Second Law: Survey and Application

Dear Colleagues,

This is an invitation to contribute to the Special Issue "Second Law: Survey and Application". The Second Law launches with two verbal formulations concerning irreversible cyclic processes of discrete systems: The principle of Kelvin (1882) and that of Clausius (1861, 1865). In the meantime, there are "as many formulations of the Second Law as there are authors" (Hutter 1977). Nevertheless, Clausius' extended inequality of irreversible processes in open discrete systems is still today a foundation of the Second Law, if it is formulated by use of time differentials. Two other versions of the Second Law are mentioned, that of Caratheodory (1909) in Bernstein's modification (1960) and that of Sears/Kestin (1963/1970). Essential tools were developed which improve our understanding of discrete systems interacting with their environment: Contact quantities belonging to the discrete system which replace controlling items of the environment; accompanying processes which elucidate the meaning of reversible processes; extension of the Second Law to negative absolute temperatures.

Classical non-linear field theories contain balance equations belonging to physically relevant quantities, such as entropy (of course in non-equilibrium). The entropy balance equation connects the time derivative of the entropy density with the entropy flux density, the entropy supply and the entropy production density. Except of the entropy supply, these quantities are constitutive fields, that means, their special shape depends on the material in consideration. In addition to the entropy balance, the balances of mass, energy and momentum and, if necessary, those of spin, internal variables and other ones for complex materials have to be taken into account. These balance equations form a system of underdetermined differential equations which has to be supplemented by constitutive equations. Dependent on the material, a state space has to be chosen which represents the domain of all fields generating a well posed system of differential equations whose solution is wanted under the constraint that the field of the entropy production density is not negative. This constraint represents the Second Law of the thermodynamical field theory.

The coinage of thermodynamical field theories is many-sided: Different state spaces are used in irreversible thermodynamics, extended thermodynamics, higher gradient and weakly nonlocal thermodynamics and thermodynamics of heat conduction, of surfaces and interfaces, and of complex systems; different exploitation procedures of the non-negative entropy production density in extended and irreversible extended thermodynamics; and different standards in relativistic thermodynamics.

Evolution criteria and GENERIC are other theoretical approaches to thermodynamics apart from thermodynamical field theories. The Second Law plays a central role in all phenomenological thermodynamical theories because it describes the macroscopic Zeitpfeil of the considered system. But on a microscopic stochastic level, there are processes of temporal negative process entropy (Jarzynski 1977), denoted as "violations" of the Second Law, a name which is not correct because the (macroscopic) Second Law is also valid in stochastic thermodynamics as a statistical mean–value.

Prof. Dr. Wolfgang Muschik
Prof. Dr. Christina Papenfuss
Guest Editors

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