Time Correlation Functions for Self-Diffusion

The self-diffusion process is discussed by Dorfman, who writes the self-diffusion current and conservation law as 

For periodic boundary conditions we will understand the particle coordinates that enter the formulas to be the infinite-checkerboard description, e.g., Axi(0 = ARi(r) e so that the transformation of (52) to (54) holds. In making this identification, one also expects to minimize effects arising from the finite size of the system. 

Equations (52), (54), and (55) are all suitable for the calculation of the Pick s law transport coefficient. Indeed, Alder and Wainwright studied the velocity autocorrelation function (53) using summation time averaging for both hard disks and hard spheres and discovered the existence of the well-known long-time tails  

TTie D t) term in (51) is called the linear super-Bumett term and we have analogous correlation function expressions for it. Corresponding to (55) we have 

In order to develop some of the considerations involved in molecular dynamics calculations of time correlation functions, we will discuss in some detail the evaluation of the super-Burnett coefficient from Eq. (58). At 

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In the next four sections, we discuss the four principal types of application of molecular dynamics. Section 3 very briefly describes the problem of the approach to equilibrium. Section 4 deals with the evaluation of equilibrium thermodynamic functions through a discussion of the dynamical equation of state. In Section 5, we consider the evaluation of equilibrium time correlation functions, detailing the application of the combined Monte Carlo-molecular-dynamics method to the time correlation functions for self-diffusion. Section 6 deals with nonequilibrium molecular dynamics and in particular with a calculation for self-diffusion. 

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