Transport processes in solids and linear-response theory

Abstract

It is argued that van Kampen's objections to linear-response theory do not apply to transport processes driven by quasi-forces (gradients in thermodynamic quantities like density, temperature, or average local velocity), but are applicable to calculations of response to real forces (externally imposed, like electric fields). A nonperturbative derivation of generalized Green-Kubo formulas is displayed. They clearly show a qualitative difference in behavior for small forces between the real and quasi-cases. A computer experiment on a simple classical model for diffusion and electric conductivity by independent electrons in a two-dimensional solid is described. The results are consistent with the predictions of linear-response theory in this case, with certain qualifications. The role of dissipation is crucial. Without it there can be no steady conduction. With a small amount of dissipation and small electric fields, the conductivity is related to the diffusivity in accord with linear-response theory, and the diffusion coefficient smoothly approaches a limit when the dissipation is reduced to zero. The key to the reconciliation of these results with van Kampen's objections is that the electric field exactly factors out of the expression for the response current, independent of any assumption of smallness or analyticity. It is shown that this is the case in both classical and quantum theory. It is noted in an appendix that the conductivity in the wind-tree model is pathological, violating linear-response theory.