That "the second law is of the nature of strong probability a*| not an absolute certainty" was already recognized by J.C. Maxwell. However, it has only been in the past two decades that theoretical, simulation, and experimental results in physics in support of that old statement have been generated. Fundamentally, there is a non-zero probability of negative entropy production rate on very small length scales and (rather) short times. The Second Law then needs to be replaced by a fluctuation theorem and, mathematically, the irreversible entropy is recognized to evolve as a submartingale.

First, we discuss the consequences of these results for modifying the axioms of continuum mechanics, arguing in favor of a framework not relying on stress, heat flux, free energy, and entropy as functionals of history of motion. Rather, we are led to stochastic generalizations of thermomechanics in the vein of either thermodynamic orthogonality or primitive thermodynamics, with explicit models formulated for Netwonian fluids having parabolic or hyperbolic heat conduction. The continuum mechanics requires the submartingale to be parametrized by spatial coordinates, so that the martingale component is a space-time random field. This field can also account for fractal and Hurst effects. Other settings we have considered so far, and which have bearing on nanoscience/nanotechnology applications, include the blow-up of acceleration wavefronts, stability of diffusion problems, micropolar fluid mechanics, and permeability of random media.