Correlation function quasi-stationary state

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. 

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. 

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)-... 

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