Continuum Mechanics Beyond the Second Law of Thermodynamics

Continuum mechanics is a branch of classical mechanics that deals with the analysis of the kinematics and the mechanical behavior of materials modeled as a continuous mass rather than as discrete particles. Modeling of objects (such as fluids and solids) as continua assumes that the substance of the object completely fills the space it occupies. However, on length scales (much) greater than that of inter-atomic distances, such models are highly accurate. Fluid and solid continuum mechanics provide bases for a multitude of models of aerodynamics, hydromechanics, structural mechanics and more, which in turn serve as guidance for design, manufacture, and operation of all engineering structures, as well as for understanding of continuum-type biological systems. While continuum mechanics is based on the laws of classical physics, including the Second Law of Thermodynamics, the experimental and theoretical results established over the past decades in statistical physics indicate that, on very small space and time scales, the entropy production rate may become negative. This motivates a generalization of continuum mechanics. In fact, in some cases (such as cholesteric liquids) the Second Law can be violated on macroscopic times – for up to a few seconds!In view of the fluctuation theorem describing such violations, it is recognized that the evolution of entropy at a continuum point is stochastically (not deterministically) conditioned by the past history. Hence, the continuum mechanics’ axiom of Clausius-Duhem inequality is replaced by a submartingale model, which, by the Doob decomposition theorem, allows classification of thermomechanical processes into four types depending on whether they are conservative or not and/or conventional continuum mechanical or not. Incidentally, the theory of martingales was developed in the Altgeld Hall by one of the great mathematicians of the 20^th century: Joseph L. Doob. His martingales, submartingales, and supermartingales have long been known to describe the games of chance and, hence, gambling and financial markets … with all their splendors and miseries. Now, stochastic generalizations of thermomechanics are constructed, with explicit models formulated for Newtonian fluids having parabolic or hyperbolic heat conduction. Several random field models of the martingale component, possibly including spatial fractal and Hurst effects, are proposed. Examples of situations where violations of the Second Law are relevant are offered by acceleration wavefronts and permeability of nano-porous media.