Second law violations, continuum mechanics, and permeability

Abstract

The violations of the second law are relevant as the length and/or time scales become very small. The second law then needs to be replaced by the fluctuation theorem and mathematically, the irreversible entropy is a submartingale. First, we discuss the consequences of these results for the axioms of continuum mechanics, arguing in favor of a framework relying on stochastic functionals of energy and entropy. We next determine a Lyapunov function for diffusion-type problems governed by stochastic rather than deterministic functionals of internal energy and entropy, where the random field coefficients of diffusion are not required to satisfy the positive definiteness everywhere. Next, a formulation of micropolar fluid mechanics is developed, accounting for the lack of symmetry of stress tensor on molecular scales. This framework is then applied to employed to show that spontaneous random fluctuations of the microrotation field will arise in Couette-and Poiseuille-type flows in the absence of random (turbulence-like) fluctuations of the classical velocity field. Finally, while the permeability is classically modeled by the Darcy law or its modifications, besides considering the violations of the second law, one also needs to account for the spatial randomness of the channel network, implying a modification of the hierarchy of scale-dependent bounds on the macroscopic property of the network.Keywords Continuum mechanics · Second law violations · Fluctuation theorem · Submartingale · Micropolar fluid · Permeability · Darcy law MotivationAs shown by theory, simulations, and experiments, the violations of the second law of thermodynamics are relevant where/when the length and/or time scales become very small. The second law must then be replaced by the fluctuation theorem [10,12,16,29,31]. [Strictly speaking, the fluctuation theorem stands for a group of such theorems (see e.g. Table 4.1 in [13])]. In effect, the second law holds on average, be it an ensemble average, or a spatial average over a sufficiently large domain, or a temporal average over a sufficiently large time interval. Interestingly, the second law violations may occur for up to several seconds (!) in, say, cholesteric liquids. While the work by physicists focuses on stochastic thermodynamics, our interest is in building these results into a continuum mechanics theory, i.e., in formulating stochastic continuum thermomechanics.From the standpoint of mathematics, the irreversible entropy is a submartingale [23]: a stochastic process conditioned by the entire past history, while it fluctuates randomly but grows on average; this is reviewed in Sect. 2. All these concepts and results have consequences for the foundations of continuum mechanics: since the fluctuation theorem is derived from the Axiom of Causality in statistical mechanics, this suggests Communicated by Victor Eremeyev, Peter Schiavone and Francesco dell'Isola.M. Ostoja-Starzewski (B)

Unified Mechanics Theory

“…Rivas and Martin-Delgado (2017) have also encountered a partial violation of the second law of thermodynamics in a quantum system known as Hofstadter lattice. Ostoja-Starzewski (2016) and Ostoja-Starzewski and Raghavan (2016) proved violations of the second law are relevant as the length and/or time scales become very small. The second law then needs to be replaced by the fluctuation theorem, and mathematically the irreversible entropy is a sub martingale.…”

Admitting Spontaneous Violations of the Second Law in Continuum Thermomechanics

“…As outlined in [13], the axioms (also called principles) of RCM need to be modified so as to admit a negative entropy production:…”

Non-equilibrium thermodynamics and stochasticity: a phenomenological look on Jarzynski’s equality

“…Performing the same protocol numerously, the values of the process works generate a probability distribution function p λ on W AB λ . The same protocol λ(t) generates by the stochastic properties of the material -introduced by the stochastic mapping L(λ, Θ) for the generalized forces [27]-different process works W AB λ which all together implement a probability distribution function p λ (W AB λ ) on the stochastic variable of the process work. Consequently, in the framework of Non-equilibrium Thermodynamics, the probability distribution function p λ (W AB λ ) is a measurable quantity which can be found out by performing a sufficently high number of Jarzynski processes always using the same protocol λ(t).…”