Continuum physics with violations of the second law of thermodynamics

As dictated by the modern statistical physics, the second law is to be replaced by the fluctuation theorem on very small length and/or time scales. This means that the deterministic continuum thermomechanics must be generalized to a stochastic theory allowing randomly spontaneous violations of the Clausius–Duhem inequality to take place anywhere in the material domain. This paper outlines possible extensions of stochastic continuum thermomechanics in coupled field problems: (i) thermoviscous fluids, (ii) thermo-elastodynamics, and (iii) poromechanics with dissipationwithin the skeleton, the fluid, and the temperature field. Linear dissipative processes are being considered, with the thermodynamic orthogonality providing the average constitutive response and the fluctuation theorem providing the violations of the second law of thermodynamics. Special attention is paid to the fact that one can develop hyperbolic theories (i.e. free of the paradox of infinite speeds of signal transmission) while working with the Fourier-type conduction for which the fluctuation theorem has already been developed.