Towards Continuum Mechanics with Spontaneous Violations of the Second Law of Thermodynamics
As dictated by modern statistical physics, the second law is to be replaced by the fluctuation theorem (FT) 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 a formulation of stochastic continuum thermomechanics, where the entropy evolves as a submartingale while the dissipation function is consistent with the FT. A summary is then given of the behavior of an atomic fluid in Couette flow, studied using a combination of kinetic theory, hydrodynamic theory, and molecular dynamics. Overall, the developed framework may be applied in many fields involving fluid flow and heat conduction on very small spatial scales.
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This material is based upon work partially supported by the NSF under grants CMMI-1462749 and IIP-1362146 (I/UCRC on Novel High Voltage / Temperature Materials and Structures).
- Doob JL (1953) Stochastic Processes. Wiley New YorkGoogle Scholar
- Edelen D (1974) Primitive thermodynamics: a new look at the Clausius-Duhem inequality. Int J Eng Sc 12(2):121–141Google Scholar
- Edelen DGB (1973) On the existence of symmetry relations and dissipation potentials. Arch Rational Mech Anal 51(3):218–227Google Scholar
- Evans DJ, Searles DJ (2002) The fluctuation theorem. Adv Phys 51(7):1529–1585Google Scholar
- Jaroszkiewicz G (2014) Principles of Discrete Time Mechanics. Cambridge University PressGoogle Scholar
- Malyarenko A, Ostoja-Starzewski M (2014) Statistically isotropic tensor random fields: correlation structures. Math Mech Complex Sys (MEMOCS) 2(2):209–231Google Scholar
- Malyarenko A, Ostoja-Starzewski M (2016) Spectral expansions of homogeneous and isotropic tensor-valued random fields. ZAMP 67(3):59Google Scholar
- Maugin GA (1999) The Thermomechanics of Nonlinear Irreversible Behaviors: An Introduction. World ScientificGoogle Scholar
- Ostoja-Starzewski M (2016) Second law violations, continuum mechanics, and permeability. Cont Mech Thermodyn 28(1-2):489Google Scholar
- Ostoja-Starzewski M (2017a) Admitting spontaneous violations of the second law in continuum thermomechanics. Entropy 19(2):78Google Scholar
- Ostoja-Starzewski M (2017b) Continuum physics with violations of the second law of thermodynamics. In: Math. Model. Sol. Mech., Springer, pp 181–192Google Scholar
- Ostoja-Starzewski M, Malyarenko A (2014) Continuum mechanics beyond the second law of thermodynamics. In: Proc. R. Soc. A, p 20140531Google Scholar
- Raghavan BV, Ostoja-Starzewski M (2017) Shear-thinning of molecular fluids in Couette flow. Phys Fluids 29(2):023,103Google Scholar
- Raghavan BV, Karimi P, Ostoja-Starzewski M (2018) Stochastic characteristics and Second Law violations of atomic fluids in Couette flow. Physica A: Statistical Mechanics and its Applications 496:90–107Google Scholar
- Searles DJ, Evans DJ (2001) Fluctuation theorem for heat flow. Int J Thermophys 22(1):123–134Google Scholar
- Ziegler H (2012) An Introduction to Thermomechanics. ElsevierGoogle Scholar
- Ziegler H, Wehrli C (1987) The derivation of constitutive relations from the free energy and the dissipation function. Adv Appl Mech 25:183–238Google Scholar