Quasi-stationary distribution, probability

Let us consider the case when the diffusion coefficient is small, or, more precisely, when the barrier height A is much larger than kT. As it turns out, one can obtain an analytic expression for the mean escape time in this limiting case, since then the probability current G over the barrier top near xmax is very small, so the probability density W(x,t) almost does not vary in time, representing quasi-stationary distribution. For this quasi-stationary state the small probability current G must be approximately independent of coordinate x and can be presented in the form... 

This conventional quasi-stationary distribution (4.68) or (4.70), however, has completely the wrong form for small n due to neglecting the factors (n - l)/n. On the other hand, it will be seen that the probability of complete extinction in the quasi-stationary solution depends decisively on the exact form for small values of n. 

The equations (4.84, 85) and their solutions are based on the approximation (4.83) which is valid for a uni-modal probability distribution only. Contrary to this assumption, however, any initially uni-modal distribution can be expected to develop first into the doubled-peaked quasi-stationary solution Pqs (n) shown in Fig. 4.10 and then to go over finally to the exact stationary solution, that of an extinct population. Consequently deviations of the true time paths of (n), and o t) from those described by (4.84, 85) are to be expected. A calculation of the development of a model population with time with the exact master equation (4.63) and with the parameters A = 0.5, n = 0.2, and bi = 0.01 confirms this expectation (Figs. 4.11, 12). The Fig. 4.11 shows the exactly calculated change with time of a distribution which starts as normal distribution but soon develops into the form of the bimodal quasi-stationary distribution Pqs(n). In Fig. 4.12 and for the same model parameters the exact paths of the mean value (n)(and the variance a t) are compared with the paths obtained by solving the approximate equations (4.84, 85). 

Lastly the ideas of Kramers [2.5] and van Kampen[l. ] will be followed in calculating the extinction process and in estimating the life time of the quasi-stationary distribution. The probabilities and t) that the population is still alive or becomes extinct at time t are given by... 

For r = 0 the quasi-stationary solution Pqs (n) is assumed to have been established according to (4.72) with ro(0) = 0 and Jii (0) = 1. Because the probability transfer from the living to the extinct state occurs slowly it is further assumed that the form of the quasi-stationary distribution is conserved during this process, while the weight of each living or non-extinct state p(n t) diminishes in proportion to This implies the assumption that ... 

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. 

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. 

To properly account for the interaction of particles of different radii in a moving and sufficiently diluted suspension (so that it is possible consider only pair interactions), it is convenient to introduce a pair (two-particle) distribution function P(r) [27], having meaning of probability to find the center of the particle of radius ai at the end of the radius vector r, given that the center of the second particle of radius aj is coincident with the origin of the chosen coordinate system. This function satisfies the Fokker-Planck quasi-stationary equation ... 

The excited atom is assumed to emit a wave shifted in frequency by equation (8.17) and the lineshape for the sample as a whole is obtained by averaging over the probability distribution of the stationary perturbers. This approximation is used to calculate the Stark broadening produced by ions in a plasma and for neutral atom broadening at pressures s 100 Torr. From equations (8.19) and (8.20) we see that the quasi-static approximation is likely to be better at high densities, where T is short, and at low temperatures. 

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