The probability flux

Now P r,p is of dimensionality I The flux vector J has the dimensionality l t and expresses the passage of walkers per luiit time and area in the J direction. 

It is important to emphasize that, again, the first of Eqs (8.115) is just a conservation law. Integrating it over some volume Q enclosed by a surface So and denoting 

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More specifically, the condition that the probability flux at the boundaries is zero and the condition that the mean mixture-fraction vector is constant in a homogeneous flow lead to natural boundary conditions (Gardiner 1990) for the mixture-fraction PDF governing equation. 

As an example of a four-environment model,144 consider the generalized mixing model proposed by Villermaux and Falk (1994) shown in Fig. 5.23. The probability exchange rates r control the probability fluxes between environments.145 The micromixing terms for the probabilities can be expressed as... 

The first terms on the right-hand sides of (6.16) and (6.17) are related to the probability flux at infinity. For all well behaved PDF and all well behaved functions18 Q(U, ),... 

The probability flux is determined by a balance of generalized mechanical and Brownian forces, of the form... 

The elastic force is given by the sum of a mechanical force —dU/dq and a corresponding Brownian force. The form of the Brownian force may be inferred by requiring that Fa vanish when /( ) = v /eq( ). in order to guarantee that the probability flux 7 vanishes in thermal equilibrium. This requirement yields an elastic force... 

Exercise. Following (4.7) the probability flux (4.2) is decomposed into a mechanical and a dissipative part. They are odd and even, respectively, with respect to time reversal. In equilibrium the dissipative part vanishes. 

In their studies of metals in Chesapeake Bay, however, Bieri et al. (1982) claim that more than 60 % of both the Pb and Mn input is retained in the bed sediments. In their recent studies of heavy metals in Delaware Bay (USA), Church, Tramontano and Murray (1984 and later personal communication) calculated retention of 92 % of the Mn, 37% of the Cu and 32 % of the Cd input to that estuary. However, losses from the estuary in that analysis were based on calculations of the probable flux out of the mouth of the Bay using a layered flow model. When sediment concentrations and accumulation rates were used, only small amounts of Mn and Cd appeared to be retained in the system (Church, personal communication). At this point we are not aware of any convincing evidence that clearly contradicts the findings regarding the behavior of Pb, Cu,Mn or Cd in Narragansett Bay. Unfortunately, the number of mass balances for these elements is so small that this is not a particularly reassuring claim. 

The right side in equation (23) has a clear physical meaning. The probability flux ji is proportional to C, the probability of finding the system in the left well. Also the probability flux is proportional to w/(2tt), the frequency with which the particle hits the barrier wall, and to the exponential tunneling factor, which is the probability of tunneling through the barrier at each hit. 

Thus, again, as in the pseudo-JT effect considered in Section 3 and, also, in the E <8> e case [7], the tunneling rate E is proportional to the probability flux through the bottleneck point of the potential barrier. Similar to equation (21), the right-side (9 > 0) ground-state WKB wave function under the barrier is... 

In the second line, the equation is written in the form of a continuity equation that, formally, is identical to the continuity equations in quantum mechanics, Eq. (4.106), and in classical statistical mechanics, Eq. (5.17). The probability flux density is identified... 

If na is the probability of finding reactants at the a-well, and ks is the rate constant for going from the a-well to the c-well, then the probability flux j across the barrier is... 

On constructing the probability flux vector Q, we start with the relation... 

The expression for the probability flux vector that we use here... 

Therefore, we assume that the barrier is high and the probability flux Q couples two compact spots in the orientational space that are localized at the poles of the unit sphere. Accordingly, on the total flux the requirement of nondivergency is imposed that is, it is assumed that in the whole coordinate interval except for the vicinities [0, f> ] and [If2,71 ] of the poles, the quantity B sin if Qn is constant. Applying this condition to defined by relation (4.37), one comes out with the equation that couples the gradients of the energy and of the distribution function ... 

On the other hand, in line with the hypothesis we exploit and in accordance with Eq. (4.39), the probability flux is... 

The limiting case Z) 0 in Eq. (4.4) is pathological. In such a case the probability flux vanishes at x = 0, thereby making the solution dq>oident on the initial position Xq. If Xq > 0, the steady state distribution is entirely bounded to the positive semiaxis. In this semiaxis region the solution is stiU described by Eq. (4.3) with D = 0. Precisely the reverse takes place when Xq < 0. Henceforth we shall always refer ourselves to the case Xq > 0. For Q > gi, the equilibrium distribution diverges at x = 0 ... 

What is the probability current of reactive trajectories This probability current is the vector field Jab x) defined in f2 AuB) which is such that, given any surface S c f2 AU B) which is the boundary of a region fis, the surface integral of Jab (x) over S gives the probability flux of reactive trajectories across S. More precisely,... 

Fig. 2. Schematic of the streamlines of the probability current of reactive trajectories out of Sa C dA, and the pushed forward region S s) at times 0 < ti < T2 < Ts < Sa- The collection of these regions, S (t) t > 0 = Ut->o

This equation defines the streamlines of Jab(x) (see Fig. 2). Eventually, every streamline x(t) must reach B for some tb > 0 and we will terminate the streamlines on dB by assuming that x(t) = x tb) G dB for t > tb (notice that Tb depends on x(0) G dA and may be different for every streamline). Using the divergence theorem again, the probability flux through the surface S t) = Uj,(o) s x(t) (that is, the push-forward of the surface Sa along the... 

The results presented so far indicates that the isocommittor functions q x) and q+ x) are essential to understand the mechanism of a reaction. However these results do not say how to compute q x) and q+ x), except via the solution of (24) and (23) which is a formidable task, even numerically, when the dimensionality of the system is large (that is, in any situation of interest). In this section, we show that the transition tube carrying most of the probability flux of reactive trajectories can be identified under the assumption that this tube is localized, in a sense made precise below. As show in Sect. 7, this is a way to make practical the probabilistic framework presented so far while standard numerical methods based on finite difference or finite element are inappropriate to determine q-(x) and q x), under the assumptions of this section one can develop algorithms to estimate these functions locally inside the tubes carrying most of the probability flux of reactive trajectories, i.e. where they matter most. 

The main claim of this section is that, if the localized tube assumption holds, then the tube U,j [o,i]C ( ) dze[o,i]D z) is the tube carrying most of the probability flux of reactive trajectories and it can be determined by an algorithmic procedure which is much simpler than solving (23) and (24) for q+ x) and q- x). 

In this chapter, we have shown why the recent transition path theory (TPT) offers the correct probabilistic framework to understand the mechanism by which rare events occur by analyzing the statistical properties of the reactive trajectories involved in these events. The main results of TPT are the probability density of reactive trajectories and the probability current (and associated streamlines) of reactive trajectories, which also allows one to compute the probability flux of these trajectories and the rate of the reaction. It was also shown that TPT is a constructive theory under the assumption that the reaction channels are local, TPT naturally leads to algorithms that allow to identify these channels in practice and compute the various quantities that TPT offers. 

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