Quasi stationary functions correlation function

The origin of this unusual behaviour is partly clarified from Fig. 6.34(a) where the relevant curves 2 demonstrate the same kind of the non-monotonous behaviour as the critical exponents above. Since, according to its definition, equation (4.1.19), the reaction rate is a functional of the joint correlation function, this non-monotonicity of curve 2 arises due to the spatial re-arrangements in defect structure. It is confirmed by the correlation functions shown in Fig. 6.34(a). The distribution of BB pairs is quasi-stationary, XB(r,t) X°(r) = exp[(re/r)3], which describes their dynamic aggregation. (The only curve is plotted for XB in Fig. 6.35(a) for t = 102 (the dotted line) since for other time values XB changes not more than by 10 per cent.) This quasi-steady spatial particle distribution is formed quite rapidly already at t 10° it reaches the maximum value of XB(r, t) 103. The effect of the statistical aggregation practically is not observed here, probably, due to the diffusion separation of mobile B particles. 

Statement 3. If the concentrations are fixed, iVa = const, Nb = const, the set of kinetic equations (8.2.17), (8.2.22) and (8.2.23) as functions of the control parameter demonstrates two kinds of motions for k k the stationary (quasi-steady-state) solution holds, whereas for k < k a regular (quasi-regular) oscillations in the correlation functions like standing waves... 

As it was said above, there is no stationary solution of the Lotka-Volterra model for d = 1 (i.e., the parameter k does not exist), whereas for d = 2 we can speak of the quasi-steady state. If the calculation time fmax is not too long, the marginal value of k = K.(a, ft, Na,N, max) could be also defined. Depending on k, at t < fmax both oscillatory and monotonous solutions of the correlation dynamics are observed. At long t the solutions of nonsteady-state equations for correlation dynamics for d = 1 and d = 2 are qualitatively similar the correlation functions reveal oscillations in time, with the oscillation amplitudes slowly increasing in time. 

Statement 2. A set (8.3.22) to (8.3.24) with fixed concentrations = P/K and N = VIP has two kinds of motions dependent on the value of parameter n. As k kq, the stationary (quasi-steady-state) solution occurs, whereas for /c < kq the correlation functions demonstrate the regular (quasiregular) oscillations of the standing wave type. The marginal magnitude is Ko = o(p,/3)-... 

In this chapter, we use a type of initial condition that is different from the waterbag used in Refs. 15 and 18, and we show that (i) probability distribution functions do not have power-law tails in quasi-stationary states and (ii) the diffusion becomes anomalous if and only if the state is neither stationary nor quasi-stationary. In other words, the diffusion is shown to be normal in quasi-stationary states, although a stretched exponential correlation function is present instead of usual exponential correlation. Some scaling laws concerned with degrees of freedom are also exhibited, and the simple scaling laws imply that the results mentioned above holds irrespective of degrees of freedom. 

The system seems quasi-stationary in Stage I. The existence of quasi-stationary states for a sufficiently long time has been questioned in Ref. 36. We will positively answer to the question by observing dependence on t of the correlation function Cp(t xj in Section VI.B. 

Using the approximate functions (19) and Eq. (12), we are able to reproduce cig(t), as shown in Fig. 12. The approximation is good in Stage I—that is, in the quasi-stationary time region—irrespective of the value of N. Consequently, there is no anomaly in diffusion in Stage I, since the diffusion is explained by stretched exponential correlation function. 

Diffusion is obtained by integrating the correlation function of momenta Cp(t x), and the correlation function is approximated by a series of stretched exponential functions Cp(t x) = Cp 0 x) exp[— (f/fCorr(x)) ]- Among the three parameters Cp(0 x), fcorr(x), and (3(x), the stretching exponent p(x) plays a crucial role to yield anomaly in diffusion. If we assume that (3(x) is a constant, we never observe anomaly in diffusion. This result is consistent with the fact that anomaly in diffusion does not appear in a (quasi-)stationary state, because the correlation functions Cp(t xj and, accordingly, (3(x) are almost invariant with respect to x. 

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