Stochastic processes random walk problem

Markov processes have no memory of earlier information. Newton equations describe deterministic Markovian processes by this definition, since knowledge of system state (all positions and momenta) at a given time is sufficient in order to determine it at any later time. The random walk problem discussed in Section 7.3 is an example of a stochastic Markov process. 

An ancient but still instructive example is the discrete-time random walk. A drunkard moves along a line by making each second a step to the right or to the left with equal probability. Thus his possible positions are the integers — oo < n < oo, and one asks for the probability pn(r) for him to be at n after r steps, starting from n = 0. While we shall treat this example in IV.5 as a stochastic process, we shall here regard it as a problem of adding variables. 

Very few stochastic lake models seem to exist in the literature. Ecological problems related to those investigated here are described stochastically by j the following tools time series, random walk, diffusion processes, differential/ equations with random parameters (taken in the wide sense), and Markovia pure jump processes. 

Nowadays, computer simulations are treated as the third fundamental discipline of interface research in addition to the two classieal ones, namely theory and experiment. Based direetly on a microscopie model of the system, eomputer simulations can, in principle at least, provide an exact solution of any physicochemical problem. By far the most common methods of studying adsorption systems by simulations are the Monte Carlo (MC) technique and the molecular dynamics (MD) method. In this ehapter, a description of simidation methods will be omitted because several textbooks and review artieles on the subject are available [274-277]. The present discussion will be restricted to elementary aspects of simulation methods. In the deterministic MD method, the moleeular trajectories are eomputed by solving Newton s equations, and a time-correlated sequenee of configurations is generated. The main advantage of this technique is that it permits the study of time-dependent processes. In MC simulation, a stochastic element is an essential part of the method the trajectories are generated by random walk in configuration space. Struetural and thermodynamic properties are accessible by both methods. 

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