Langevin Equation for General Harmonic Heat Bath. II

概要

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Statistical dynamical motion of a heavy particle in an n-dimensional (n-D) cubic lattice is investigated on the basis of the Langevin-equation formalism presented in the preceding paper I. The random force which is closely related to the memory functions characterizing the stochastic process {u_O(t),P_O(t)} of an isotopic impurity, is shown to be determined only by the dynamics of the reference system with an appropriately chosen Hamiltonian. Through our formalism, the memory functions can be calculated in closed forms. For a simple cubic lattice Rubin's result for the Brownian motion in 1D and 3D lattices can be derived from our results by applying the first-order long time approximation. It is shown that for bodycentered cubic and face-centered cubic lattices, the Brownian motion exhibits an oscillatory behaviour in the same approximation as in the case of 3D simple cubic lattice. Finally the effect of regular simple cubic lattices with temperature T_l and T_γ attached to a finite massdisordered chain on the motion of the latter is investigated. The result is that the model which is usually considered in discussing the heat conduction can be justified under certain conditions.
1973-04-25