2002:i:3:p:357-386
    DOI: 10.1016/S0378-4371(01)00479-4
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    Cited by:

    1. McComb, W.D. & Quinn, A.P., 2003. "Two-point, two-time closures applied to forced isotropic turbulence," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 317(3), pages 487-508.
    2. Frenkel, Gad, 2015. "Droplet shape fluctuations in agitated emulsions—Beyond the dilute limit," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 427(C), pages 251-261.

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