The aim of the paper is to understand how the inclusion of more and more time-scales into a stochastic stationary Markovian process affects its conditional probability. To this end, we consider two Gaussian processes: (i) a short-range correlated process with an infinite set of time-scales bounded from below, and (ii) a power-law correlated process with an infinite and unbounded set of time-scales. For these processes we investigate the equal position conditional probability P(x,t|x,0) and the mean First Passage Time T(L). The function P(x,t|x,0) can be considered as a proxy of the persistence, i.e. the fact that when a process reaches a position x then it spends some time around that position value. The mean First Passage Time can be considered as a proxy of how fast is the process in reaching a position at distance L starting from position x. In the first investigation we show that the more time-scales the process includes, the larger the persistence. Specifically, we show that the power-law correlated process shows a slow power-law decay of P(x,t|x,0) to the stationary pdf. By contrast, the short range correlated process shows a decay dominated by an exponential cut-off. Moreover, we also show that the existence of an infinite and unbouded set of time-scales is a necessary and not sufficient condition for observing a slow power-law decay of P(x,t|x,0). In the second investigation, we show that for large values of L the more time-scales the process includes, the larger the mean First Passage Time, i.e. the slowest the process. On the other hand, for small values of L, the more time-scales the process includes, the smaller the mean First Passage Time, i.e. when a process statistically spends more time in a given position the likelihood that it reached nearby positions by chance is also enhanced.
We introduce a single generative mechanism with which it is able to describe diverse non-stationary diffusions. A non-stationary Markovian replication process for steps is considered, for which we analytically derive time-evolution of the probability distribution of the walker's displacement and the generalized telegrapher equation with time-varying coefficients, and find that diffusivity can be determined by temporal changes of replication of a immediate step. By controlling the replications, we realize the diverse diffusions such as alternating diffusions, superdiffusions, subdiffusions, and…
We investigate the stochastic behavior of a two-temperature Langevin system with non-Markovian thermal reservoirs. The model describes an overdamped Brownian particle in a quadratic potential and coupled to heat baths at different temperatures. The reservoirs are characterized by Gaussian white and colored noises and a dissipation memory kernel. The stationary states present non-trivial average rotational motion influenced by stochastic torques due to harmonic, friction and fluctuating thermal forces. However, the Markovian limit leads to a vanishing average torque produced by fluctuating ther…
How much time does it take for a fluctuating system, such as a polymer chain, to reach a target configuration that is rarely visited -- typically because of a high energy cost ? This question generally amounts to the determination of the first-passage time statistics to a target zone in phase space with lower occupation probability. Here, we present an analytical method to determine the mean first-passage time of a generic non-Markovian random walker to a rarely visited threshold, which goes beyond existing weak-noise theories. We apply our method to polymer systems, to determine (i) the first…
The persistence of a stochastic variable is the probability that it does not cross a given level during a fixed time interval. Although persistence is a simple concept to understand, it is in general hard to calculate. Here we consider zero mean Gaussian stationary processes in discrete time $n$. Few results are known for the persistence $P_0(n)$ in discrete time, except the large time behavior which is characterized by the nontrivial constant $θ$ through $P_0(n)\sim θ^n$. Using a modified version of the Independent Interval Approximation (IIA) that we developed before, we are able to calculat…
In this study, we analytically formulated the path integral representation of the conditional probabilities for non-Markovian kinetic processes in terms of the free energy of the thermodynamic system. We carry out analytically the time-fractional kinetic equations for these processes. Thus, in a simple way, we generalize path integral solutions of the Markovian to the non-Markovian cases. We conclude that these pedagogical results can be applied to some physical problems such as the deformed ion channels, internet networks and non-equilibrium phase transition problems.
The breaking of the continuous time-translation symmetry manifests, in Markovian open quantum systems, through the emergence of non-stationary dynamical phases. Systems that display nonequilibrium transitions into these phases are referred to as time-crystals, and they can be realized, for example, in many-body systems governed by collective dissipation and long-ranged interactions. Here, we provide a complete analytical characterization of a boundary time-crystal phase transition. This involves exact expressions for the order parameter and for the dynamics of quantum fluctuations, which, in t…
We present a nonlinear and non-Markovian random walk model for stochastic movement and the spatial aggregation of living organisms that have the ability to sense population density. We take into account social crowding effects for which the dispersal rate is a decreasing function of the population density and residence time. We perform stochastic simulations of random walk and discover the phenomenon of self-organized anomaly (SOA) which leads to a collapse of stationary aggregation pattern. This anomalous regime is self-organized and arises without the need for a heavy tailed waiting time dis…
Sensors often serve at least two purposes: predicting their input and minimizing dissipated heat. However, determining whether or not a particular sensor is evolved or designed to be accurate and efficient is difficult. This arises partly from the functional constraints being at cross purposes and partly since quantifying the predictive performance of even in silico sensors can require prohibitively long simulations. To circumvent these difficulties, we develop expressions for the predictive accuracy and thermodynamic costs of the broad class of conditionally Markovian sensors subject to unifi…
In applications spaning from image analysis and speech recognition, to energy dissipation in turbulence and time-to failure of fatigued materials, researchers and engineers want to calculate how often a stochastic observable crosses a specific level, such as zero. At first glance this problem looks simple, but it is in fact theoretically very challenging. And therefore, few exact results exist. One exception is the celebrated Rice formula that gives the mean number of zero-crossings in a fixed time interval of a zero-mean Gaussian stationary processes. In this study we use the so-called Indepe…
Employing the optimal fluctuation method (OFM), we study the large deviation function of long-time averages $(1/T)\int_{-T/2}^{T/2} x^n(t) dt$, $n=1,2, \dots$, of centered stationary Gaussian processes. These processes are correlated and, in general, non-Markovian. We show that the anomalous scaling with time of the large-deviation function, recently observed for $n>2$ for the particular case of the Ornstein-Uhlenbeck process, holds for a whole class of stationary Gaussian processes.
We investigate the effects of markovian resseting events on continuous time random walks where the waiting times and the jump lengths are random variables distributed according to power law probability density functions. We prove the existence of a non-equilibrium stationary state and finite mean first arrival time. However, the existence of an optimum reset rate is conditioned to a specific relationship between the exponents of both power law tails. We also investigate the search efficiency by finding the optimal random walk which minimizes the mean first arrival time in terms of the reset ra…
