Probability distribution evolution times

Yamada and Kawasaki [68, 69] proposed a nonequilibrium probability distribution that is applicable to an adiabatic system. If the system were isolated from the thermal reservoir during its evolution, and if the system were Boltzmann distributed at t — x, then the probability distribution at time t would be... 

The evolution of the probability distribution over time consists of adiabatic development and stochastic transitions due to perturbations from the reservoir. As above, use a single prime to denote the adiabatic development in time A, r —> r, and a double prime to denote the final stochastic position due to the influence of the reservoir, T —> T . The conditional stochastic transition probability may be taken to be... 

The theory of first passage times enables one to take a master equation like eq. (15.12) that describes the evolution of a probability distribution P t) for a population, and derive from it an equation for the probability distribution of times T t) for reaching a population m at time t when we start out with n at time zero. For the Szabo model of eq. (15.12), we find... 

Again denoting the adiabatic evolution over the intermediate time A, by a prime, Iv = r(A( r), the adiabatic change in the even exponent that appears in the steady-state probability distribution is... 

This is equal and opposite to the adiabatic change in the odd exponent. (More detailed analysis shows that the two differ at order Af, provided that the asymmetric part of the transport matrix may be neglected.) It follows that the steady-state probability distribution is unchanged during adiabatic evolution over intermediate time scales ... 

EVOLUTION TIMES OF PROBABILITY DISTRIBUTIONS AND AVERAGES—EXACT SOLUTIONS OF THE KRAMERS PROBLEM... 

We suppose that at initial instant t = 0 all Brownian particles are located at the pointx = xo, which corresponds to the initial condition W(x, 0) = 8(x — xo). The initial delta-shaped probability distribution spreads with time, and its later evolution strongly depends on the form of the potential profile (p(x). We shall consider the problem for the three archetypal potential profiles that are sketched in Figs. 3-5. 

Monotony Condition. Let us turn back to the monotony condition of the variations of P t) or W( , t). If, for example, the point l is arranged near a0, where the initial probability distribution W(x, 0) = 8(x — xo) is located, the probability density W( , t) early in the evolution may noticeably exceed the final value W( , oo). For such a situation the relaxation time 0( ) according to (5.76) may take not only a zero value, but also a negative one. In other words,... 

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