Symmetry Properties of the Relaxation Equations

The kinetic coefficients can be expressed in terms of ordinary time-correlation functions. Such relations are called Green-Kubo relations (see Section 1 l.B). 

One consequence of Eq. (11.4.5) follows immediately. In the neighborhood of certain points of instability such as the gas-liquid critical point or order disorder phase transitions, the susceptibilities corresponding to the fluctuations in the order parameters become very large. Thus if A does not increase as rapidly as/, the corresponding relaxation rates Twill become small. This phenomenon is called critical slowing of the fluctuations. There has been much recent work on this phenomena (Swinney, 1974). 

The set of variables in Eqs. (11.4.2) and (11.4.3) must include all of the slowly relaxing variables. When the Hamiltonian has certain symmetry properties, the set of Eqs. (11. 3.26) and (11.4.2) can be separated into groups of uncoupled equations. Since, in general, we do not know how to compute the time-correlation functions (F(r), F+(0)), the elements of f should be regarded as quantities to be determined from a comparison between theory and experiment. However, symmetry can be used to relate the off-diagonal elements of V to each other and thereby to reduce the number of independent quantities. 

In this section we show how the symmetry of the Hamiltonian can be used to simplify the relaxation equations. We also present several important theorems involving time-correlation functions and memory functions. We begin by discussing time reversal symmetry. 

Since the Hamiltonian H of a conservative system is a quadratic function of the momenta, H is invariant to this transformation. The equilibrium distribution function Po(-T) is a functional of H so that it is invariant to this transformation. The Liouvillian, on the other hand, contains terms such as 

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To illustrate the general applicability of the relaxation equations of Section 11.4 let us study the simple case of a single conserved variable A(q, t) which has the form given by Eq. (11.5.32). The property aj of the jth molecule is presumed to have definite time-reversal symmetry and parity. 

Having discussed the symmetry properties of the primary variables 5aap(q, t), let us now pass to the derivation of the relaxation equations that describe evolution of 5a... 

The extraction of numerical values for the local densities of state at the Fermi energy from NMR resonance position and relaxation rate requires of course a number of hypotheses. Some of them (such as knowledge of the resonance position corresponding to zero total shift the breakup of the density of states into parts of different symmetry, etc.) already come into play when we try to parameterize data for the bulk metal [58]. Here we mention only the additional ones used to go to the local version of the equations. It is assumed that the hyperfine fields and exchange integrals are a kind of atomic properties that do not vary when the atom is put in one environment or another, whether it is deep inside the particle or on its surface. The approximation is probably reasonable when the atomic volume stays approximately... 

Clearly then, projection operator techniques are useful for the derivation of relaxation equations and for the derivation of formulas for transport coefficients in terms of microscopic properties. The various symmetry properties discussed here are useful for reducing a set of dynamical variables into subsets that are statistically independent of each other. 

The macroscopic nematodynamic equations describe the dynamics of the slowly relaxing variables, which usually are either connected with conservation laws or with the Goldstone modes of the spontaneously broken symmetries. To formulate them we wUl follow the traditional approach [65-67] rather than the one based more directly on the principles of hydrodynamics and irreversible thermodynamics [68]. In the nematic state isotropy is spontaneously broken and the averaged molecular alignment singles out an axis whose orientation defines the director n, i. e. an object that has the properties of a unit vector with n = -n. The static properties are conveniently expressed in terms of a free energy density whose orientational elastic part is given by [69]... 

On the other hand the proponents of the electronic theory ignore viscosity and use instead an estimate of in equation (2) made from proton relaxation times in the solvent, though this method still uses a bulk property to estimate a local rate of motion. An acceptable value can be given to r, but an adjustable counterion symmetry parameter is still needed to obtain reasonable predicted relaxation times. 

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