Theory of relaxation

Discontinuous Theory of Relaxation Oscillations.—This theory was developed in the USSR by a number of physicists between 1930 and 1937. We give a condensed account of this work for further details see Reference 4 or Reference 6 (pp. 610 to 630). 

Direct and inverse collisions, 12 Discontinuous theory of relaxation oscillations, 385... 

Usually, the most general nonspecific effects of dipole-orientational and electronic polarization of the medium are discussed, and the results of the theory of relaxational shifts developed under the approximation of a continuous dielectric medium may be used.(86 88) The shift of the frequency of the emitted light with time is a function of the dielectric constant e0, the refractive index n, and the relaxation time xR ... 

The work on theory of relaxation in paramagnetic systems performed at Stockholm University has been supported by the Swedish Natural Science Research Council, Swedish Research Council and the Wenner-Gren Foundations. We wish to acknowledge the courtesy of the publishers who permitted reproductions of the figures published in their journals. 

Green MS, Tobolsky AV. A new approach to the theory of relaxing polymeric media. J Chem Phys 1946 14 80-92. 

The essential difference between the two transition probability densities lies in the fact that for the gaussian distribution pw r, ) the different moments E[Xm], m = 1, 2,. . . , n, exist, while for the Cauchy distribution pc(j, x) they do not exist. The Levy distributions characterized by p(t, k) = exp -a k qT) with 0< <2U 127 128 play a prominent role in the theory of relaxation processes.129 133... 

The theory of relaxation processes for a macromolecular coil is based, mainly, on the phenomenological approach to the Brownian motion of particles. Each bead of the chain is likened to a spherical Brownian particle, so that a set of the equation for motion of the macromolecule can be written as a set of coupled stochastic equations for coupled Brownian particles... 

The most advanced theories of relaxation phenomena in a system of entangled macromolecules is based on the dynamics of a single macromolecule. Dynamics of the tagged macromolecule is simplified by the assumption that the neighbouring macromolecules can be described as a uniform structureless medium and all important interactions can be reduced to intramolecular interactions. The dynamic equation for a macromolecule can be written as a modification of equation (2.1) for dynamics of macromolecule in viscous liquid... 

Although the microscopic theory remains to be the real foundation of the theory of relaxation phenomena in polymer systems, the mesoscopic approach has and will not lose its value. It will help to understand the laws of diffusion and relaxation of polymers of various architecture. The information about the microstructure and microdynamics of the material can be incorporated in the form of constitutive relation, thus, allowing to relate different linear and non-linear effects of viscoelasticity to the composition and chemical structure of polymer liquid. 

Equation (153) is the semiclassical limit of the quantum approach of indirect damping. Now, the question may arise as to how Eq. (153) may be viewed from the classical theory of relaxation in order to make a connection with the semiclassical approach of Robertson and Yarwood, which used the classical theory of Brownian motion. 

H. J. Bauer, Theory of Relaxation Phenomena in Gases, in Physical Acoustics, Vol. II, edited by W. P. Mason, Academic, New York, 1965. 

CONTINUED FRACTIONS IN THE THEORY OF RELAXATION The first few even approximants are... 

Out of the detailed mathematical aspects, some of them summarized in this section, there is a more general physical concept at the heart of the theory of error bounds. It is a fact that the memory function formalism provides in a natural manner a framework by which Ae short-time behavior, via the kernel of the integral equations, makes its effects felt in the long-time tail. The mathematical apparatus of continued fractions can adequately describe memory effects, and this explains the central role of this tool in the theory of relaxation. 

---