One-dimensional harmonic liquid: A Fokker-Planck description of fluctuations from the nonequilibrium steady state

Abstract

The one-dimensional harmonic liquid, subject to a temperature gradient, is studied using the Fokker-Planck (FP) equation. A formalism is set up for solution of the FP equation and calculation of (1) the phase-space distribution function for the nonequilibrium steady state (NESS), (2) the conditional probability for evolution through phase space in the NESS, (3) phase-function averages in the NESS, (4) the correlation of phase functions in the NESS, etc. For the harmonic liquid this formalism can be implemented without approximation. Some properties of the harmonic liquid in equilibrium are examined to illustrate use of the formalism; so are some phase-function averages in the NESS. The displacement-displacement correlation function $D(k,\ensuremath{\omega})$ in the NESS is calculated and found to have the well-established, interesting amplitude and frequency dependence. The completeness of the Fokker-Planck description makes it possible to identify the physical processes responsible for the behavior of $D(k,\ensuremath{\omega})$ and the dynamic structure factor $S(k,\ensuremath{\omega})$. The interesting features of light scattering from a liquid subject to a temperature gradient, seen in $S(k,\ensuremath{\omega})$ or $D(k,\ensuremath{\omega})$, are due to light scattering from width fluctuations induced by the gradient; the interesting features in light scattering from a liquid-supporting shear are due to an attenuation mechanism that arises because of the velocity field induced by the shear. The results in this paper constitute a partial demonstration of the usefulness of the method employed in handling the Fokker-Planck equation.