- Access by Xinjiang University
Langevin equation for the extended Rayleigh model with an asymmetric bath
Phys. Rev. E 69, 021112 – Published 27 February, 2004
DOI: https://doi.org/10.1103/PhysRevE.69.021112
Abstract
In this paper a one-dimensional model of two infinite gases separated by a movable heavy piston is considered. The nonlinear Langevin equation for the motion of the piston is derived from first principles for the case when the thermodynamic parameters and/or the molecular masses of gas particles on the left and right sides of the piston are different. Microscopic expressions involving time correlation functions of the force between bath particles and the piston are obtained for all parameters appearing in the nonlinear Langevin equation. It is demonstrated that the equation has stationary solutions corresponding to directional fluctuation-induced drift in the absence of systematic forces. In the case of ideal gases interacting with the piston via a quadratic repulsive potential, the model is exactly solvable and explicit expressions for the kinetic coefficients in the nonlinear Langevin equation are derived. The transient solution of the nonlinear Langevin equation is analyzed perturbatively and it is demonstrated that previously obtained results for systems with the hard-wall interaction are recovered.
References (9)
- A.V. Plyukhin and J. Schofield, Phys. Rev. E 68, 041107 (2003).
- S. Kambayashi and Y. Hiwatari, Phys. Rev. E 49, 1251 (1994), and references therein.
- See, for example, Eq. (50) in Ref. [1]
- E. Lieb, Physica A 263, 491 (1999); Ch. Gruber, Europhys. Lett. 20, 259 (1999); Ch. Gruber, Séverine Pache, and Annick Lesne, J. Stat. Phys. 108, 669 (2002), ibid.N. Chernov and J.L. Lebowitz, 109, 507 (2002); ibid.N. Chernov, J.L. Lebowitz, and Ya. Sinai, 109, 529 (2002); ibid.C. Boldrighini, S. Frigio, and D. Tognetti, 108, 703 (2002).
- J. Piasecki and Ch. Gruber, Physica A 265, 463 (1999), ibid.Ch. Gruber and J. Piasecki, 268, 412 (1999).
- Ch. Gruber and L. Frachebourg, Physica A 272, 392 (1999); E. Kestemont, C. Van den Broeck, and M.M. Mansour, Europhys. Lett. 49, 143 (2000); T. Munakata and H. Ogawa, Phys. Rev. E 64, 036119 (2001).
- P. Mazur and I. Oppenheim, Physica A 50, 241 (1970).
- It is shown in Ref. [1] that for the case of the piston in a bath of ideal gas particles, appears in the expansion of the generalized Langevin equation via the parameter where N is a number of bath particles in the interaction shell around the piston (provided The actual parameter controlling separation of time scales in this case is therefore
- This assumption justifies the parabolic potential model and is similar to that used in the linear theory of lattice vibrations when displacements of atoms are assumed to be small compared to the lattice spacing.