Probability fluxes

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. 

Note that the E-model is asymmetric in the sense that environment 1 always grows in probability, while environment 2 always decreases. In general, this is not a desirable feature for a CFD-based micromixing model, and can be avoided by adding a probability flux from environment 1 to environment 2, or by using three environments and letting environment 2 represent pure fluid that mixes with environment 1 to form environment 3. Examples of these models are given in Tables 5.1-5.5 at the end of this section. 

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. 

This formula demonstrates that the tunneling splitting is determined—like the imaginary part of metastable state energy (A.20)—as a normalized probability flux through the dividing line. In the present case this flux corresponds to coherent probability oscillations between the wells rather than exponential decrease of the survival probability in the well, so A is a real value. 

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. 

Fig. 3. The four-vertex Petersen graph representing the tunneling paths between the trigonal wells in the T t2 case. The arrows show the direction of probability flux from the corresponding wells.

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... 

The tunneling integral, IIl2, is shown to be proportional to density of probability flux, j = — rf>0(0)(()), at the bottle-neck point, 2 = 0,... 

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... 

In Section 5.2, we used an expression for the flux operator. In most quantum mechanics textbooks expressions for the probability current density, or probability flux density, are given in terms of the wave function in the coordinate representation. We need an expression for the flux density operator without reference to any particular representation, and since it is rarely found in the textbooks, let us in the following derive this expression. 

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 ... 

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