Spectrum fluctuations

Spp(A) which is the liquid structure factor discussed earlier in section A2.2.5.2. The density fluctuation spectrum is... 

In this section we discuss the frequency spectrum of excitations on a liquid surface. Wliile we used linearized equations of hydrodynamics in tire last section to obtain the density fluctuation spectrum in the bulk of a homogeneous fluid, here we use linear fluctuating hydrodynamics to derive an equation of motion for the instantaneous position of the interface. We tlien use this equation to analyse the fluctuations in such an inliomogeneous system, around equilibrium and around a NESS characterized by a small temperature gradient. More details can be found in [9, 10]. 

The upper end of the range in t] is in excellent agreement with what has been deduced from WMAP measurements of the angular fluctuation spectrum of the cosmic microwave background (see Fig. 4.3 and Section 4.9). 

Balloon and WMAP satellite missions provide details of angular fluctuation spectrum of MWB, giving precise estimates of cosmological parameters. 

It follows that the total Gibbs energy change will be negative, and a fluctuation will be stable, when a wavelength component of the fluctuation spectrum corresponds to the relationship... 

It is again impossible to choose ax(q) such that nonlinear case and that a more fundamental starting point is indispensable. 

The first two terms of Eq. (23) constitute the Fokker-PIanck equation (Eq. (II)). However, since Eq. (23) is not an expansion in powers of a small quantity, there is no a priori justification for breaking off after two terms. It has also been demonstrated by an example that there may be solutions of Eq. (15) which do not obey Eq. (23), but they hardly contribute to the fluctuation spectrum.9... 

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. 

Then the fluctuation spectrum of any linear combination Z = Cj Yj is Sz(co) = Lij Sij((D)CiCj. Find a similar formula for r complex components. 

Exercise. Let the charge Q on a condenser be described by a master equation. The fluctuation spectrum in equilibrium obeys... 

This is the general formula (in the linear noise approximation) for the autocorrelation function of the fluctuations in a stable stationary state. Hence it is possible to write down the fluctuation spectrum in an arbitrary system without solving any specific equations. This fact is the basis of the customary noise theory. 

Note that experimentally one cannot determine a and b by measuring only [Pg.262]

It is useful to find a quantity that could serve us as a measure of these density fluctuations. Its simplest characteristic is the dispersion of a number of particles N in some volume V i.e., (N2) — (N)2. The distinctive feature of the classical ideal gas is a simple relation between the dispersion and macroscopic density (TV2) - (TV)2 = (IV) = nV. Moreover, all other fluctuation characteristics of the ideal gas, related to the quantity (Nm, could also be expressed through (TV) or density n. Therefore, in the model of ideal gas the density n is the only parameter characterizing the fluctuation spectrum. Such the particle distribution is called the Poisson distribution. It could be easily generalized for the many-component system, e.g., a mixture of two ideal gases. Each component is characterized here by its density, nA and nB density fluctuations of different components are statistically independent, (IVAIVB) = (Na)(Nb). 

The only problem necessary for developing the condensation theory is to add to the above-mentioned equation of the state the equation defining the function x(r)- Unfortunately, it turns out that the exact equation for the joint correlation function, derived by means of basic equations of statistical physics, contains f/iree-particle correlation function x 3), which relates the correlations of the density fluctuations in three points of the reaction volume. The equation for this three-particle correlations contains four-particle correlation functions and so on, and so on [9], This situation is quite understandable, since the use of the joint correlation functions only for description of the fluctuation spectrum of a system is obviously not complete. At the same time, it is quite natural to take into account the density fluctuations in some approximate way, e.g., treating correlation functions in a spirit of the mean-field theory (i.e., assuming, in particular, that three-particle correlations could be expanded in two-particle ones). 

However, a question arises - could similar approach be applied to chemical reactions At the first stage the general principles of the system s description in terms of the fundamental kinetic equation should be formulated, which incorporates not only macroscopic variables - particle densities, but also their fluctuational characteristics - the correlation functions. A simplified treatment of the fluctuation spectrum, done at the second stage and restricted to the joint correlation functions, leads to the closed set of non-linear integro-differential equations for the order parameter n and the set of joint functions x(r, t). To a full extent such an approach has been realized for the first time by the authors of this book starting from [28], Following an analogy with the gas-liquid systems, we would like to stress that treatment of chemical reactions do not copy that for the condensed state in statistics. The basic equations of these two theories differ considerably in their form and particular techniques used for simplified treatment of the fluctuation spectrum as a rule could not be transferred from one theory to another. 

The said allows us to understand the importance of the kinetic approach developed for the first time by Waite and Leibfried [21, 22]. In essence, as is seen from Fig. 1.15 and Fig. 1.26, their approach to the simplest A + B —0 reaction does not differ from the Smoluchowski one However, coincidence of the two mathematical formalisms in this particular case does not mean that theories are basically identical. Indeed, the Waite-Leibfried equations are derived as some approximation of the exact kinetic equations due to a simplified treatment of the fluctuational spectrum a complete set of the joint correlation functions x(rJ) for kinds of particles is replaced by the only function xab (a t) describing the correlation of chemically reacting dissimilar particles. Second, the equation defining the correlation function X = Xab(aO is linearized in the function x(rJ)- This is analogous to the... 

Chapter 5 deals with derivation of the basic equations of the fluctuation-controlled kinetics, applied mainly to the particular bimolecular A + B 0 reaction. The transition to the simplified treatment of the density fluctuation spectrum is achieved by means of the Kirkwood superposition approximation. Its accuracy is estimated by means of a comparison of analytical results for some test problems of the chemical kinetics with the relevant computer simulations. Their good agreement permits us to establish in the next Chapters the range of the applicability of the traditional Waite-Leibfried approach. 

As it was first noted by Zeldovich [33] it is not easy to distinguish experimentally between exponents 1 and 3/4 (equations (2.1.8) and (2.1.77)). The approach just presented cannot be applied to charged reactants since their electrostatic attraction cuts off spatial fluctuation spectrum at the Debye radius. 

At last, note logical inconsistency of the method presented. Non-uniform concentration distribution, corresponding to the Poisson fluctuation spectrum (2.1.42), is introduced through initial condition imposed on Z(r,t) - see (2.1.71), (2.1.72). However, equation (2.1.42) disagrees with the starting kinetic equation (2.1.40) the solution of the latter in the absence of reaction, Fi = 0, is Ci(r, t — oo) = nj(0). Consequently, we can find dispersion of a number of particles within an arbitrary volume ... 

The fluctuation spectrum just discussed differs considerably from (2.1.72). Let us estimate a role of similar particle aggregation in their decay kinetics after the source is switched off at time t = 0. Decay is described by (2.1.62), (2.1.63) (Da = Db). For concentration difference (2.1.64) we have the initial condition Z(r, 0) = Zo(r). In this case the fluctuation spectrum corresponds to (2.2.29) ... 

Its calculation with the fluctuation spectrum (2.2.31) could be performed employing the scheme used in Section 2.1.2, which results in [84]... 

Note that its asymptotics (2.2.34) gives essentially slower decay than (2.1.77) observed for the Poisson initial distribution. In the d = 2 case the integral in (2.2.33) should be cut off at km n oc S x/2 where S is surface square practically complete disappearance of particles takes place after t S/D. Consider briefly the applicability of (2.2.34). Use of (2.2.34) for the initial fluctuation spectrum continues infinitely similar particle aggregation takes a very prolonged time. Under finite excitation time, the peculiarity (2.2.31) at small k is not pronounced, (2.2.34) is not universal and it plays the role of the intermediate asymptotics and holds at t [Pg.93]

The relevant calculations performed by Ovchinnikov and Burlatsky [84] showed that - in line with general physical ideas - the peculiarity of the fluctuation spectrum at small k disappears due to Coulomb repulsion. Automatically it transforms the long-time asymptotics into that known in formal kinetics (2.1.1). 

Its obvious peculiarity as compared with the standard chemical kinetics, equation (2.1.10), is the emergence of the fluctuational second term in r.h.s. The stochastic reaction description by means of equation (2.2.37) permits us to obtain the equation for dispersions crjj which, however, contains higher-order momenta. It leads to the distinctive infinite set of deterministic equations describing various average quantities, characterizing the fluctuational spectrum. 

Since the Poisson fluctuation spectrum results in cr = (N), the second term in r.h.s. of (2.3.13) defines the deviation of fluctuations from the Poisson... 

In fact, the latter is a functional of the correlation function of dissimilar particles, i.e., to calculate K(t) we need to know either Y(r, t) or p. In its turn, equation (4.1.16) demonstrates that these latter are coupled with three-point densities etc. Therefore, to solve the problem, we have to cut off the infinite equation hierarchy, thus only approximately describing the fluctuation spectrum. Usually it is done by means of the complete Kirkwood superposition approximation, equations (2.3.62) and (2.3.63), or the shortened approximation, equations (2.3.64) and (2.3.65). 

If there is no interaction between similar reactants (traps) B, they are distributed according to the Poisson relation, Ab (r, t) = 1. Besides, since the reaction kinetics is linear in donor concentrations, the only quantity of interest is the survival probability of a single particle A migrating through traps B and therefore the correlation function XA(r,t) does not affect the kinetics under study. Hence the description of the fluctuation spectrum of a system through the joint densities A (r, ), which was so important for understanding the A4-B — 0 reaction kinetics, appears now to be incomplete. The fluctuation effects we are interested in are weaker here, thus affecting the critical exponent but not the exponential kinetics itself. It will be shown below that adequate treatment of these weak fluctuation effects requires a careful analysis of many-particle correlations. 

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