Relaxation equations periodic perturbations

The kinetics of formation and disintegration of micelles has been studied for about thirty years [106-130] mainly by means of special experimental methods, which have been proposed for investigation of fast chemical reaction in liquids [131]. Most of the experimental methods for micellar solutions study the relaxation of small perturbations of the aggregation equilibrium in the system. Small perturbations of the micellar concentration can be generated by either fast mixing of two solutions when one of them does not contain micelles (method of stopped flow [112]), or by a sudden shift of the equilibrium by instantaneous changes of the temperature (temperature jump method [108, 124, 129, 130]) or pressure (pressure jump method [1, 107, 116, 122, 126]). The shift of the equilibrium can be induced also by periodic compressions or expansions of a liquid element caused by ultrasound (methods of ultrasound spectrometry [109-111, 121, 125, 127]). All experimental techniques can be described by the term relaxation spectrometry [132] and are characterised by small deviations from equilibrium. Therefore, linearised equations can be used to describe various processes in the system. 

The forcing functions used to initiate chemical relaxations are temperature, pressure and electric held. Equilibrium perturbations can be achieved by the application of a step change or, in the case of the last two parameters, of a periodic change. Stopped-flow techniques (see section 5.1) and the photochemical release of caged compounds (see section 8.4) can also be used to introduce small concentration jumps, which can be interpreted with the linear equations discussed in this chapter. The amplitudes of perturbations and, consequently of the observed relaxations, are determined by thermodynamic relations. The following three equations dehne the dependence of equilibrium constants on temperature, pressure and electric held respectively, in terms of partial differential equations and the difference equations, which are convenient approximations for small perturbations ... 

In Equation 2.10 B is a constant and the A/s represent the relaxation amplitudes that are proportional to the square of the isentropic volume change 0/° given by Equation 2.9 and that contains both 07 and AH°. Indeed, the propagation of ultrasonic waves in fluids gives rise to harmonic changes of p and T. The investigated equilibria are shifted periodically by these two perturbations, thus the ultrasonic relaxation amplitude dependence on... 

The equations of motion of a molecular system formally represent a coupled set of nonlinear differential equations. (The nonlinearity comes from the complicated distance-dependence of the pair-potentials.) It is a property of nonlinear differential equations that they are extremely sensitive to small differences in their initial conditions. In nature, these small differences are most generally created by the perturbations of the surroundings while in the computer simulations they are produced by the finite accuracy of the numerical computation. The sensitivity is manifested in the fast increase of these initial differences nearby trajectories separate exponentially until the system boundaries force them to turn back. This mechanism quickly mixes the trajectories and after a short initial period the behavior of the system forgets its past. This obviously happens for equilibrium systems when their macroscopic properties relax to fixed average values. It also occurs for NESS systems because after short transients their distribution function also becomes stationary. ... 

The solutions to equation (1) may be written down immediately in terms of a matrix exponential. For the two-dimensional NOESY experiment, with appropriate normalization, the initial conditions (at the beginning of the mixing period ) can be written as a unit matrix, i.e., the two-dimensional pulse sequence is equivalent to repeated relaxation experiments in which each spin in turn is displaced from equilibrium. The two-dimensional NOE cross-peak intensity at chemical shifts corresponding to spins i and j is then related to the magnetization of spin i for the experiment in which spin j was initially perturbed. After a mixing time tm, this is just exp(—Rrm),. ... 

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