Path-dependent evolution

Two examples of path-dependent micromechanical effects are models of Swegle and Grady [13] for thermal trapping in shear bands and Follansbee and Kocks [14] for path-dependent evolution of the mechanical threshold stress in copper. 

The lone remaining aspect of this topic that requires additional discussion is the fact that the mechanical threshold stress evolution is path-dependent. The fact that (df/dy)o in (7.41) is a function of y means that computations of material behavior must follow the actual high-rate deformational path to obtain the material strength f. This becomes a practical problem only in dealing with shock-wave compression. 

Depending on the values of the probability parameters of the branching process two paths of evolution for every population are possible either it will degenerate in some generation or it will infinitely grow. The probability of... 

Equation (4.21) states that the dynamics of the forward-rate process, beginning with the initial rate/(0, J), are specified by the set of Brownian motion processes and the drift parameter. For practical applications, the evolution of the forward-rate term structure is usually derived in a binomial-type path-dependent process. Path-independent processes, however, have also been used, as has simulation modeling based on Monte Carlo techniques (see Jarrow (1996)). The HJM approach has become popular in the market, both for yield-curve modeling and for pricing derivative instruments, because it matches yield-curve maturities to different volatility levels realistically and is reasonably tractable when applied using the binomial-tree approach. 

These institutional/organizational standards tend to be resource-driven and path dependent. Their evolution depends in part on historical... 

In the PPF, the first factor Pi describes the statistical average of non-correlated spin fiip events over entire lattice points, and the second factor P2 is the conventional thermal activation factor. Hence, the product of P and P2 corresponds to the Boltzmann factor in the free energy and gives the probability that on<= of the paths specified by a set of path variables occurs. The third factor P3 characterizes the PPM. One may see the similarity with the configurational entropy term of the CVM (see eq.(5)), which gives the multiplicity, i.e. the number of equivalent states. In a similar sense, P can be viewed as the number of equivalent paths, i.e. the degrees of freedom of the microscopic evolution from one state to another. As was pointed out in the Introduction section, mathematical representation of P3 depends on the mechanism of elementary kinetics. It is noted that eqs.(8)-(10) are valid only for a spin kinetics. 

Saio, Kato, and Nomoto (1988) recently examined under what conditions a massive star undergoes a blue-red-blue evolution. The evolution of a star of initial mass 20 M0 star in the HR diagram is shown in Figure 1 from the zero-age main-sequence through carbon ignition at the center. The metallicity in the envelope was assumed to be Z = 0.005 and the Schwarzschild criterion was adopted. The star shows the three types of evolutionary path (A, B, C) depending on the mass loss, metallicity, and the change in the helium abundance Y in the envelope. 

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