Spectral density fluctuations systems

All that remains to be done for determining the fluctuation spectrum is to compute the conditional average, Eq. (31). However, this involves the full equations of motion of the many-body system and one can at best hope for a suitable approximate method. There are two such methods available. The first method is the Master Equation approach described above. Relying on the fact that the operator Q represents a macroscopic observable quantity, one assumes that on a coarse-grained level it constitutes a Markov process. The microscopic equations are then only required for computing the transition probabilities per unit time, W(q q ), for example by means of Dirac s time-dependent perturbation theory. Subsequently, one has to solve the Master Equation, as described in Section TV, to find both the spectral density of equilibrium fluctuations and the macroscopic phenomenological equation. 

Describing SR in terms of a susceptibility is particularly advantageous for systems that are in thermal equilibrium, or in quasiequilibrium. In such cases the fluctuation-dissipation relations [9] can be used to express the susceptibility in terms of the spectral density of fluctuations in the absence of the periodic driving. This was used explicitly in the case of noise-protected heterodyning. It is true in general that the analysis of fluctuations is greatly facilitated by the presence of thermal equilibrium when the conditions of detailed balance and of the time reversal symmetry are satisfied [44]. 

One of the most striking chaotic phenomena in generic Hamiltonian systems is the onset of nonstationary fluctuations with extremely long-time memories [5-8], Usually, the nonstationarity is characterized by the infrared catastrophe that is, the power spectral density function S(f) satisfies S(f) V(v > 1) for f O. 

The power available within a molecular system to induce transitions by virtue of its molecular tumbling is referred to as the spectral density J(co) (Section 2.5) and this provides a measure of how the relaxation rates Wq, W] and W2 vary as a function of tumbling rates. This is illustrated schematically in Fig. 8.6 for three different correlation times. An alternative description of the spectral density is that it represents the probability of finding a fluctuating magnetic component at any given frequency as a result of the motion and as such the area under each of the curves of Fig. 8.6 must then be equal. Thus, for a molecule with a short tc (rapid tumbling) there exists an almost... 

After discussing the effect of the flow on the average concentration, we turn to the analysis of the statistics of the concentration fluctuations in such decay-type reaction systems. Even in the case of linear decay, where the average concentration is not affected by mixing, the fluctuations depend on the advecting flow. The distribution of the fluctuations over different length scales can be characterized by the power spectral density of the concentration field (taken along a one-dimensional section C(x), for simplicity) defined as... 

The Nyquist relation is no longer valid for nonequilibrium systems. When a nonzero mean electrical current flows through the system, the spectral density of electrical current fluctuations may differ by orders of magnitude from the fluctuations calculated from eq 1. 

A noise that has a clearly distinct origin from noise discussed in previous sections is the electric noise that originates in modulation of ion transport by fluctuations in system conductance. These temporal fluctuations can be measured, at least in principle, even in systems at equilibrium. Such a measurement was conducted by Voss and Clark in continuous metal films (44). The idea of the Voss and Clark experiment was to measure low-frequency fluctuations of the mean-square Johnson noise of the object. In accordance with the Nyquist formula, fluctuations in the system conductance result in fluctuations in the spectral density of its equilibrium noise. Measurement of these fluctuations (that is, measurement of the noise of noise) yields information on conductance fluctuations of the system without the application of any external perturbations. The samples used in these experiments require rather large amplitude conductance fluctuations to be distinguished from Johnson noise fluctuations because of the intrinsic limitation of statistics. Voss and... 

There are several motional processes that take place simultaneously and may cause spin relaxation in liquid crystals. To incorporate the time scales and amplitudes of various physical motions in liquid crystals, it is necessary to consider different coordinate systems. Because of thermal fluctuations of the director, the orientation of the director has both spatial and temporal variations. A local (or instantaneous) director n(r ) may be defined to represent the direction of preferred orientation of the molecules in the neighborhood of any point in the sample. Thus, an additional coordinate system is needed to specify the local director n(r ). The average director no is obtained by spatially averaging the local directors over the sample at any particular instant. Now the motional spectral densities are given by Eq. (5.31) ... 

Relaxation measurements provide another way to study dynamical processes over a large dynamic range in both thermotropic and lyotropic liquid crystals (see Sec. 2.6 of Chap. Ill of Vol. 2A). The two basic relaxation times of a spin system are the spin-lattice or longitudinal relaxation time 7] and the spin-spin or transverse relaxation time T2. A detailed description, however, requires a more precise definition of the relaxation times. For spin 7=1, for instance, two types of spin-lattice relaxation must be distinguished, related to the relaxation of Zeeman and quadrupolar order with rates 7j"2 and Jfg. The relaxation rates depend on spectral density functions which describe the spectrum of fluctuating fields due to molecular motions. A detailed discussion of spin relaxation is beyond the scope of this... 

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