Symmetries and criticality of generalised van der Waals models

Abstract

We consider a family of thermodynamic models such that the energy density can be expressed as an asymptotic expansion in the scale formal parameter and whose terms are suitable functions of the volume density. We examine the possibility to construct solutions for the Maxwell thermodynamic relations relying on their symmetry properties and deduce the critical properties implied in terms of the dynamics of coexistence curves in the space of thermodynamic variables.

Complete integrability and equilibrium thermodynamics of biaxial nematic systems with discrete orientational degrees of freedom

“…Hence, phase transitions can be studied through the analysis of critical points of the equations (3.3). Similarly to the thermodynamic models studied in [85–87,90], order parameters ml can be viewed as solutions to a nonlinear integrable system of hydrodynamic type where coupling constants x, y, z, w and t play the role of, respectively, space and time variables. In this framework, state curves within the critical region of a phase transition are the analogue of shock waves of the hydrodynamic flow.…”

“…As pointed out in a number of papers [10,[82][83][84][85][86][87][88][89][90][91] the nature of the PDEs derived for the orientational order parameters suggests a natural interpretation of the singularities as classical shocks propagating in the space of thermodynamic variables. This allows to explain and, qualitatively, predict some features of the phase diagram based on the general properties of shock waves, as for example the occurrence of tricritical points as a collision and merging mechanism of two shock waves.…”

On solutions to a novel non-evolutionary integrable 1 + 1 PDE

“…170). But from results in [17] regarding the symmetry properties one realises that no local transformations can map (1) into these equations. Notwithstanding, the model can be linearised through the Cole-Hopf transformation (4) with real-valued parameter σ and potential φ, whose actual value and meaning may be suggested from the specific problem 3 .…”

“…Because of the aforementioned reasons, we are going to look for basic distinguishing attributes exhibited by solution to (1), by paying attention, in particular, to the presence of singularities. To do this, we exploit the knowledge of the symmetries identified in [17], so to perform a suitable reduction leading to a nonlinear ordinary differential equation (ODE). Such an ODE describes the solutions on the orbits of a one-dimensional symmetry subgroup, generated by the simultaneous Galilei and scaling transformation for the x, t, v variables, and it turns out to be connected with the Painlevé III (PIII) equation.…”

“…Implications for the corresponding functions v(x, t) solving equation (1) are in section 5. The most general case is considered in section 6 where, also bearing in mind findings 3 In the real fluid case dealt with in [1], for instance, σ turns out to be connected to the universal gas constant (or equivalently to the Boltzmann thermodynamic constant [17]) while φ plays the role of the statistical partition function. 4 The connections between statistical field models and the heat or KG equations for the underlying partition function have been pointed out in literature for archetypical complex systems such as the Curie-Weiss model for spins, the van der Waals model for real gases and generalisations, and the Maier-Saupe model and biaxial generalisations for nematic liquid crystals [18][19][20][21][22][23][24].…”