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

One-dimensional thermodynamic instabilities are phase transitions, not prohibited by Landau's argument because the energy of the domain wall 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 properties of the corresponding singular transfer integral (TI) equation. The present work 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.