Transient bimodality

The investigation of stochastic bistable reactions showed that processes leading to the stationary state might be decomposed in certain situations by 

In particular, a stochastic model of thermal explosion in a chemical system was studied by Baras. Frankowicz (1984). Their main results are (i) during the ignition regime the probability distribution function displays transient bimodality (ii) the distribution of ignition times displays a long time tail (iii) the mean ignition time calculated from the stochastic model is significantly shorter than that obtained from the deterministic model. 

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Baras, F. Frankowicz, M. (1984). Transient bimodality a new mechanism of stochasticity in explosive chemical reactions. In Nomequilibrium dynamics in chemical systems, eds C. Vidal A. Pacault (Springer Series in Synergetics, Vol. 27), p. 250. Springer Verlag, Berlin. 

Transient Bimodality A New Mechanism of Stochasticity in Explosive Chemical Reactions... 

In the supercritical region the probability distribution function displays transient bimodality. 

A stochastic description of explosion phenomena is set up, both for isothermal and for exothermic reaction mechanisms. Numerical simulations and analytic study of the master equation show the appearence of long tail and multiple humps in the probability distribution, which subsist for a certain period of time. During this interval the system displays chaotic behavior, reflecting the random character of the ignition process. An estimate of the onset time of transient bimodality is carried out in terms of the size of the system, the intrinsic parameters, and the initial condition. The implications of the results in combustion are discussed. 

A section by a line parallel to the X axis gives the probability profile for a given t, while a section by a line parallel to the t axis gives the evolution of P for given X. From this picture one can easily determine the way the most probable values evolves in time. The existence of a two-hump distribution will be reflected by a curve exhibiting limit points and hysteresis. Again however, contrary to ordinary situations in which hysteresis behavior appears when a parameter is varied, in the present case both the new branches of most probable values and the hysteresis region will be observed as time follows its course. e can refer to this as transient bimodality. 

The results just described are not. universal. In particular, the occurrence of transient bimodality depends on the size of the system, on its intrinsic parameters (Uo> ry 6, 6 ), and on the initial conditions. Generally speaking, for very large systems and for fixed parameters and initial conditions bimodality is bound to disappear, but the dependence of this trend on size is a weak one. Denoting by At the time interval of bimodality, one finds that... 

A fully quantitative treatment of the above intuitive ideas is difficult at the present time, for two reasons in the thermal case the death rate depends exponentially on the state variable and in the chemical case one deals with a birth and death process with highly nonlinear transition probabilities whose time-dependent behavior remains poorly known, despite recent significant progress [2,5] In a preceding paper [ ] we circumvented this difficulty in the thermal case by adopting an idealized piecewise linear representation of the transition rates, which captures their essential features while allowing a rather exhaustive analytic treatment. Here we present an alternative description using the full form of the transition rates, and the more limited aim we fix to ourselves is to determine the critical time beyond which transient bimodality is expected to occur. 

Conversely, a necessary condition for transient bimodality to occur is that a2 reaches values which are of the order of some power of e, and subsequently changes sign from negative to positive values. In such a case one would have to push the expansion of ( > at least up to fourth order terms. For suitable values of the coefficients the function - ( )(0) would represent a "stochastic potential" having two minima and a maximum. In that sense the evolution of our system could be viewed as the motion of a "particle" in a time-dependent potential, which is similar to the deterministic one (Fig. 4) for the initial and final stages but is qualitatively different from it for intermediate times. 

Eldering, A., and R. M. Glasgow, Short-Term Particulate Matter Mass and Aerosol-Size Distribution Measurements Transient Pollution Episodes and Bimodal Aerosol-Mass Distributions, Atmos. Environ., 32, 2017-2024 (1998). 

Here Cq = c(z = 0,t) and Dg is the effective diffusion coefficient (porosity and tortuosity effects are incorporated in Dg). If the upstream (high) pressure is constant and much larger than the downstream (low) pressure, the slope of the asymptote will correspond to the steady state and so it is possible to determine the diffusivity under both steady state and transient conditions from a single permeation experiment. With a narrow and unimodal pore size distribution both methods yield reasonable consistent values. Large discrepancies point to strong microstructural effects (bimodal broad distribution, many dead ends, many defects). 

The NaPSS/LaCl3 dynamics is characterized by a bimodal decay of the concentration correlation function at very low salt concentration. Two dynamic modes (fast and slow) are present. The fast process is interpreted as a diffusion of blobs of a transient network in the semidilute regime. The origin of the slow mode is not always understood. It can be interpreted as the diffusion of multichain domains (clusters). Due to the large dimensions of the domains, the slow mode does not correspond to a pure diffusive motion. 

The approximate change in the size distribution that will have taken place after a 12-h intermission of new particle production is shown in Fig. 7-5 by the dashed curve. It results in a bimodal size distribution, which in addition to the accumulation peak now contains a transient peak caused by the incomplete coagulation of Aitken particles. We have previously designated this transient the nucleation mode. The ensuing size distribution gives a better representation of the natural aerosol, even though its resemblance to the size spectra in Fig. 7-1 is still marginal. 

We have previously described that 3,4-dihydroxy-phenylacetic acid (PAA) directly decreased NOS activity by 40%, while caffeic acid did not. Furthermore, PAA increased iNOS mRNA while it concurrently reduced eNOS mRNA expression (Kampa et al. 2004). We have also reported that catechin, epicatechin, quercetin, and resveratrol decrease NO secretion by prostate cancer cell lines (LNCaP, PC3, and DU145) and inhibit NO production by T47D breast cancer cells. Although a concurrent decrease in total NOS (eNOS and iNOS) activity was observed after treatment for 24 h or longer, these agents induced a transient early increase in NOS activity. This bimodal effect indicated a possible dual action of polyphe-... 

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