Dynamics of nonlinear systems: The heavy damping limit

The Smoluchowski equation for diffusion through phase space is solved in terms of the eigen-states of the relaxation operator equation. Formally exact expressions for single-particle averages, two-particle averages, the linear response, etc., are found. An Einstein relation is established between the diffusion constant and the mobility that describes a displacement current. This formal apparatus is able to be implemented if the eigenvalues and eigenstates of the relaxation operator equation are available. The many-particle wave functions that solve the relaxation operator equation are found by employing a variational ansatz. The low-lying excited states of the relaxation operator problem are able to be built up from the many-particle ground state and from the single-particle states that describe the equilibrium statistical mechanics. A number of problems are examined to illustrate the content of the formal solution to the Smoluchowski equation and to provide a test of the adequacy of the variational ansatz. These problems are (a) two single-particle problems (a particle in a harmonic potential and a particle in a sinusoidal potential), (b) two exactly soluble problems, i.e., problems for which the relaxation operator equation can be solved exactly (a linear chain of harmonically coupled particles and this chain with each particle in a harmonic external potential), and (c) the ${\phi }^4$ chain and the sine-Gordon chain. Comparison of the exact solution and the variational solution to (b) shows the variational solution to give a good description of the low-lying excited states. For the latter two problems the linear response, diffusion constant, dynamic structure factor, etc., are calculated. Particular attention is given to the role of kinks in the dynamics of the ${\phi }^4$ chain and the sine-Gordon chain.