Suggestions for Experimental Tests of the Master Equation

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). 

There are other indirect methods. Consider a one-variable system (or an effectively one variable). Let the system have multiple stationary states and in Fig. 11.1, taken from [1], we show a schematic diagram of the hysteresis loop in such systems. 

Several experiments can be suggested to test aspects of the predictions of the master equation. To construct a diagram as in Fig. 11.1 from the master equation we need to know or guess rate coefficients and the reaction mechanism of the system. For the experiments we need to measure the concentration c of a given species as the the influx coefficient is varied. Thus we establish the solid lines by experiment. If we can form a CCECS, as discussed in Sects. 11.2 and 11.3 of this chapter, then we can locate the combined system at point 1 on line A by imposing a given current flow. This point is a stable stationary state of the combined system. If the imposed current is stopped (the electrochemical system is disconnected) then the chemical system will return deterministically 

The same approach works for the displacement of a system by imposition of an influx of a given species, see Sect. 11.4 of this chapter. 

Equistability of a homogeneous stable stationary state on the upper branch of the hysteresis loop, labelled I in Fig. 11.1, with a homogeneous stable stationary state on the lower branch, labelled II, occurs at one value of the influx coefficient k within the loop. Say that point occms at the location of line A. The predictions of the stationary solution of the master stochastic master equation are (a) the minimmn of the bimodal stationary probability distribution is located on the separatrix, and (b) at equistability the probability of fluctuations P(c) obeys the condition 

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