Time-convolutionless projection operator formalism for elimination of fast variables. Applications to Brownian motion

The relation between the time-convolutionless projection operator formalism and the methods developed by Kubo and van Kampen in the context of linear stochastic differential equations is discussed in detail. It is shown how this formalism may be used to develop a systematic procedure for the elimination of “fast variables”. This procedure is applied to the case of a free Brownian particle and to that of a heavily damped Brownian particle in the presence of an external force to obtain a reduced description in terms of the position variable: the results are found to be in complete agreement with the previous works on this problem. The rotational Brownian motion of an asymmetrical top in the presence of an external torque is also considered and an equation for the reduced probability distribution is derived which contains all the previous works on this problem as special cases.

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