The sine-Gordon chain. II. Nonequilibrium statistical mechanics

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Abstract
A calculation of the current carried on the sine-Gordon chain in an external field with viscous damping is formulated in terms of the solution of a Smolouchowski equation. The Smolouchowski equation is solved by employing its relationship to the BBGKY hierarchy and by using transfer-integral techniques. The method of solution is not specific to the sine-Gordon chain and can be usefully employed in examining other nonlinear systems. The solution of the Smolouchowski equation takes the form of a self-consistent equation for the single-particle distribution function; the solution describes the system for all values of the external field. At low field, the current is proportional to the field and due to thermally activated phase solitons; at high field, the current is proportional to the field and the conductivity many orders of magnitude greater than the low-field conductivity. A region of very nonlinear conduction separates the low-field and high-field regimes. It is possible to determine the effects of various perturbations on the conductivity of the sine-Gordon chain.