Balakrishnan V, Bena I, Van Den Broeck C
Centre for Nonlinear Phenomena and Complex Systems, Université Libre de Bruxelles, C.P. 231, Boulevard du Triomphe, 1050 Brussels, Belgium.
Phys Rev E Stat Nonlin Soft Matter Phys. 2002 Mar;65(3 Pt 1):031102. doi: 10.1103/PhysRevE.65.031102. Epub 2002 Feb 8.
We consider the motion of a test particle in a one-dimensional system of equal-mass point particles. The test particle plays the role of a microscopic "piston" that separates two hard-point gases with different concentrations and arbitrary initial velocity distributions. In the homogeneous case when the gases on either side of the piston are in the same macroscopic state, we compute and analyze the stationary velocity autocorrelation function C(t). Explicit expressions are obtained for certain typical velocity distributions, serving to elucidate in particular the asymptotic behavior of C(t). It is shown that the occurrence of a nonvanishing probability mass at zero velocity is necessary for the occurrence of a long-time tail in C(t). The conditions under which this is a t(-3) tail are determined. Turning to the inhomogeneous system with different macroscopic states on either side of the piston, we determine its effective diffusion coefficient from the asymptotic behavior of the variance of its position, as well as the leading behavior of the other moments about the mean. Finally, we present an interpretation of the effective noise arising from the dynamics of the two gases, and thence that of the stochastic process to which the position of any particle in the system reduces in the thermodynamic limit.
我们考虑一个由等质量点粒子组成的一维系统中测试粒子的运动。该测试粒子起到微观“活塞”的作用，它将两种具有不同浓度和任意初始速度分布的硬点气体分隔开。在均匀情况下，即活塞两侧的气体处于相同的宏观状态时，我们计算并分析稳态速度自相关函数(C(t))。对于某些典型的速度分布，我们得到了明确的表达式，特别用于阐明(C(t))的渐近行为。结果表明，在(C(t))中出现长时间尾部的情况下，在零速度处出现非零概率质量是必要的。确定了出现(t^{-3})尾部的条件。转向活塞两侧具有不同宏观状态的非均匀系统，我们根据其位置方差的渐近行为以及关于均值的其他矩的主导行为来确定其有效扩散系数。最后，我们对由两种气体的动力学产生的有效噪声进行了解释，并由此对系统中任何粒子在热力学极限下位置所简化成的随机过程进行了解释。
