The concept of entropy in nonequilibrium macroscopic systems is investigated in the light of an extended equation of motion for the density matrix obtained in a previous study. It is found that a time-dependent information entropy can be defined unambiguously, but it is the time derivative or entropy production that governs ongoing processes in these systems. The differences in physical interpretation and thermodynamic role of entropy in equilibrium and nonequilibrium systems is emphasized and the observable aspects of entropy production are noted. A basis for nonequilibrium thermodynamics is also outlined. † Because there are numerous 'entropies' defined in different contexts, we shall denote the experimental equilibrium entropy of Clausius as S without further embellishments, such as subscripts. † Requiring the equilibrium system to have no 'memory' of its past precludes 'mysterious' effects such as those caused by spin echos.
This book describes the scattering of waves, both scalar and electromagnetic, from impenetrable and penetrable spheres. Although the scattering of plane waves from spheres is an old subject, there is little doubt that it is still maturing as a broad range of new applications demands an understanding of finer details. In this book attention is focused primarily on spherical radii much larger than incident wavelengths, along with the asymptotic techniques required for physical analysis of the scattering mechanisms involved. Applications to atmospheric phenomena such as the rainbow and glory are included, as well as a detailed analysis of optical resonances. Extensions of the theory to inhomogeneous and nonspherical particles, collections of spheres, and bubbles are also discussed. This book will be of primary interest to graduate students and researchers in physics (particularly in the fields of optics, the atmospheric sciences and astrophysics), electrical engineering, physical chemistry and some areas of biology.
Abstract. The concept of entropy in nonequilibrium macroscopic systems is investigated in the light of an extended equation of motion for the density matrix obtained in a previous study. It is found that a time-dependent information entropy can be defined unambiguously, but it is the time derivative or entropy production that governs ongoing processes in these systems. The differences in physical interpretation and thermodynamic role of entropy in equilibrium and nonequilibrium systems is emphasized and the observable aspects of entropy production are noted. A basis for nonequilibrium thermodynamics is also outlined. IntroductionThe empirical statement of the Second Law of thermodynamics by Clausius (1865) iswhere S is the total entropy of everything taking part in the process under consideration, and the entropy for a single closed system is defined to within an additive constant bywhere C(T ) is a heat capacity. The integral in (2) is to be taken over a reversible path, a locus of thermal equilibrium states connecting the macroscopic states 1 and 2, which is necessary because the absolute temperature T is not defined for other than equilibrium states; dQ represents the net thermal energy added to or taken from the system in the process. As a consequence, entropy is defined in classical thermodynamics only for states of thermal equilibrium. Equation (1) states that in the change from one state of thermal equilibrium to another along a reversible path the total entropy of all bodies involved cannot decrease; if it increases, the process is irreversible. That is, the integral provides a lower bound on the change in entropy. This phenomenological entropy is to be found from experimental measurements with calorimeters and thermometers, so that by construction it is a function only of the macroscopic parameters defining the macroscopic state of a system, S(V, T, N ), say, where V and N are the system volume and particle number, respectively. It makes no reference to microscopic variables or probabilities, nor can any explicit time dependence be justified in the context of classical thermodynamics. Equation (1) is a statement of macroscopic phenomenology that cannot be proved true solely as a consequence of the microscopic dynamical laws of physics, as appreciated already by 1 Boltzmann (1895): "The Second Law can never be proved mathematically by means of the equations of dynamics alone." (Nor, for that matter, can the First Law!) Theoretical definitions of entropy were first given in the context of statistical mechanics by Boltzmann and Gibbs, and these efforts culminated in the formal definition of equilibrium ultimately given by Gibbs (1902) in terms of his variational principle. In Part I (Grandy, 2003, preceding paper † ) we observed that the latter is a special case of a more general principle of maximum information entropy (PME), and in equilibrium it is that maximum subject to macroscopic constraints that is identified with the experimental entropy of (2). One of the dominant concerns in statistical mechanic...
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