Langevin equation relaxation time calculations

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. 

No calculations yet have tackled this problem in all its complexity. Typically, it is assumed that the photodissociation produces some initial distribution of pairs, and then the subsequent time evolution of the unreacted pair probability is calculated. Even this more modest program has been carried out only at the diffusion and Langevin equation levels. We briefly comment on these results, since they indicate the magnitude of the solvent and velocity relaxation effects. 

The MD simulation eoupled with the Langevin thermostat simulates Rouse dynamics of a polymer chain. The Rouse relaxation time scales with the number of monomers on a chain as and it is necessary to perform at least cN (where constant c depends on the value of the integration time step At) integrations of the equation of motion for a ehain to completely renew its eonfiguration. During each time step, At, N N — l)/2 calculations of forces between monomers are performed. The CPU time required to do cN integrations of the equations of motion will grow with the number of monomers on a chain as xmd — N - Thus, the computational efficiency of MD simulation has the same N dependence as a MC simulation with only local moves. 

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