Reciprocity relations in driven dissipative systems

Abstract

Almost a century ago Onsager found a universal feature of all dissipative systems describing approach to thermodynamic equilibrium, the reciprocal relations. They hold for linear differential equations describing an evolution of macrovariables in vicinity of equilibrium states. It was assumed implicitly that underlying microdynamics is Hamiltonian and ergodic. The reversibility of Hamiltonian equations was one of the key features of microdynamics yielding Onsager’s relations. There are many situations where macrovariables, which are the slow parameters of microdynamics, describe, in fact, the mesoscopic phenomena. Thus, an additional course graining is to be done to get the equations of macroscopic theory. In particular, such are the problems of dislocation-mediated plasticity, where elimination of atomic degrees of freedom yields dislocation dynamics, and an additional averaging is needed to obtain the equations of macroscopic plasticity. This averaging problem is qualitatively different from the classical problems of nonequilibrium thermodynamics because the equations to be averaged are neither Hamiltonian nor ergodic, they are dissipative and irreversible. Besides, in driven systems there might be no equilibrium states. However, the basic question remains the same: Are there universal relations which the averaged equations must respect? In this paper, a positive answer to this question is given: the dependence of rates on forces must be potential in the limit of small and large forces. For small forces, this yields the relations which look quite similar to Onsager’s relations, though their origin caused by dissipative irreversible dynamics is quite different. In the limit of small forces a variational principle for dissipative potential is obtained; it reveals quite non-trivial reciprocities of macroscopic interactions. An immediate consequence of these results is the potentiality of the strain rate-stress constitutive relations in dislocation-mediated plasticity for small and large stresses. The reciprocity relations are illustrated by dislocation double-slip of a single crystal.