[en] We prove that a transitive topological Markov chain has almost-periodic points of all D-periods. Moreover, every D-period is realized by continuously many distinct minimal sets. We give a simple constructive proof of the result which asserts that any transitive topological Markov chain has periodic points of almost all periods, and study the structure of the finite set of positive integers that are not periods.

[en] Feedback control engineers have been interested in multiple-input-multiple-output (MIMO) extensions of single-input-single-output (SISO) results of various kinds due to its rich mathematical structure and practical applications. An outstanding problem in quantum feedback control is the extension of the SISO theory of Markovian feedback by Wiseman and Milburn [Phys. Rev. Lett. 70, 548 (1993)] to multiple inputs and multiple outputs. Here we generalize the SISO homodyne-mediated feedback theory to allow for multiple inputs, multiple outputs, and arbitrary diffusive quantum measurements. We thus obtain a MIMO framework which resembles the SISO theory and whose additional mathematical structure is highlighted by the extensive use of vector-operator algebra.

[en] This tutorial article showcases the many varieties and uses of quantum walks. Discrete time quantum walks are introduced as counterparts of classical random walks. The emphasis is on the connections and differences between the two types of processes (with rather different underlying dynamics) for producing random distributions. We discuss algorithmic applications for graph-searching and compare the two approaches. Next, we look at quantization of Markov chains and show how it can lead to speedups for sampling schemes. Finally, we turn to continuous time quantum walks and their applications, which provide interesting (even exponential) speedups over classical approaches. (author)

[en] A set of conditions is formulated under which the deterministic motion evolves into direction of non decreasing values of suitably chosen probability distributions in the limit of large systems. In contrast to former treatment this derivation is not restricted to processes homogeneous in time nor to Markoffian processes. Some examples are presented. (orig.)

[en] We discuss the selection procedure in the framework of mutation models. We study the regulation for stochastically developing systems based on a transformation of the initial Markov process which includes a cost functional. The transformation of initial Markov process by cost functional has an analytic realization in terms of a Kimura-Maruyama type equation for the time evolution of states or in terms of the corresponding Feynman-Kac formula on the path space. The state evolution of the system including the limiting behavior is studied for two types of mutation-selection models

[en] We consider a Brownian particle that switches between two different diffusion states (D ₀, D ₁) according to a two-state Markov chain. We further assume that the particle’s position is reset to an initial value X _r at a Poisson rate r, and that the discrete diffusion state is simultaneously reset according to the stationary distribution ρ _n, n = 0, 1, of the Markov chain. We derive an explicit expression for the non-equilibrium steady state (NESS) on $\mathbb{R}$, which is given by the sum of two decaying exponentials. In the fast switching limit the NESS reduces to the exponential distribution of pure diffusion with stochastic resetting. The effective diffusivity is given by the mean $\stackrel{¯<!--\; ??\; -->}{D}={\mathrm{\rho <!--\; ??\; -->}}_0{D}_0+{\mathrm{\rho <!--\; ??\; -->}}_1{D}_1$. We then determine the mean first passage time (MFPT) for the particle to be absorbed by a target at the origin, having started at the reset position X _r > 0. We proceed by calculating the survival probability in the absence of resetting and then use a last renewal equation to determine the survival probability with resetting. Similar to the NESS, we find that the MFPT depends on the sum of two exponentials, which reduces to a single exponential in the fast switching limit. Finally, we show that the MFPT has a unique minimum as a function of the resetting rate, and explore how the optimal resetting rate depends on other parameters of the system. (paper)

[en] For a two-level atom in a lossy cavity, a scheme to manipulate the non-Markovian speedup dynamics has been proposed in the controllable environment (the lossy cavity field). We mainly focus on the effects of the qubit–cavity detuning Δ and the qubit–cavity coupling strength κ on the non-Markovian speedup evolution of an open system. By controlling the environment, i.e., tuning Δ and κ, two dynamical crossovers from Markovian to non-Markovian and from no-speedup to speedup are achieved. Furthermore, it is clearly found that increasing the coupling strength κ or detuning Δ in some cases can make the environmental non-Markovianity stronger and hence can lead to faster evolution of the open system. (paper)

[en] For a general two-cluster network, a new methodology of the cluster-based generalized quantum kinetic expansion (GQKE) is developed in the matrix formalism under two initial conditions: the local cluster equilibrium and system-bath factorized states. For each initial condition, the site population evolution follows exactly a distinct closed equation, where all the four terms involved are systematically expanded over inter-cluster couplings. For the system-bath factorized initial state, the numerical investigation of the two models, a biased (2, 1)-site system and an unbiased (2, 2)-site system, verifies the reliability of the GQKE and the relevance of higher-order corrections. The time-integrated site-to-site rates and the time evolution of site population reveal the time scale separation between intra-cluster and inter-cluster kinetics. The population evolution of aggregated clusters can be quantitatively described by the approximate cluster Markovian kinetics

[en] We prove that the normalisation of the stationary state of the multi-species asymmetric simple exclusion process (mASEP) is a specialisation of a Koornwinder polynomial. As a corollary we obtain that the normalisation of mASEP factorises as a product over multiple copies of the two-species ASEP. (paper)

[en] Non-Markovian local-in-time master equations give a relatively simple way to describe the dynamics of open quantum systems with memory effects. Despite their simple form, there are still many misunderstandings related to the physical applicability and interpretation of these equations. Here, we clarify these issues both in the cases of quantum and classical master equations. We further introduce the concept of a classical non-Markov chain signified through negative jump rates in the chain configuration. (paper)