  We study dynamics and thermodynamics of ion 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 lipids,
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 corresponding 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 long
water-filled channels.
