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The minimal principle for the current distribution in microstructures and the resistance in long incoherently coupled chains

As in the Landauer - Büttiker approach to transport, a transmitter is, at fixed energy, characterized by its reflection and transmission coefficients. Generalizing a prior approach, we establish a variational principle to determine, without magnetic field, the distribution of the current over the different channels if the transmitter is coupled incoherently to its surroundings. For transmitters coupled with each other, a general demand of additivity defines the form of the variational functional. Contacts are treated as special transmitters, and, as a test, the results of the standard model are reproduced. A typical serial resistance is defined as the resistance within a long incoherently coupled chain, with no regard to contacts and reservoirs. This resistance is strictly additive. It is shown that well within the chain a relaxed current and density distribution is established that is independent of the conditions at the ends of the chain. This distribution coincides with the optimal distribution that minimizes the resistance of each of the single transmitters that are the building blocks of the chain. For a sufficiently long chain, the serial resistance is determined by the linear dependence of the inverse total transmission on the number of single transmitters. The relaxational behaviour of a long chain implies corresponding features of the reflection and transmission matrices in the asymptotic regime, especially a factorization of the transmission matrix, expressing memory loss with respect to the ingoing and outgoing channels.