Relaxation time linear response theory

The above theory is usually called the generalized linear response theory because the linear optical absorption initiates from the nonstationary states prepared by the pumping process [85-87]. This method is valid when pumping pulse and probing pulse do not overlap. When they overlap, third-order or X 3 (co) should be used. In other words, Eq. (6.4) should be solved perturbatively to the third-order approximation. From Eqs. (6.19)-(6.22) we can see that in the time-resolved spectra described by x"( ), the dynamics information of the system is contained in p(Af), which can be obtained by solving the reduced Liouville equations. Application of Eq. (6.19) to stimulated emission monitoring vibrational relaxation is given in Appendix III. 

The linear response theory [50,51] provides us with an adequate framework in order to study the dynamics of the hydrogen bond because it allows us to account for relaxational mechanisms. If one assumes that the time-dependent electrical field is weak, such that its interaction with the stretching vibration X-H Y may be treated perturbatively to first order, linearly with respect to the electrical field, then the IR spectral density may be obtained by the Fourier transform of the autocorrelation function G(t) of the dipole moment operator of the X-H bond ... 

For systems close to equilibrium the non-equilibrium behaviour of macroscopic systems is described by linear response theory, which is based on the fluctuation-dissipation theorem. This theorem defines a relationship between rates of relaxation and absorption and the correlation of fluctuations that occur spontaneously at different times in equilibrium systems. 

In the framework of the standard linear response theory, the relaxation times of precession are determined in the a< 1 limit according to the generic formula x = (ay//)-1. Setting the field equal to its resonance value, we get... 

In the present section, it is demonstrated how the linear response of an assembly of noninteracting polar Brownian particles to a small external field F applied parallel and perpendicular to the bias field Fo may be calculated in the context of the fractional noninertial rotational diffusion in the same manner as normal rotational diffusion [8]. In order to carry out the calculation, it is assumed that the rotational Brownian motion of a particle may be described by a fractional noninertial Fokker-Planck (Smoluchowski) equation, in which the inertial effects are neglected. Both exact and approximate solutions of this equation are presented. We shall demonstrate that the characteristic times of the normal diffusion process, namely, the integral and effective relaxation times obtained in Refs. 8, 65, and 67, allow one to evaluate the dielectric response for anomalous diffusion. Moreover, these characteristic times yield a simple analytical equation for the complex dielectric susceptibility tensor describing the anomalous relaxation of the system. The exact solution of the problem reduces to the solution of the infinite hierarchies of differential-recurrence equations for the corresponding relaxation functions. The longitudinal and transverse components of the susceptibility tensor may be calculated exactly from the Laplace transform of these relaxation functions using linear response theory [72]. 

The time dependent solvation funetion S(t) is a directly observed quantity as well as a convenient tool for numerical simulation studies. The corresponding linear response approximation C(t) is also easily eomputed from numerical simulations, and can also be studied using suitable theoretical models. Computer simulations are very valuable both in exploring the validity of such theoretical calculations, as well as the validity of linear response theory itself (by comparing S(t) to C(t)). Furthermore they can be used for direct visualization of the solute and solvent motions that dominate the solvation process. Many such simulations were published in the past decade, using different models for solvents such as water, alcohols and acetonitrile. Two remarkable outcomes of these studies are first, the close qualitative similarity between the time evolution of solvation in different simple solvents, and second, the marked deviation from the simple exponential relaxation predicted by the Debye relaxation model (cf Eq. [4.3.18]). At least two distinct relaxation modes are... 

Such numerical simulations have played an important role in the development of our understanding of solvation dynamics. For example, they have provided the first indication that simple dielectric continuum models based on Debye and Debey-like dielectric relaxation theories are inadequate on the fast timescales that are experimentally accessible today. It is important to keep in mind that this failure of simple theories is not a failure of linear response theory. Once revised to describe reliably response on short time and length scales, e.g. by using the full k and (O dependent dielectric response function e(k,o , and sufficiently taking into account the solvent structure about the solute, linear response theory accounts for most observations of solvation dynamics in simple polar solvents. 

One of the most important principles of linear response theory relates the system s response to an externally imposed perturbation, which causes it to depart from equilibrium, to its equilibrium fluctuations. Indeed, the system response to a small perturbation should not depend on whether this perturbation is a result of some external force, or whether it is just a random thermal fluctuation. Spontaneous concentration fluctuations, for instance, occur all the time in equilibrium systems at finite temperatures. If the concentration c at some point of a liquid at time zero is (c) + 3c(r, t), where (c) is an average concentration, concentration values at time t + 8t dXt and other points in its vicinity will be affected by this. The relaxation of the spontaneous concentration fluctuation is governed by the same diffusion equation that describes the evolution of concentration in response to the external imposition of a compositional heterogeneity. The relationship between kinetic coefficients and correlations of the fluctuations is derived in the framework of linear response theory. In general, a kinetic coefficient is related to the integral of the time correlation function of some relevant microscopic quantity. 

In liquids and dense gases where collisions, intramolecular molecular motions and energy relaxation occur on the picosecond timescales, spectroscopic lineshape studies in the frequency domain were for a long time the principle source of dynamical information on the equilibrium state of manybody systems. These interpretations were based on the scattering of incident radiation as a consequence of molecular motion such as vibration, rotation and translation. Spectroscopic lineshape analyses were intepreted through arguments based on the fluctuation-dissipation theorem and linear response theory (9,10). In generating details of the dynamics of molecules, this approach relies on FT techniques, but the statistical physics depends on the fact that the radiation probe is only weakly coupled to the system. If the pertubation does not disturb the system from its equilibrium properties, then linear response theory allows one to evaluate the response in terms of the time correlation functions (TCF) of the equilibrium state. Since each spectroscopic technique probes the expectation value... 

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