Quasi-stationary distribution

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... 

We apply this approach now and demonstrate its applicability in more details later in this section. For the quasi-stationary distribution on the cycle we get C(j-T+ = c k /k (K/reaction network is transformed... 

In the foregoing treatment it is implicitly assumed that the system relaxes to the quasi-stationary distribution Pqs(x, t) on a time scale Tqs that is much shorter than the time scale ts = 1/ s characterizing the evolution to Ps(x). The following remarks on the Kramers method are of importance ... 

Note that equation (5.1) follows immediately from Fick s laws on the assumption of a quasi-stationary distribution of the concentration of components within the diffusion boundary layer. Indeed, if in this layer cAl t 0, then the second Fick law yields cA x const. It means that the distribution of the concentration of component A within this layer is close to linear (Fig. 5.1). Anywhere outside of this layer, the concentration of A is assumed to be the same and equal to an instantaneous value, c. This implies sufficiently intensive agitation of the liquid. In such a case, the flow of A atoms across the diffusion boundary layer under the condition of constancy of the surface area of the dissolving solid is... 

Thus, wehavechangedonlytwostepsinthe L(ifshitz)S(lezov)W(agner) - scheme [13], namely, the description of the quasi-stationary distribution of the concentration around a randomly chosen particle in the mean-field approximation, and the expression for flux in the flux balance at the moving precipitate boundaries. Instead of considering the standard steady-state equation DV cb = 0, we have analyzed a system of two coupled steady-state equations, for B atoms and for vacancies [14]... 

At times t Tw> quasi-resonant VV processes and VT relaxation are frozen thus, there is only resonant VV transfer in A + A and B + B collisions. This results in the onset of quasi-stationary distributions over A and B vibrational states characterized by different vibrational temperatures T and T . 

The situation changes considerably when the rate of the VV energy exchange in collisions of relaxing molecules exceeds that of VT processes. Then, at the first stage, the quasi-resonant W exchange results in a quasi-stationary distribution with the total number of vibrational quanta equal to Ef/hco. This quasi-stationary distribution function, sometimes called Treanor distribution [485], is of 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 ... 

In fact, exactly these parameters have been chosen for the comparison of the two approximations Pqs( ) and p (n) of the quasi-stationary distribution (Fig. 4.10). Using them again in (4.93, 94) to determine the mean life time of the population. 

Figure 4.15. Comparison of two quasi-stationary distributions 1) the triple birth process distribution (—) with parameters 0.1533, = 0 for g 4= 3,...

Calculating the mean life times of the quasi-stationary distributions n) and pfj n) for the Hanson [4.17] observations presented at the end of Sect. 4.3.1, = 1460 years and = 79 years are obtained. 

Finally, we attack the problem of the transport coefficients, which, by definition, are calculated in the stationary or quasi-stationary state. The variation of the distribution functions during the time rc is consequently rigorously nil, which allows us to calculate these coefficients from more simple quantities than the generalized Boltzmann operators which we call asymptotic cross-sections or transport operators. 

In addition to the general problem of the kinetics of the approach towards equilibrium, the statistical mechanics of irreversible phenomena concern in particular the study of transport phenomena. The latter are calculated in a stationary or quasi-stationary form (the distribution functions do not vary or vary in hydrodynamic fashion). Therefore, let us consider (see, for... 

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. 

One is in the presence of a two-time-scale description. In the short part, the system relaxes nonexponentially to a quasi-stationary state with the characteristic time r° [Eq. (4.220)] depending on the initial condition, and the spatial distribution shows large variations. Then, on a longer-time... 

It is reasonable to assume that in the vast majority of cases encountered in reactive processing Re < < 1 and (H/L)Re < < 1. Thus, we can consider the flow to be quasi-stationary and that temperature changes occur quickly after alterations in the temperature and degree of conversion distributions.202 Now we can rewrite the system of balance equations in the following dimensionless form ... 

The system of equations with initial and boundary conditions formulated above allows us to find the velocity distributions and pressure drop for the filled part of the mold. In order to incorporate effects related to the movement of the stream front and the fountain effect, it is possible to use the velocity distribution obtained285 for isothermal flow of a Newtonian liquid in a semi-infinite plane channel, when the flow is initiated by a piston moving along the channel with velocity uo (it is evident that uo equals the average velocity of the liquid in the channel). An approximate quasi-stationary solution can be found. Introduction of the function v /, transforms the momentum balance equation into a biharmonic equation. Then, after some approximations, the following solution for the function jt was obtained 285... 

Modeling EM solitary waves in a plasma is quite a challenging problem due to the intrinsic nonlinearity of these objects. Most of the theories have been developed for one-dimensional quasi-stationary EM energy distributions, which represent the asymptotic equilibrium states that are achieved by the radiation-plasma system after long interaction times. The analytical modeling of the phase of formation of an EM soliton, which we qualitatively described in the previous section, is still an open problem. What are usually called solitons are asymptotic quasi-stationary solutions of the Maxwell equations that is, the amplitude of the associated vector potential is either an harmonic function of time (for example, for linear polarization) or it is a constant (circular polarization). Let s briefly review the theory of one-dimensional RES. 

Quasi-stationary models assume that, besides their spatial distribution, the temperature of the fast electrons also does not evolve during the ion acceleration process. These issues do not seem to provide a major problem when determining the maximum ion energy, since the acceleration of these ions (which will be the most energetic) takes place over a time scale over which the temperature and the entire distribution do not vary appreciably [93]. 

The temperature distribution in a reacting mixture is stabilized when the rate of loss of heat by conduction or convection from any volume element is equal to that produced by the reaction itself in that volume element. In the case that the rate of heat loss cannot compensate for the rate of heat production, a stationary or quasi-stationary temperature distribution is impossible and the temperature of the reaction mixture increases exponentially, causing the reaction rate to do likewise, and a thermal explosion results. This is illustrated in Fig. XIV. 1, which follows... 

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