Scale invariance and some limits in transport phenomenology: Existence of a spontaneous scale

Abstract

In transport phenomenology it is a common practice to express equations for continuous quantities such as fluxes, current densities among others, in a dimensionless fashion, i.e. independent of scales. If from the physical point of view one respects the microscopic origin of fluids, then these equations, when scaled to the microscopic or particle level such as the mean free path or the mean inter-particle distance, should break scale invariance or invariance under dilatation transformation. In this chapter we show the possibility using a dimensionally extended space, where the extra dimensions allow us to define an asymmetry together with a normalization by a closed line integral defined by commutators of Poincaré and diffeomorphic conformal displacements applied to a density current, respectively. The asymmetry plays the role of an indicator of (spontaneous) symmetry breaking. We show how starting from a model without a reference scale that a scale emerges through spontaneous SO(4, 2)-symmetry breaking.