Jensen’s inequality

Jarzynski s identity, (5.8), immediately leads to the second law in the form of (5.6) because of Jensen s inequality, (e x) > e. Moreover, in the limit of an infinitely fast transformation, r —> 0, we recover free energy perturbation theory. In that limit, the configurations will not relax during the transformation. The average in... 

This is a case of Jensen s inequality, which states that, for a convex function, a chord between any two points is always below the curve that is, the property of the average is always higher than the average of the two properties. On the other hand, it is not advantageous to do mixing when the property is a concave function of the composition, so that the mixture property is less than the arithmetic average. 

The equality in Eq. (16) immediately implies, by using Jensen s inequality, the following inequality. 

This inequality is called Jensen s inequality after the Danish mathematician who first introduced it. 

Proof. With the definition of conditional entropy and Jensen s inequality,... 

By Jensen s inequality it is easy to show that (C)Da>i < (C)Dalower frequency of annihilation encounters and slower overall decay when the reactants are uniformly distributed, compared to the case of a distribution concentrated in some small regions. Therefore, the steady state average concentration, or total amount, is an increasing function of the stirring rate in this nonlinear annihilation process. Such change of the overall reactivity of a second order reaction due to chaotic mixing was observed experimentally by Paireau and Tabeling (1997). 

Jensen s inequality (10] states that given a convex function R, arbitrary function c and probability density p(6). 

As follows from the general Jensen s inequality, the acceptor-gated kinetics proceeds faster than the donor-gated kinetics for an initial equilibrium distribution of the two gate states [378g], Such an asymmetry in the ET kinetics diminishes for faster gate interconversion rates and for low acceptor concentrations. 

The crucial step in the argument is to use the fact that the function x log x is convex for x O. That is, for every two points pi and P2 on the curve y = x log x, the points on the arc of y between pi and p2 lie below the chord joining pi and P2, as illustrated in Fig. 8. For such a function Jensen s inequality applies. This inequality states that if / and are positive functions, then... 

It can be shown using Jensen s Inequality [17] that, if the execution time function r(0 is convex (i.e. the gradient is constant or increases over calendar time), the total number of failures, rifl edh when the fix delay is included is bounded by ... 

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