Steady-state probability distribution

The aim of this section is to give the steady-state probability distribution in phase space. This then provides a basis for nonequilibrium statistical mechanics, just as the Boltzmann distribution is the basis for equilibrium statistical mechanics. The connection with the preceding theory for nonequilibrium thermodynamics will also be given. 

The odd contribution to the nonequilibrium steady-state probability distribution is just the exponential of this entropy change. Hence the full nonequilibrium steady-state probability distribution is... 

This result is a very stringent test of the present expression for the steady-state probability distribution, Eq. (160). There is one, and only one, exponent that is odd, linear in Xr, and that satisfies the Green-Kubo relation. 

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 ... 

The steady-state probability distribution for a system with an imposed temperature gradient, pss(r p0, pj), is now given. This is the microstate probability density for the phase space of the subsystem. Here the reservoirs enter by the zeroth, (10 = 1 /k To, and the first, (i, = /k T, temperatures. The zeroth energy moment is the ordinary Hamiltonian,... 

Monte Carlo heat flow simulation, 68—70 steady-state probability distribution, nonequilibrium statistical mechanics, 40-43... 

Nonequilibrium statistical mechanics Green-Kubo theory, 43-44 microstate transitions, 44-51 adiabatic evolution, 44—46 forward and reverse transitions, 47-51 stationary steady-state probability, 47 stochastic transition, 464-7 steady-state probability distribution, 39—43 Nonequilibrium thermodynamics second law of basic principles, 2-3 future research issues, 81-84 heat flow ... 

It should be noted that besides being widely used in the literature definition of characteristic timescale as integral relaxation time, recently intrawell relaxation time has been proposed [42] that represents some effective averaging of the MFPT over steady-state probability distribution and therefore gives the slowest timescale of a transition to a steady state, but a description of this approach is not within the scope of the present review. 

If a potential profile is of the type I (see Fig. 3) when (p(x) goes to plus infinity fast enough at x > oo, there is the steady-state probability distribution... 

First, the current state of affairs is remarkably similar to that of the field of computational molecular dynamics 40 years ago. While the basic equations are known in principle (as we shall see), the large number of unknown parameters makes realistic simulations essentially impossible. The parameters in molecular dynamics represent the force field to which Newton s equation is applied the parameters in the CME are the rate constants. (Accepted sets of parameters for molecular dynamics are based on many years of continuous development and checking predictions with experimental measurements.) In current applications molecular dynamics is used to identify functional conformational states of macromolecules, i.e., free energy minima, from the entire ensemble of possible molecular structures. Similarly, one of the important goals of analyzing the CME is to identify functional states of areaction network from the entire ensemble of potential concentration states. These functional states are associated with the maxima in the steady state probability distribution function p(n i, no, , hn). In both the cases of molecular dynamics and the CME applied to non-trivial systems it is rarely feasible to enumerate all possible states to choose the most probable. Instead, simulations are used to intelligently and realistically sample the state space. 

Perform a computer simulation of the CME for the system of Equation (11.25), using the parameters given in the legend of Figure 11.6. (Assume that A, B, and C are held at fixed numbers.) From the simulated stochastic trajectory, can you reproduce the steady state probability distribution shown in Figure 11.6 ... 

If k(r) represents the probability that molecules A and B, separated by a distance r in a solution, will react and P r) represents the steady-state probability distribution for finding a pair of molecules A and B distant r apart, show that P(r) is close to the random value in a nonreacting solution only when k r) is very small for small r. Show that when the probability of reaction of A and B on an encounter is very high, P(r) has very strong deviations from equilibrium (i.e., there is very much less chance of finding A near B). 

Figure 4. Steady state probability distribution

Fig. 28.3. The steady state probability distribution for a switch modulated by a monomeric transcription factor. The two peak behavior at low u) (a highly non-adiabatic system) comes from the extinction of the transcription factor population...

Consider first the barrier dynamics problem, which is defined by replacing the potential barrier by an inverted parabola, Eq. (2.4), and by looking for a steady-state probability distribution which satisfies Eqs. (2.6) and (2.8). Here we follow the treatment of Hanggi and Mojtabai. Equations (5.1) and (5.2), with K(x) = Eg — are used to obtain (with a procedure due to... 

Here I is the unit matrix and is the frequency square matrix in the space of nonreactive modes Equation (6.1) is a generalized Langevin equation of the form used in treating the one-dimensional case in Section V, and leads to the result of Eq. (5.25) (with m = 1) for the steady-state probability distribution of the reactive mode near the barrier. In the present multidimensional treatment it is convenient to redefine the distribution according to... 

Ching, W. K., Zhang, S., Ng, M. K., and Akutsu, T. (2007). An approximation method for solving the steady-state probability distribution of probabilistic Boolean networks. Bioinformatics, 23 1511. 

Steady-state probability distribution would be obtained by the mathematical detailed balance equation ... 

In the chemical master equation, the steady-state probability distribution of the equUihrium steady state is a Poisson distribution. For Schlogl s model steady-state probability distributions become... 

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