Screening like-charges in one-dimensional Coulomb systems: Exact results

TL;DR

This paper provides exact analytical results for a one-dimensional Coulomb system demonstrating how like-charges can attract each other mediated by counterions, highlighting the influence of system parameters on long-range interactions.

Contribution

It derives explicit partition functions and analyzes the effects of charge asymmetry and counterion number parity in a simplified 1D Coulomb model.

Findings

Exact partition function obtained for the 1D Coulomb system.

Long-range attraction depends on the parity of the number of counterions.

Effective forces and counterion profiles are explicitly characterized.

Abstract

The possibility that like-charges can attract each other under the mediation of mobile counterions is by now well documented experimentally, numerically, and analytically. Yet, obtaining exact results is in general impossible, or restricted to some limiting cases. We work out here in detail a one dimensional model that retains the essence of the phenomena present in higher dimensional systems. The partition function is obtained explicitly, from which a wealth of relevant quantities follow, such as the effective force between the charges or the counterion profile in their vicinity. Isobaric and canonical ensembles are distinguished. The case of two equal charges screened by an arbitrary number $N$ of counterions is first studied, before the more general asymmetric situation is addressed. It is shown that the parity of $N$ plays a key role in the long range physics.