![]() Thermodynamic equilibrium is only approached for systems enclosed in a vessel of finite volume. Indeed, in thermodynamics too one finds that a plume of gas emitted into free space will similarly diffuse, becoming ever more dilute without ever approaching an equilibrium state. However, it is not reasonable to see this as a serious defect. Because this holds for every choice of ρ 0, there is no stationary distribution in this case. Which becomes gradually lower, smoother and wider in the course of time, but does not approach any stationary probability density. But if the set Y is infinite or continuous this is not always true. Still, it turns out that, due to a theorem of Perron (1907) and Frobenius (1909), every stochastic matrix indeed has a eigenvector, with exclusively non-negative components, and eigenvalue 1 (see e.g. Further, even if an eigenvector with the corresponding eigenvalue exists, it is not automatically suitable as a probability distribution because its components might not be positive. Note that T or L are not necessarily Hermitian (or, rather, since we are dealing with real matrices, symmetric), so that the existence of eigenvectors is not guaranteed by the spectral theorem. Ad (i).Ī stationary state as defined by (190), can be seen as an eigenvector of T t with eigenvalue 1, or, in the light of (183), an eigenvector of L for the eigenvalue 0. How this behaviour is compatible with the time symmetry of Markov processes. How to motivate the assumptions needed in this approach or how to make judge their (in)compatibility with an underlying time deterministic dynamics and (v) Harder questions, which we postpone to the next sub section 7.5, are: (iv)
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