Reciprocal Relations in Irreversible Processes. II.
TLDR
In this article, a general reciprocal relation applicable to transport processes such as the conduction of heat and electricity, and diffusion, is derived from the assumption of microscopic reversibility, and certain average products of fluctuations are considered.Abstract:
A general reciprocal relation, applicable to transport processes such as the conduction of heat and electricity, and diffusion, is derived from the assumption of microscopic reversibility. In the derivation, certain average products of fluctuations are considered. As a consequence of the general relation $S=k logW$ between entropy and probability, different (coupled) irreversible processes must be compared in terms of entropy changes. If the displacement from thermodynamic equilibrium is described by a set of variables ${\ensuremath{\alpha}}_{1},\ensuremath{\cdots},{\ensuremath{\alpha}}_{n}$, and the relations between the rates ${\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{\ensuremath{\alpha}}}_{1},\ensuremath{\cdots},{\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{\ensuremath{\alpha}}}_{n}$ and the "forces" $\frac{\ensuremath{\partial}S}{d{\ensuremath{\alpha}}_{1}},\ensuremath{\cdots},\frac{\ensuremath{\partial}S}{d{\ensuremath{\alpha}}_{n}}$ are linear, there exists a quadratic dissipation-function, $2\ensuremath{\Phi}(\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{\ensuremath{\alpha}},\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{\ensuremath{\alpha}})\ensuremath{\equiv}\ensuremath{\Sigma}{\ensuremath{\rho}}_{j}{\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{\ensuremath{\alpha}}}_{\mathrm{ij}}{\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{\ensuremath{\alpha}}}_{i}=\frac{\mathrm{dS}}{\mathrm{dt}}=\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{S}(\ensuremath{\alpha},\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{\ensuremath{\alpha}})\ensuremath{\equiv}\ensuremath{\Sigma}(\frac{\ensuremath{\partial}S}{d{\ensuremath{\alpha}}_{j}}){\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{\ensuremath{\alpha}}}_{j}$ (denoting definition by $\ensuremath{\equiv}$). The symmetry conditions demanded by microscopic reversibility are equivalent to the variation-principle $\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{S}(\ensuremath{\alpha},\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{\ensuremath{\alpha}})\ensuremath{-}\ensuremath{\Phi}(\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{\ensuremath{\alpha}},\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{\ensuremath{\alpha}})=\mathrm{maximum},$ which determines ${\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{\ensuremath{\alpha}}}_{1},\ensuremath{\cdots},{\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{\ensuremath{\alpha}}}_{n}$ for prescribed ${\ensuremath{\alpha}}_{1},\ensuremath{\cdots},{\ensuremath{\alpha}}_{n}$. The dissipation-function has a statistical significance similar to that of the entropy. External magnetic fields, and also Coriolis forces, destroy the symmetry in past and future; reciprocal relations involving reversal of the field are formulated.read more
Citations
More filters
Journal ArticleDOI
MR diffusion tensor spectroscopy and imaging.
TL;DR: Once Deff is estimated from a series of NMR pulsed-gradient, spin-echo experiments, a tissue's three orthotropic axes can be determined and the effective diffusivities along these orthotropic directions are the eigenvalues of Deff.
Journal ArticleDOI
Reaction-rate theory: fifty years after Kramers
TL;DR: In this paper, the authors report, extend, and interpret much of our current understanding relating to theories of noise-activated escape, for which many of the notable contributions are originating from the communities both of physics and of physical chemistry.
Journal ArticleDOI
The fluctuation-dissipation theorem
TL;DR: In this article, the linear response of a given system to an external perturbation is expressed in terms of fluctuation properties of the system in thermal equilibrium, which may be represented by a stochastic equation describing the fluctuation, which is a generalization of the familiar Langevin equation in the classical theory of Brownian motion.
Journal ArticleDOI
Selforganization of matter and the evolution of biological macromolecules
TL;DR: The causes and effect of cause and effect, and the prerequisites of Selforganization, are explained in more detail in the I.IA.
Journal ArticleDOI
Synthetic molecular motors and mechanical machines.
TL;DR: The exciting successes in taming molecular-level movement thus far are outlined, the underlying principles that all experimental designs must follow, and the early progress made towards utilizing synthetic molecular structures to perform tasks using mechanical motion are highlighted.