Stochastic potential

Shepherd, T.D., Hernandez, R. Ghemical reaction dynamics with stochastic potentials beyond the high-friction limit, J. Ghem. Phys. 2001,115,2430. 

Stochastic bifurcation problems in chemical systems have been analysed by Lemarchand (1980). He proposed a systematic expansion of the free-energy-like quantity ( stochastic potential ) in powers of... 

Critical dynamic analysis using renormalisation techniques were presented by Walgraef et aL, 1982. A method based on the systematic study of the Taylor expansion of the stochastic potential was applied for reaction-diffusion systems exhibiting Hopf bifurcation (Fraikin Lemarchand, 1985). The Poissonian representation technique of Gardiner Chaturvedi (1977) is also an efficient procedure for evaluating the parameters of fluctuations. A more rigorous derivation of reaction-diffusion equations with fluctuations were given by De Masi et al (1985). 

The thermodynamics of transport properties, diffusion, thermal conduction and viscous flow is taken up in Chap. 8, and non-ideal systems are treated in Chap. 9. Electrochemcial experiments in chemical systems in stationary states far from equilibrium are presented in Chap. 10, and the theory for such measurements in Chap. 11 in which we show the determination of the introduced thermodynamic and stochastic potentials from macroscopic measurements. 

Theory of Determination of Thermodynamic and Stochastic Potentials from Macroscopic Measurements... 

We have several purposes in mind for this chapter First we present a development of the theory for the determination of thermodynamic and stochastic potentials, (f), for non-equilibrium systems from electrochemical measurements second, a parallel development for neutral (not ionic) systems in general and third, the presentation of suggestions for testing the consistency of the master equation with such measurements [1]. 

For the first purpose we choose a chemical reaction system with some ionic species, as for example the minimal bromate reaction, for which we presented some experiments in Chap. 10. The system may be in equilibrium or in a nonequilibrium stationary state. An ion selective electrode is inserted into the chemical system and coimected to a reference electrode. The imposition of a current flow through the electrode coimection drives the chemical system (CS) away from its initial stationary state to a new stationary state of the combined chemical and electrochemical system (CCECS), analogous to driving the CS away from equilibrium in the same maimer. A potential difference is generated by the imposed current, which consists of a Nernstian term dependent on concentrations only, and a non-Nernstian term dependent on the kinetics. We shall relate the potential difference to the stochastic potential for this we need to know the ionic species present and their concentrations, but we do not need to know the reaction mechanism of the chemical sj tem, nor rate coefficients. 

For the second purpose we offer a suggestion for reaction systems with or without ionic species for an indirect method of determining the stochastic potential from macroscopic measurements. We impose an influx of any of the stable intermediate chemical species into the system (CS), and thus displace the CS from its initial stationary state to a new stationary state of the combined CS and the imposed influx. We measure the concentrations of species in the new stationary state and repeat this experiment for different imposed influx rates. We can fit these measurements to an assumed reaction... 

Determination of the Stochastic Potential in Chemical Systems with Imposed Fluxes... 

Now we impose a flux of a chemical species present in the system and inquire on the effect of that imposition on the stochastic potential of the system. For that we need to go from the deterministic kinetic equations to a stochastic... 

Thus from measurements with imposed flux we obtain the derivative of the stochastic potential for the system without imposed flux, but we need kinetic information, the rate coefficients in as well. 

For multi-variable systems this approach is more difficult the determination of the stochastic potential requires sufficient measurements to determine rate coefficients and then the numerical solution of the stationary form of the master equation. Details of this procedure are described in Appendix A of [1]. 

A direct test of the master equation for systems in non-equilibrium stationary states comes from the measurements of concentration fluctuations such measurements have not been made yet. Some other tests of the master equation are possible based on the earlier sections in this chapter, where we can compare measurements of the stochastic potential with numerical solutions of the master equation (which requires knowledge of rate coefficients and the reaction mechanism of the system). 

Approximately, at equistability the height of the probability peak at point 2 equals that at point 4. To either side of the value of k at equistability, the peak of the more stable stationary state is higher than the other peak. A comparison of deterministic and stochastic calculations (not experiments) has been discussed in a different context, that of viewing the stochastic potential as an excess work [4,5]. 

In [4] further studies are presented on fluctuations near limit cycles, on the basis of approximate solutions of the master equation (rather than the Fokker-Planck equation). In [5] there is an analysis of fluctuations (the stochastic potential) for a periodically forced limit cycle, with references to earlier work. Both these articles are intensive mathematical treatments. 

Some progress has been made in the direction of applying the thermodynamic and stochastic theory of rate processes presented here to disordered systems. In some cases [35] it is possible to construct a stochastic potential with the properties the same as that for ordered systems discussed in Chaps. 2-11. A general set of fluctuation-dissipation relations has been derived that establishes a connection between the expression of the average kinetic curve,... 

U( x )) is the stochastic potential. This expression supposes 0 i.e. (the volume V of the system becomes infinite) the concentrates on the deterministic stationary profiles. . = Xj = x or x (system uniformly metastcible or stable) the potential should then have minima for these two particular distributions. The assumption (19) is of course based on the fact that the deterministic results must be recovered from the stochastic theory when V 00. 

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. 

Our results suggest that the above dynamics can be viewed as an evolution in a stochastic potential whose qualitative aspect depends on time at the beginning it is similar to the deterministic potential, but subsequently it deforms (the deformation depending on the volume and initial conditions) and develops a second minimum. This minimum is responsible for the transient "stabilization" of the maximum of P(X,t) before the inflexion point. As the tunneling towards the other minimum on the stable attractor goes on, the first minimum disapears and the asymptotic form of the stochastic potential, determining the stationary properties of P(X,t), reduces again to the deterministic one. This phenomenon of "phase transition in time" is somewhat reminiscent of spinodal decomposition. 

The dominant term in the exponential, referred as stochastic potential U, enjoys at the steady-state the remarkable Lyapunov property [20,21]... 

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