Energetics of the undamped stochastic harmonic oscillator

The harmonic oscillator is one of fundamental models in physics. In stochastic thermodynamics, such models are usually accompanied with both stochastic and damping forces, acting as energy counter-terms. Here, on the other hand, we study properties of the undamped harmonic oscillator driven by additive noises. Consequently, the popular cases of Gaussian white noise, Markovian dichotomous noise and Ornstein–Uhlenbeck noise are analyzed from the energy point of view employing both analytical and numerical methods. In accordance to one’s expectations, we confirm that energy is pumped into the system. We demonstrate that, as a function of time, initially total energy displays abrupt oscillatory changes, but then transits to the linear dependence in the long-time limit. Kinetic and potential parts of the energy are found to display oscillatory dependence at all times

Escape from finite intervals : numerical studies of order statistics

The subdiffusive systems are characterized by the diverging mean residence time. The escape of a subdiffusive particle from finite intervals cannot be characterized by the mean exit time. The situation significantly changes when instead of a single subdiffusive particle there is an ensemble of subdiffusive particles. In such a case, if the ensemble of particles is large enough, the mean minimal first escape time (first exit time of the fastest particle) is well defined quantity and the minimal first exit time distribution has fast decaying power-law asymptotics. Consequently, the increase in the number of particles facilitates escape kinetics and shortenes the system’s lifetime

Universal character of escape kinetics from finite intervals

We study a motion of an anomalous random walker on finite intervals restricted by two absorbing boundaries. The competition between anomalously long jumps and long waiting times leads to a very general kind of behavior. Trapping events distributed according to the power-law distribution result in occurrence of the Mittag–Leffler decay pattern which in turn is responsible for universal asymptotic properties of escape kinetics. The presence of long jumps which can be distributed according to non-symmetric heavy tailed distributions does not affect asymptotic properties of the survival probability. Therefore, the probability of finding a random walker within a domain of motion decays asymptotically according to the universal pattern derived from the Mittag–Leffler function, which describes decay of single modes in subdiffusive dynamics

Multimodal stationary states under Cauchy noise

A L\'evy noise is an efficient description of out-of-equilibrium systems. The presence of L\'evy flights results in a plenitude of noise-induced phenomena. Among others, L\'evy flights can produce stationary states with more than one modal value in single-well potentials. Here, we explore stationary states in special double-well potentials demonstrating that a sufficiently high potential barrier separating potential wells can produce bimodal stationary states in each potential well. Furthermore, we explore how the decrease in the barrier height affects the multimodality of stationary states. Finally, we explore a role of the multimodality of stationary states on the noise induced escape over the static potential barrier.Comment: 10 pages, 11 figure

Deterministic force-free resonant activation

Combined action of noise and deterministic force in dynamical systems can induce resonant effects. Here, we demonstrate a minimal, deterministic-force-free, setup allowing for occurrence of resonant, noise induced effects. We show that in the archetypal problem of escape from finite intervals driven by $\alpha$-stale noise with the periodically modulated stability index, depending on the initial direction of the modulation, resonant-activation-like or noise-enhanced-stability-like phenomena can be observed.Comment: 8 page

Underdamped, anomalous kinetics in double-well potentials

The noise driven motion in a bistable potential acts as the archetypal model of various physical phenomena. Here, we contrast the overdamped dynamics with the full (underdamped) dynamics. For the overdamped particle driven by a non-equilibrium, $\alpha$-stable noise the ratio of forward and backward transition rates depends only on the width of a potential barrier separating both minima. Using analytical and numerical methods, we show that in the regime of full dynamics, contrary to the overdamped case, the ratio of transition rates depends both on widths and heights of the potential barrier. The analytical formula for the ratio of transition rates is corroborated by extensive numerical simulations.Comment: 9 page

Escape from the potential well : accelerating by shaping and noise tuning

Noise driven escape from the potential well is the basic component of various noise induced effects. The efficiency of the escape process or time scales matching is responsible for occurrence of the stochastic resonance and (stochastic) resonant activation. Here, we are extending the discussion on how the structure of the potential can be used to optimize the mean first passage time. It is demonstrated that corrugation of the potential can be beneficial under action of the weak Gaussian white noise. Furthermore, we show that the noise tuning can be more effective than shaping the potential. Therefore, action of the tuned additive α-stable noise can accelerate the escape kinetics more than corrugation of the potential. Finally, we demonstrate that mean first passage time from a potential well can be a non-monotonous function of the stability index α

Optimization of escape kinetics by reflecting and resetting

Stochastic restarting is a strategy of starting anew. Incorporation of the resetting to the random walks can result in the decrease of the mean first passage time, due to the ability to limit unfavorably meandering, sub-optimal trajectories. In the following manuscript we examine how stochastic resetting influences escape dynamics from the $(-\infty,1)$ interval in the presence of the single-well power-law $|x|^\kappa$ potentials with $\kappa>0$. Examination of the mean first passage time is complemented by the analysis of the coefficient of variation, which provides a robust and reliable indicator assessing efficiency of stochastic resetting. The restrictive nature of resetting is compared to placing a reflective boundary in the system at hand. In particular, for each potential, the position of the reflecting barrier giving the same mean first passage time as the optimal resetting rate is determined. Finally, in addition to reflecting, we compare effectiveness of other resetting strategies with respect to optimization of the mean first passage time.Comment: 7 page

Weighted Axelrod model: different but similar

The Axelrod model is a cellular automaton which can be used to describe the emergence and development of cultural domains, where culture is represented by a fixed number of cultural features taking a discrete set of possible values (traits). The Axelrod model is based on two sociological phenomena: homophily (a tendency for individuals to form bonds with people similar to themselves) and social influence (the way how individuals change their behavior due to social pressure). However, the Axelrod model does not take into account the fact that cultural attributes may have different significance for a given individual. This is a limitation in the context of how the model reflects mechanisms driving the evolution of real societies. The study aims to modify the Axelrod model by giving individual features different weights that have a decisive impact on the possibility of aligning cultural traits between (interacting) individuals. The comparison of the results obtained for the classic Axelrod model and its modified version shows that introduced weights have a significant impact on the course of the system development, in particular, increasing the final polarization of the system and increasing the time needed to reach the final state.Comment: 7 pages. 7 figure

Bounding energy growth in friction-less stochastic oscillators

The paper presents analytical and numerical results on energetics of non-harmonic, undamped, single-well, stochastic oscillators driven by additive Gaussian white noises. Absence of damping and the action of noise are responsible for lack of stationary states in such systems. We explore properties of average kinetic, potential and total energies along with the generalized equipartition relations. It is demonstrated that in the friction-less dynamics nonequilibrium stationary states can be produced by stochastic resetting. For an appropriate resetting protocol average energies become bounded. If the resetting protocol is not characterized by finite variance of renewal time intervals stochastic resetting can only slow down the growth of average energies but does not bound them. Under special conditions regarding frequency of resets, ratios of average energies follow the generalized equipartition relations