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We consider the motion of a test particle in a one-dimensional system ofequal-mass point particles. The test particle plays the role of a microscopic"piston" that separates two hard-point gases with different concentrations andarbitrary initial velocity distributions. In the homogeneous case when thegases on either side of the piston are in the same macroscopic state, wecompute 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 shownthat the occurrence of a non-vanishing probability mass at zero velocity isnecessary for the occurrence of a long-time tail in C(t). The conditions underwhich this is a $t^{-3}$ tail are determined. Turning to the inhomogeneoussystem with different macroscopic states on either side of the piston, wedetermine its effective diffusion coefficient from the asymptotic behavior ofthe variance of its position, as well as the leading behavior of the othermoments about the mean. Finally, we present an interpretation of the effectivenoise arising from the dynamics of the two gases, and thence that of thestochastic process to which the position of any particle in the system reducesin the thermodynamic limit.Comment: 22 files, 2 eps figures. Submitted to PR

Velocity correlations, diffusion, and stochasticity in a one-dimensional system