Most probable paths

In this derivation, the diffusion coefficient which is used is really a parameter, since it is not certain which gas diffusion rate is controlling, that of hydrogen into a pore, or that of water vapour out of die pore. The latter seems to be the most probable, but the path of diffusion will be vety tormous drrough each pore and drerefore the length of the diffusion path is ill-defined. 

Computations have shown that in the quantum region it is possible to have various most probable transition paths (ranging from the classical minimum energy path (MEP) to the straight-line one-dimensional tunneling of early models), depending on the PES geometry. 

From the very simple WKB considerations it is clear that the tunneling rate is proportional to the Gamov factor exp —2j[2(F(s(0) — )] ds, where s Q) is a path in two dimensions Q= 61)62 ) connecting the initial and flnal states. The most probable tunneling path , or instanton, which renders the Gamov factor maximum, represents a compromise of two competing factors, the barrier height and its width. That is, one has to optimise the instanton path not only in time, as has been done in the previous section, but also in space. This complicates the problem so that numerical calculations are usually needed. 

Tortuosity t is basically a correction factor applied to the Kozeny equation to account for the fact that in a real medium the pores are not straight (i.e., the length of the most probable flow path is longer than the overall length of the porous medium) ... 

The path variables of the PPM corresponds to the cluster probabilities of the CVM by which the free energy is minimized to obtain the most probable state. Likewise, under a set of constraints, the PPF is maximized with respect to the path variables for each time step, which yields the optimized set of path variables. Since a set of path variables, + At), relates cluster probabilities t and at time t + At... 

It is clear that if S> Xj, particle-particle collision happens most probably and X(9,P) = Xh. If Sparticle-wall collision happens within the travel distance of <5/cos 9. Then, the local free path of the test particle, X, is the average oiX 9,0) over the whole ranges of 6 and j3. That is. 

Figure 10. The smooth solid profile denotes the barrier along the most probable path. Thick horizontal lines at low energies and the shaded area at energies above the threshold represent energy levels available at size N. The jagged curves above the most probable path demonstrate generic paths, while the jagged curve beneath the most probable path shows the actual (lowest barrier) path, which would be followed if Scot > kgT/2ti.

Interestingly, the permeability coefficients of mannitol in the two cell types are identical, most probably for different reasons, since the physical dimensions of the Caco-2 and MDCK monolayers (Table 8) are markedly different. Compared to the MDCK cell monolayer, the Caco-2 cell monolayer has a taller cell height, a shorter length in tight junctions, longer tortuous path lengths, and smaller slit width in lateral space. One recognizes that... 

Fig. 2. Most probable flow paths of injected brine in MT2RD to the production sectors.

Figure 12. (Upper panel) Path entropy i(w) (Middle panel) path free-energy (w) = w — Ts(w), and (lower panel) Lagrange multipher X(w) equal to the inverse of the path temperature 1/7 (m ). is the most probable work value given by y(w P) = X,(rv P) = 0 or = 1 is the value of the work that has to be sampled to recover free energies from nonequilibrium work values using the JE. This is given by y(w() = l/T or d> (w() = 0 Wrev and Wdis are the reversible and average dissipated work, respectively. (From Ref. 117.)...

The most probable work can be determined by finding the extremum of the path... 

The mean free path X, of a molecule in air can be calculated from the sizes of the molecules involved. The most probable collision partners for a trace molecule (such as CFC-12) in air are molecular nitrogen (N2) and oxygen (02). The trace molecule i is hit whenever its center gets closer to the center of an air molecule than the critical distance, rcrit = r, + rair (Fig. 18.8). Picturing the molecules as spheres, the molecular radius r, can be estimated from the collision cross-section A listed in chemical handbooks such as the Tables of Physical and Chemical Constants (Longman, London, 1973) ... 

Equation (9.53) depicts the most probable path of the formation of 15 through protolysis of the C—C a bond ... 

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