Quasi stationary functions

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. 

In the standard overdamped version of the Kramers problem, the escape of a particle subject to a Gaussian white noise over a potential barrier is considered in the limit of low diffusivity—that is, where the barrier height AV is large in comparison to the diffusion constant K [14] (compare Fig.6). Then, the probability current over the potential barrier top near xmax is small, and the time change of the pdf is equally small. In this quasi-stationary situation, the probability current is approximately position independent. The temporal decay of the probability to find the particle within the potential well is then given by the exponential function [14, 22]... 

The resulting rate can be estimated as logT 4>q(G/Gq)x If o < 1, this reduces to log T 4>o(G/Go)Ith/If- In the opposite limit, the estimation for the rate reads log r 4>o(G/Gq ) l< (It J h), F being a dimensionless function 1. It is important to note that these expressions match the quantum tunneling rate log Jr Uqt/K (G/Gq)< provided eVr h. Therefore the quasi-stationary approximation is valid when the quantum tunneling rate is negligible and the third factor mentioned in the introduction is not relevant. For equilibrium systems, the situation corresponds to the well-known crossover between thermally activated and quantum processes at k Tr h [9]. 

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... 

A.M. Dykhne, A.V. Chaplik, Normalization of the Wave Functions of Quasi Stationary... 

Table 13 Quasi-stationary concentrations of HCN in masonry in per-cent of saturation, as a function of daily exposure time to HCN...

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. 

In these expressions the M s a,re functions of the concentrations of the different ordinary molecules in the mixture and therefore vary slowly with time, but they are not explicit functions of the time. This is what we mean by saying that the concentrations are quasi-stationary. 

Classifying variables into fast or slow is a typical approach in chemical kinetics to apply the method of (quasi)stationary concentrations, which allows the initial set of differential equations to be largely reduced. In the chemically reactive systems near thermodynamic equihbrium, this means that the subsystem of the intermediates reaches (owing to quickly changing variables) the stationary state with the minimal rate of entropy production (the Rayleigh Onsager functional). In other words, the subsys tern of the intermediates becomes here a subsystem of internal variables. 

To find what composition the polymer will have at a given monomer composition, an equation for the ratio of the monomer consumption rates as a function of the concentrations of the monomers is needed. With eqns 10.93 and the Bodenstein approximation of quasi-stationary behavior of either propagating center, Ma or Mb, one obtains... 

Instead of the quasi-stationary state assumption of Kramers, he assumed only that the density of particles in the vicinity of the top of the barrier was essentially constant. Visscher included in the Foldcer-Planck equation a source term to accoimt for the injection of particles so as to compensate those escaping and evaluated the rate constant in the extreme low-friction limit. Blomberg considered a symmetric, piecewise parabolic bistable potratial and obtained a partial solution of the Fokker-Hanck equation in terms of tabulated functions by requiring this piecewise analytical solution to be continuous, the rate constant is obtained. The result differs from that of Kramers only when the potential has a sharp, nonharmonic barrier. 

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. 

Now let us check whether the distribution functions have power-law tails or not in quasi-stationary states. The time shown in Fig. 6d is t/N17 = 0.0048,... 

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. 

Figure 12. Time series of the mean-square displacement of the phases aj t). N = 100,1000, and 10,000 from top to bottom. The vertical axis is the original scale only for N = 10,000, and is multiplied by 103 and 106 for N = 1000 and 100, respectively, just for a graphical reason. In Stage I where the system is quasi-stationary, the numerical results are approximated by solid curves that are obtained from Eq. (12) using functions in Eq. (19). After the system reaches equilibrium, diffusion becomes normal. Anomaly in diffusion is observed only in Stage II. The two short vertical lines on each curve show the end of Stages I and II, which correspond to the ones found in Fig. 2. [Reproduced with permission from Y. Y. Yamaguchi, Phys. Rev. E 68, 066210 (2003). Copyright 2004 by the American Physical Society.]...

The length of the first quasi-stationary stage and relaxation time reaching the third equilibrium stage increases as a nontrivial power-type function of degrees of freedom N, namely x NlJ. This time scale is also confirmed by observing temporal evolution of single-particle distribution of momenta. The distributions taken at the same scaled time (i.e., t/NlJ = constant) are well-superposed irrespective of values of N, while a trivial time scale x N is numerically excluded. 

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. 

Both in quasi-stationary and in equilibrium stages, fcorT is proportional to N, and (3 is almost constant. These simple scaling laws imply that fitting by stretched exponential functions is valid irrespective of degrees of freedom. 

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