Stochastic approaches

For half a century different theories have been and continue to be proposed to model chromatography and to explain the migration and separation of analytes in the column. The best known are those employing a statistical approach (stochastic theory), the theoretical plate model or a molecular dynamics approach. 

In the modeling of catalytic reactions at the molecular level, the stochastic approach is also fruitful along with the simulation based on equations (the deterministic approach). Stochastic simulations (the dynamic Monte Carlo method) makes it possible to penetrate into the microlevel and monitor detailed changes in the adsorption layer, and explain the observed phenomena. 

One approach to a mathematically well defined performance measure is to interpret the amplitude values of a processed signal as realizations of a stochastic variable x which can take a discrete number of values with probabilities P , n = 1,2,..., N. Briefly motivated in the introduction, then an interesting quality measure is the entropy H x) of the amplitude distribu-... 

A second type of relaxation mechanism, the spin-spm relaxation, will cause a decay of the phase coherence of the spin motion introduced by the coherent excitation of tire spins by the MW radiation. The mechanism involves slight perturbations of the Lannor frequency by stochastically fluctuating magnetic dipoles, for example those arising from nearby magnetic nuclei. Due to the randomization of spin directions and the concomitant loss of phase coherence, the spin system approaches a state of maximum entropy. The spin-spin relaxation disturbing the phase coherence is characterized by T. ... 

This discussion suggests that even the reference trajectories used by symplectic integrators such as Verlet may not be sufficiently accurate in this more rigorous sense. They are quite reasonable, however, if one requires, for example, that trajectories capture the spectral densities associated with the fastest motions in accord to the governing model [13, 15]. Furthermore, other approaches, including nonsymplectic integrators and trajectories based on stochastic differential equations, can also be suitable in this case when carefully formulated. 

Application of a Stochastic Path Integral Approach to the Computations of an Optimal Path and Ensembles of Trajectories ... 

The two sources of stochasticity are conceptually and computationally quite distinct. In (A) we do not know the exact equations of motion and we solve instead phenomenological equations. There is no systematic way in which we can approach the exact equations of motion. For example, rarely in the Langevin approach the friction and the random force are extracted from a microscopic model. This makes it necessary to use a rather arbitrary selection of parameters, such as the amplitude of the random force or the friction coefficient. On the other hand, the equations in (B) are based on atomic information and it is the solution that is approximate. For ejcample, to compute a trajectory we make the ad-hoc assumption of a Gaussian distribution of numerical errors. In the present article we also argue that because of practical reasons it is not possible to ignore the numerical errors, even in approach (A). 

Another difference is related to the mathematical formulation. Equation (1) is deterministic and does not include explicit stochasticity. In contrast, the equations of motion for a Brownian particle include noise. Nevertheless, similar algorithms are adopted to solve the two differential equations as outlined below. The most common approach is to numerically integrate the above differential equations using small time steps and preset initial values. 

Other methods which are applied to conformational analysis and to generating multiple conformations and which can be regarded as random or stochastic techniques, since they explore the conformational space in a non-deterministic fashion, arc genetic algorithms (GA) [137, 1381 simulation methods, such as molecular dynamics (MD) and Monte Carlo (MC) simulations 1139], as well as simulated annealing [140], All of those approaches and their application to generate ensembles of conformations arc discussed in Chapter II, Section 7.2 in the Handbook. 

A Stochastic Approach to Combinatorial Optimization and Neural Computing. John Wiley Sons, New York, 1989. 

Another way is to reduce the magnitude of the problem by eliminating the explicit solvent degrees of freedom from the calculation and representing them in another way. Methods of this nature, which retain the framework of molecular dynamics but replace the solvent by a variety of simplified models, are discussed in Chapters 7 and 19 of this book. An alternative approach is to move away from Newtonian molecular dynamics toward stochastic dynamics. 

The form of the stochastic transfer function p x) is shown in figure 10.7. Notice that the steepness of the function near a - 0 depends entirely on T. Notice also that this form approaches that of a simple threshold function as T —> 0, so that the deterministic Hopfield net may be recovered by taking the zero temperature limit of the stochastic system. While there are a variety of different forms for p x) satisfying this desired limiting property, any of which could also have been chosen, this sigmoid function is convenient because it allows us to analyze the system with tools borrowed from statistical mechanics. 

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