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

Abstract

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.

Minimum entropy production in microstructures

“…In section 4 we apply this principle, rather academical in its general form, to a regular network with bulk-like properties. This generalizes our prior treatment [8] of a chain. Future work along these lines should contribute to a realistic modelling of extended structures with pronounced quantum interference effects that are broken by inelastic processes.…”

“…Instead, we can use a variational principle: the entropy production is defined for any distribution of internal currents and/or densities and the correct distribution gives the minimum entropy production for the whole network. (This statement contradicts a point made in the introduction of [8].) The imagined states with higher entropy production are rather possible for the isolated scatterers but impossible if their mutual coupling is properly taken into account.…”

“…This principle applies directly to transport in the case of incoherently coupled scatterers of arbitrary internal complexity. It has been constructed at first for a transmitter between two resistive leads [6], then generalized for a network of scatterers and shown to reproduce correctly the well-known basic equations of the standard model [7,8], i.e. employing ideal leads, contacts and reservoirs.…”

“…employing ideal leads, contacts and reservoirs. Finally the current distribution well within a long chain was proven to be exactly the distribution that minimizes the variational functional [8]. The general form of this functional has been determined by a demand of additivity with respect to the different channels as well as to the coupled scatterers.…”

“…In this respect we have tentatively alluded to [8] the principle of minimum entropy production. In the papers mentioned, we have coined the denotation 'dissipation functional' or, in the two-terminal case, 'resistance functional'.…”