Langevin equation approximate expressions

The relaxation equations for the time correlation functions are derived formally by using the projection operator technique [12]. This relaxation equation has the same structure as a generalized Langevin equation. The mode coupling theory provides microscopic, albeit approximate, expressions for the wavevector- and frequency-dependent memory functions. One important aspect of the mode coupling theory is the intimate relation between the static microscopic structure of the liquid and the transport properties. In fact, even now, realistic calculations using MCT is often not possible because of the nonavailability of the static pair correlation functions for complex inter-molecular potential. 

The time-dependent classical statistical mechanics of systems of simple molecules is reviewed. The Liouville equation is derived the relationship between the generalized susceptibility and time-correlation function of molecular variables is obtained and a derivation of the generalized Langevin equation from the Liouville equation is given. The G.L.E. is then simplified and/or approximated by introducing physical assumptions that are appropriate to the problem of rotational motion in a dense fluid. Finally, the well-known expressions for spectral intensity of infrared and Raman vibration-rotation bands are reformulated in terms of time correlation functions. As an illustration, a brief discussion of the application of these results to the analysis of spectral data for liquid benzene is presented. 

Of course, the problem of determining the time dependence of B(, p ) remains. At present, the physically interesting variables can a best be approximately modelled, for dense phases. However, many of these models are (or can be) obtained by simplifyin a rigorous, general equation of motion known as the generalized Langevin equation (GLE). We now show how this expression can be derived from the Liouville equation. 

This expression may determine the relationship between ea as a function of Aa in intense fields, e.g., for x > 3, when the approximation L(x) A = 1 - l/xA applies. However, it assumes classical statistics, so it will require a quantum correction for x values in this range, assuming that the Langevin function is still applicable. For smaller values of xA, Eq. (48) may be solved graphically. A more transparent expression for small xA values would be useful. Equation (42) with m = 1/2 leads to a quartic equation in ea which may be simplified by expansion to a cubic expression. A solution is the use of Booth s approximation ea and e0 n2, with substitution ofEq. (47) into Eq. (42). This gives a simple and useful approximation ... 

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