We study dynamics and thermodynamics of ion transport in narrow, water-filled channels, considered as effective 1D Coulomb systems. The long range nature of the inter-ion interactions comes about due to the dielectric constants mismatch between the water and the surrounding medium, confining the electric filed to stay mostly within the water-filled channel. Statistical mechanics of such Coulomb systems is dominated by entropic effects which may be accurately accounted for by mapping onto an effective quantum mechanics. In presence of multivalent ions the corresponding quantum mechanics appears to be non-Hermitian. In this review we discuss a framework for semiclassical calculations for the effective non-Hermitian Hamiltonians. Non-Hermiticity elevates WKB action integrals from the real line to closed cycles on a complex Riemann surfaces where direct calculations are not attainable. We circumvent this issue by applying tools from algebraic topology, such as the Picard-Fuchs equation. We discuss how its solutions relate to the thermodynamics and correlation functions of multivalent solutions within narrow, water-filled channels.
Bibliographical noteFunding Information:
A.K. was supported by NSF grants DMR-2037654. T.G. acknowledges funding from the Institute of Science and Technology (IST) Austria, and from the European Union?s Horizon 2020 research and innovation program under the Marie Sk?odowska-Curie Grant Agreement No. 754411.
Funding: A.K. was supported by NSF grants DMR-2037654. T.G. acknowledges funding from the Institute of Science and Technology (IST) Austria, and from the European Union’s Horizon 2020 research and innovation program under the Marie Skłodowska-Curie Grant Agreement No. 754411.
© 2021 by the authors. Licensee MDPI, Basel, Switzerland.
- Algebraic topology
- Ion transport
- Non-Hermitian Hamiltonians
- Semiclassical methods
- Statistical mechanics
PubMed: MeSH publication types
- Journal Article