Stationary distribution

A stationary ensemble density distribution is constrained to be a functional of the constants of motion (globally conserved quantities). In particular, a simple choice is pip, q) = p (W (p, q)), where p (W) is some fiinctional (fiinction of a fiinction) of W. Any such fiinctional has a vanishing Poisson bracket (or a connnutator) with Wand is thus a stationary distribution. Its dependence on (p, q) through Hip, q) = E is expected to be reasonably smooth. Quanttun mechanically, p (W) is die density operator which has some fiinctional dependence on the Hamiltonian Wdepending on the ensemble. It is also nonnalized Trp = 1. The density matrix is the matrix representation of the density operator in some chosen representation of a complete orthononnal set of states. If the complete orthononnal set of eigenstates of die Hamiltonian is known ... 

Thus, the neglect of the off-diagonal matrix elements allows the change from mixed states of the nuclear subsystem to pure ones. The motion of the nuclei leads only to the deformation of the electronic distribution and not to transitions between different electronic states. In other words, a stationary distribution of electrons is obtained for each instantaneous position of the nuclei, that is, the elechons follow the motion of the nuclei adiabatically. The distribution of the nuclei is described by the wave function x (R i) in the potential V + Cn , known as the proper adiabatic approximation [41]. The off-diagonal operators C n in the matrix C, which lead to transitions between the states v / and t / are called operators of nonadiabaticity and the potential V = (R) due to the mean field of all the electrons of the system is called the adiabatic potential. 

We studied vacancy segregation near interphase and antiphase boundaries using the MFA and PCA approaches described in Sec. 6 below. For the A-B alloy with vacancies, the stationary distribution of mean vacancy occupations =< > can be explicitly... 

In what follows the illustration of this method is concerned with equation (1) from (Section 1) capable of describing the stationary distribution of temperature over a homogeneous bar 0 < a < 1. The equation of the heat balance can be written on the segment ... 

Assuming the MPC dynamics is ergodic, the stationary distribution is microcanonical and is given by... 

Figure 5.37 depicts the stationary distribution of the electroactive substance (the reaction layer) for kc—> oo. The thickness of the reaction layer is defined in an analogous way as the effective diffusion layer thickness (Fig. 2.12). It equals the distance [i of the intersection of the tangent drawn to the concentration curve in the point x = 0 with the line c = cA/K,...

Let us consider the case when the diffusion coefficient is small, or, more precisely, when the barrier height A is much larger than kT. As it turns out, one can obtain an analytic expression for the mean escape time in this limiting case, since then the probability current G over the barrier top near xmax is very small, so the probability density W(x,t) almost does not vary in time, representing quasi-stationary distribution. For this quasi-stationary state the small probability current G must be approximately independent of coordinate x and can be presented in the form... 

If the barrier is high, the probability density near xmln will be given approximately by the stationary distribution ... 

We have obtained the relaxation time—that is, the time of attainment of the equilibrium state or, in other words, the transition time to the stationary distribution W( , oo) at the point in the rectangular potential profile. This time depends on the delta-function position xo and on the observation point location . 

We consider a stationary distribution, where the velocity distribution of the fluid is at equilibrium ... 

The stationary theory deals with time-independent equations of heat conduction with distributed sources of heat. Its solution gives the stationary temperature distribution in the reacting mixture. The initial conditions under which such a stationary distribution becomes impossible are the critical conditions for ignition. 

Under fuel cell operation, a finite proton current density, 0, and the associated electro-osmotic drag effect will further affect the distribution and fluxes of water in the PEM. After relaxation to steady-state operation, mechanical equilibrium prevails locally to fix the water distribution, while chemical equilibrium is rescinded by the finite flux of water across the membrane surfaces. External conditions defined by temperature, vapor pressures, total gas pressures, and proton current density are sufficient to determine the stationary distribution and the flux of water. 

Under steady-state operation with a constant water flux through the membrane, mechanical equilibrium of water will prevail locally at external membrane faces and inside fhe membrane that involves the balance of local liquid, gas, capillary, and elastic pressures. This condition corresponds to a stationary distribution of wafer in the membrane. However, the condition of chemical equilibrium, Equafion (6.5), will be violafed due to the chemical flux of species. 

The parallel-replica method [5] is perhaps the least glamorous of the AMD methods, but is, in many cases, the most powerful. It is also the most accurate AMD method, assuming only first-order kinetics (exponential decay) i.e., for any trajectory that has been in a state long enough to have lost its memory of how it entered the state (longer than the correlation time icorr, the time after which the system is effectively sampling a stationary distribution restricted to the current state), the probability distribution function for the time of the next escape from that state is given by... 

Slightly highertemperature(80-120°C), lower cost membrane materials for more efficient waste heat utilization for cogeneration in stationary/distributed applications or as process heat in a fuel reformer, reducing radiator size for transportation applications and for reduced carbon monoxide (CO) management requirements. 

Hatano and Sasa [56] have derived an interesting result for nonequilibrium transitions between steady states. Despite the generality of the Hatano-Sasa approach, explicit computations can be worked out only for harmonic traps. In the present example the system starts in a steady state described by the stationary distribution of Eq. (60) and is driven away from that steady state by varying the speed of the trap, v. The stationary distribution can be written in the frame system that moves solidly with the trap. If we definey (f) = x — x (f) thenEq. (60) becomes... 

In the first case, the limit (for t- co) distribution for the auxiliary kinetics is the well-studied stationary distribution of the cycle A A , +2, described in Section 2 (ID-QS), (15). The set A j+], A . c+2, , n is the only ergodic component for the whole network too, and the limit distribution for that system is nonzero on vertices only. The stationary distribution for the cycle A i+] A t+2. ., A A t+i approximates the stationary distribution for the whole system. To approximate the relaxation process, let us delete the limiting step A A j+] from this cycle. By this deletion we produce an acyclic system with one fixed point, A , and auxiliary kinetic equation (33) transforms into... 

We apply this approach now and demonstrate its applicability in more details later in this section. For the quasi-stationary distribution on the cycle we get C(j-T+ = c k /k (K/reaction network is transformed... 

Again we should analyze, whether this new cycle is a sink in the new reaction network, etc. Finally, after a chain of transformations, we should come to an auxiliary discrete dynamical system with one attractor, a cycle, that is the sink of the transformed whole reaction network. After that, we can find stationary distribution by restoring of glued cycles in auxiliary kinetic system and applying formulas (11)-(13) and (15) from Section 2. First, we find the stationary state of the cycle constructed on the last iteration, after that for each vertex Ay that is a glued cycle we know its concentration (the sum of all concentrations) and can find the stationary distribution, then if there remain some vertices that are glued cycles we find distribution of concentrations in these cycles, etc. At the end of this process we find all stationary concentrations with high accuracy, with probability close to one. 

To construct the approximation to the basis of stationary distributions of iV, it is sufficient to apply the described algorithm to distributions concentrated on a single fixed point A , cj = for every i. 

Suppose V n = 0, — 1, -2,. .. is a set of real, normalized random variables subject to a stationary distribution with covariance... 

As a further specialization suppose that q(x) is constant in an interval (— T, T) and vanishes outside it. The constant is necessarily equal to v/2 T= p and represents the average number of events per unit time. In the limit T -> oo, v -> oo, with fixed p, one approaches a stationary distribution of dots, called shot noise The fact that stationary distributions can only be described by a limiting process is another drawback of the present treatment of random dots which will be overcome in the next section. 

Note that for a strictly stationary distribution both integrals diverge since the number of contributing terms u(xa) is infinite. One therefore has to... 

Remark. Consider a Markov process that can be visualized as a particle jumping back and forth among a finite number of sites m, with constant probabilities per unit time. Suppose it has a single stationary distribution psn, with the property (5.3). After an initial period it will be true that, if I pick an arbitrary t, the probability to find the particle at n is ps . That implies that psn is the fraction of its life that the particle spends at site n, once equilibrium has been reached. This fact is called ergodicity. For a Markov process with finitely many sites ergodicity is tantamount to indecom-posability. ) In (VII.7.13) a more general result for the times spent at the various sites is obtained. 

The case of a decomposable matrix (2.6) merely means that one has two non-interacting systems, governed by two M-equations with matrices A and B, respectively. A non-trivial example is a system in which all transitions conserve energy each energy shell E has its own M-equation and its own stationary distribution . The stationary solutions of the total M-equation are linear superpositions of them with arbitrary coefficients nEi... 

Show that there is no stationary distribution and that all states are transient (compare... 

From this follows first that there cannot be more than one stationary distribution. Secondly, that any other solution tends to it. This is the desired result. 

The following calculation proves the approach to the stationary distribution in an alternative way, which is more familiar to physicists. In addition it provides some information about how this approach takes place, viz., in such a way that a certain functional of the distribution increases monotoni-cally. In fact, it turns out that there are many such functionals. The reasons why one of them has a special role in physics and is honored with the name entropy will be discussed. 

Exercise. Find the stationary distribution for the asymmetric random walk with reflecting boundary described for n 1 by (2.13) with a > fi, together with the special boundary equation... 

Exercise. In the case of a half-infinite interval the stationary distribution is given by (3.8) and (3.9) with N = oo. Study the condition on rn,gn for (3.9) to converge. Find an example in which these conditions are not satisfied. What is the physical consequence of the lack of convergence ... 

---