Abstract
In this chapter, we introduce a mathematical framework, stochastic processes, which is used to describe small stochastic systems.
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Notes
- 1.
We denote the initial state by \(w^0\) for convenience.
- 2.
Of course, the other past states \(w^{n-2},\ldots , w^0\) can affect the present distribution through the latest state \(w^{n-1}\).
- 3.
If \(T_{ij}\) is negative, then applying T to a probability distribution with \(p^{n-1}_j=1\) and \(p^{n-1}_k=0\)(\(k\ne j\)), the i-th element of the probability distribution at the n-th step is negative: \(p_i^n=T_{ij}<0\).
- 4.
If \(\sum _j T_{ji}=c\ne 1\) for some i, by setting \(p_i^{n-1}=1\) and \(p^{n-1}_k=0\) for \(k\ne i\), we have \(\sum _j p_j^{n}=\sum _j T_{ji}=c\ne 1\).
- 5.
By taking \( \Delta t\) sufficiently small, all the matrix elements of T become nonnegative.
- 6.
We implicitly assume that function \(R_{ij}(t)\) is continuous almost everywhere.
- 7.
A matrix with connectivity is also called irreducible.
- 8.
Perron-Frobenius theorem states that a nonnegative matrix A with strong connectivity with eigenvalues \(\left| \lambda _1\right| \ge \left| \lambda _2\right| \ge \cdots \) satisfies (i) \(\lambda _1\) is real, (ii) the first inequality is strict (i.e., \(\lambda _1>\left| \lambda _2\right| \)), (iii) we can set the corresponding eigenvector to \(\lambda _1\) such that all the vector components are real and positive.
- 9.
A \(2\times 2\) transition rate matrix with connectivity can be expressed as \(R=\begin{pmatrix}-a&{}b\\ a&{}-b \end{pmatrix}\) with \(a,b>0\). This matrix has a unique positive vector \(\boldsymbol{v}=\begin{pmatrix}b/(a+b)\\ a/(a+b)\end{pmatrix}\) which satisfies \(R\boldsymbol{v}={\boldsymbol{0}}\).
- 10.
Connectivity ensures that for any i there exists \(j\ne i\) such that \(R_{j,i}>0\). Then, \(R_{i,i}=-\sum _{j\ne i}R_{j,i}\) and \(R_{j,i}\ge 0\) for \(j\ne i\) implies that all the diagonal elements of R are strictly negative; \(R_{i,i}<0\) for any i.
- 11.
This property is also called primitiveness or aperiodic.
- 12.
It is easy to show that a transition probability matrix with strong connectivity has connectivity.
- 13.
This minimum satisfies \(\mu <1/K\).
- 14.
We here use the fact that \(\left| \boldsymbol{v}\right| _1=0\) holds only when \(\boldsymbol{v}={\boldsymbol{0}}\).
- 15.
Here, we, of course, should restrict a class of possible functions to moderate (e.g., measurable) ones.
- 16.
In the following definitions, we assume that a proper limit of the right-hand side of Eq. (2.30) exists. This assumption is satisfied in physically plausible settings.
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Shiraishi, N. (2023). Stochastic Processes. In: An Introduction to Stochastic Thermodynamics. Fundamental Theories of Physics, vol 212. Springer, Singapore. https://doi.org/10.1007/978-981-19-8186-9_2
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DOI: https://doi.org/10.1007/978-981-19-8186-9_2
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