Thermodynamic instabilities in one dimensional particle lattices: a finite-size scaling approach

TL;DR

This paper investigates phase transitions in one-dimensional particle lattices, specifically in the Peyrard-Bishop DNA model, using finite-size scaling and numerical methods to demonstrate the occurrence of thermodynamic instabilities.

Contribution

It introduces a finite-size scaling approach with Gauss-Hermite quadratures to analyze phase transitions in the Peyrard-Bishop model of DNA denaturation.

Findings

Demonstrates the occurrence of a phase transition in the model.

Derives critical exponents for the transition.

Employs a novel numerical approach to analyze the transfer integral spectrum.

Abstract

One-dimensional thermodynamic instabilities are phase transitions not prohibited by Landau's argument, because the energy of the domain wall (DW) which separates the two phases is infinite. Whether they actually occur in a given system of particles must be demonstrated on a case-by-case basis by examining the (non-) analyticity properties of the corresponding transfer integral (TI) equation. The present note deals with the generic Peyrard-Bishop model of DNA denaturation. In the absence of exact statements about the spectrum of the singular TI equation, I use Gauss-Hermite quadratures to achieve a single-parameter-controlled approach to rounding effects; this allows me to employ finite-size scaling concepts in order to demonstrate that a phase transition occurs and to derive the critical exponents.