Fluctuation parameter

The crucial step in the Onsager argument concerns the correlation between fluctuations that occur, not simultaneously, but at times r apart. Such a correlation reflects the effect of one fluctuating parameter on another, of a second kind, that occurs at time r later. Alternatively it shows the effect on one fluctuating parameter, of a fluctuation of the other kind that occurred at time — r earlier. 

M.R. Jumaev Evolution of random pulses at nonlinear systems with fluctuating parameters -Dissertation for Doctor in Science in Theoretical Physics (2002) M.R. Jumaev NATO Advanced Research Workshop, Vortex dynamics and high temperature superconductors , Tashkent, Uzbekistan, 16-23 May 2002 (2002). M.R. Jumaev Evolution of random and regular pulses at nonlinear systems with constant and fluctuating parameters , Bukhara, Universitet (2004). 

As an excellent, simple example of how fluctuating parameters can affect a reacting system, one can examine how the mean rate of a reaction would differ from the rate evaluated at the mean properties when there are no correlations among these properties. In flow reactors, time-averaged concentrations and temperatures are usually measured, and then rates are determined from these quantities. Only by optical techniques or very fast response thermocouples could the proper instantaneous rate values be measured, and these would fluctuate with time. 

This value includes the expansion factor which was calculated separately in item (g) but not the fluctuation parameter with this correction the bed height is... 

To obtain the radial distance correlation function one has to take the average of many system configurations. The fluctuation parameter g can be derived from these functions easily. 

Fig. 7. Distribution of the coordination number, shown for different fluctuation parameters g. For g = 0.06, a bimodal distribution results (see Fig. 8)...

Y. I. Kljatskin, Statistical Description of Dynamic Systems with Fluctuating Parameters, Nauka, Moscow, 1975. 

Early investigations were concerned mostly with physical properties, somewhat with particle characteristics but little with particle group behavior. Even so, significant results were obtained. For instance, the distinction between L/S fluidization and G/S fluidization, viz., particulate and aggregative, and the provision of criteria for such distinction (Wilhelm and Kwauk, 1948 Harrison et ai, 1961 Romero and Johanson, 1962), most of which were based on the Froude number. Other criteria were then proposed involving fluctuating parameters in fluidization (Rietema, 1967), for instance, pressure drop or voidage. 

More detailed information about the structure of turbulent flows in a circular (noncircular) tube and a plane channel, as well as various relations for determining the average velocity profile and the drag coefficient, can be found in the books [138, 198, 268, 276, 289], which contain extensive literature surveys. Systematic data for rough tubes and results of studying fluctuating parameters of turbulent flow can also be found in the cited references. 

Analogously, for any other fluctuating value, the average square fluctuation equals the ratio of kT to the second derivative of work (free energy) of fluctuations with respect to the fluctuating parameter. We will utilize this approach in describing the optical properties of disperse systems (further down in this chapter), the electric properties of aerosols (see Chapter VIII), and the conditions of the formation of critical emulsions (see Chapter VI,2). 

In agreement with the general theory of fluctuations (see Chapter V), the average value for the squared radius of such fluctuations, r2, is determined by the second derivative of the work of fluctuation with respect to the fluctuating parameter, i.e., with respect to radius in the present case ... 

Figure 4.16 Idealized plot showing the fluctuation parameters rf and...

If we integrate Equation 152 twice with respfx t to the fluctuating parameter we get a Hamiltonian in the fashion of the Landau phenomenological theory (section 2.5)... 

Here both integrals can be interpreted as ordinary Riemann integrals is the nonspecified fluctuating parameter). 

A most interesting consequence of the unified description of fluctuations given by 1 2.6) shows up explicitly when considering the equations satisfied by the moments of P(N,t). For the sake of clarity, and although such equations can be written quite generally, here we restrict ourselves to the two first moments for the example considered in (2.7)-(2.8) and with X+=0. Inclusion of a fluctuating parameter in W+ gives similar results. In terms of the variable x=N/V we find[4]... 

Using expansion of all fluctuating parameters into Fourier series inside volume V, i.e. 

The new SmA2-SmA(j phase boundary terminates at a critical point C where the jump in wavevector goes to zero. Although the fluctuating parameter is a scalar (layer thickness) it will be shown in this chapter, Sec. 6.1.7.4 that the new critical point C is not expected to belong to the Ising universality class [85]. 

Fig. 2. Phase portrait of the homogeneous limit cycle of the Brusselator model in the presence of fluctuations. Parameter values a = 2,b = 5.2, = 2,500.

Ratto, M., Paladino, O. Analysis of controlled CSTR models with fluctuating parameters and uncertain parameters. Chem. Eng. J. 79, 13-21 (2000)... 

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