Sum Rules, Kramers-Kronig Relations, and Transport Coefficients in Charged Systems

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Abstract
A number of difficulties in previous treatments of the properties of linear homogeneous charged systems are discussed. With respect to the Kramers-Kronig relations and sum rules, it is argued that since the nonlocal electrical conductivity is the ratio of the current to the total field, a quantity which is physical but not necessarily arbitrary, the usual proof of the Kramers-Kronig relation fails. A valid proof, based on the dynamical properties of the electromagnetic field, is presented for the transverse electrical conductivity. On the other hand, it is shown by counterexample that there can be no general proof for the conventionally defined longitudinal electrical conductivity. Part of the difficulty may be ascribed to the failure of the conventional definition to account properly for the change in chemical potential which accompanies a longitudinal spatially varying field and charge distribution. A nonlocal quantity more closely related to the ratio of the longitudinal current to the longitudinal field plus the chemical potential gradient is shown to have more desirable causality properties. Rigorous expressions for the other transport coefficients (the thermopower and thermal conductivity) of simple charged systems are also obtained. These expressions differ substantially from the usually rigorous integral expressions associated with Kubo because of the long-range electromagnetic forces. A compilation of sum rules is included and a number of recurring pitfalls noted.