Abstract
Stochastic processes are introduced as ‘time’ dependent processes depending on randomness where ‘time’ is a totally ordered set. Each stochastic process is coupled to a pdf which describes the probability of a realization of the process at ‘time’ \(t\). The general characterization of these processes is discussed in detail. The focus moves on to a specific class of stochastic processes, the Markov processes. It has the remarkable property that a future realization of the process depends solely on its current realization (Markov property). A huge class of processes in physics and related sciences are Markovian in nature. The Wiener process and the Poisson process are discussed as typical examples. This is followed by the introduction of Markov-chains together with their classification. The role of detailed balance in stochastics and physics is elucidated. This explains why Markov-chains are the back-bone of Monte-Carlo sampling techniques. In a last point continuous-time Markov-chains are described.
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Notes
- 1.
This form is equivalent to the above definition of the Wiener process, in particular to the requirement of normally distributed increments with variance \(t_2 - t_1\).
- 2.
Alternatively, we may draw d\( y\) from a normal distribution with variance \(1\) and multiply it by \(\sqrt{\mathrm{d} t}\). This follows from a simple transformation of variables.
- 3.
As an example we quote Fermi’s golden rule, where the transition rate \(w_{nn'}\) from unperturbed states \(n\) to \(n'\) is of the form
$$\begin{aligned} w_{n n'} = \frac{2 \pi }{\hbar } \vert H_{nn'}' \vert \rho (E_n), \end{aligned}$$where \(H_{nn'}'\) are the matrix element of the perturbation Hamiltonian \(H'\) and \(\rho (E_n)\) denotes the density of states of the unperturbed system.
References
Chow Y.S., Teicher, H.: Probability theory, 3rd edn. Springer Texts in Statistics. Springer, Berlin (1997)
van Kampen, N.G.: Stochastic Processes in Physics and Chemistry. Elsevier, Amsterdam (2008)
Breuer, H.P., Petruccione, F.: Open Quantum Systems. Clarendon Press, Oxford, UK (2010)
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Stickler, B.A., Schachinger, E. (2014). Some Basics of Stochastic Processes. In: Basic Concepts in Computational Physics. Springer, Cham. https://doi.org/10.1007/978-3-319-02435-6_16
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DOI: https://doi.org/10.1007/978-3-319-02435-6_16
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