The persistence behavior for fluctuating steps on the $Si(111)$ $(\sqrt3 \times \sqrt3)R30^{0} - Al$ surface was determined by analyzing time-dependent STM images for temperatures between 770 and 970K. The measured persistence probability follows a power law decay with an exponent of $θ=0.77 \pm 0.03$. This is consistent with the value of $θ= 3/4$ predicted for attachment/detachment limited step kinetics. If the persistence analysis is carried out in terms of return to a fixed reference position, the measured persistence probability decays exponentially. Numerical studies of the Langevin equation used to model step motion corroborate the experimental observations.
We present a scheme to accurately calculate the persistence probabilities on sequences of $n$ heights above a level $h$ from the measured $n+2$ points of the height-height correlation function of a fluctuating interface. The calculated persistence probabilities compare very well with the measured persistence probabilities of a fluctuating phase-separated colloidal interface for the whole experimental range.
Finite heat reservoir capacity and temperature fluctuations lead to modification of the well known canonical exponential weight factor. Requiring that the corrections least depend on the one-particle energy, we derive a deformed entropy, K(S). The resulting formula contains the Boltzmann-Gibbs, the Renyi and the Tsallis formulas as particular cases. For extreme large fluctuations (compared to the Gaussian case) a new, parameter-free entropy - probability relation emerges. This formula and the corresponding canonical equilibrium distribution are nearly Boltzmannian for high probability, but dev…
For many stochastic processes, the probability $S(t)$ of not-having reached a target in unbounded space up to time $t$ follows a slow algebraic decay at long times, $S(t)\sim S_0/t^θ$. This is typically the case of symmetric compact (i.e. recurrent) random walks. While the persistence exponent $θ$ has been studied at length, the prefactor $S_0$, which is quantitatively essential, remains poorly characterized, especially for non-Markovian processes. Here we derive explicit expressions for $S_0$ for a compact random walk in unbounded space by establishing an analytic relation with the mean first…
We investigate the persistence properties of critical d-dimensional systems relaxing from an initial state with non-vanishing order parameter (e.g., the magnetization in the Ising model), focusing on the dynamics of the global order parameter of a d'-dimensional manifold. The persistence probability P(t) shows three distinct long-time decays depending on the value of the parameter ζ= (D-2+η)/z which also controls the relaxation of the persistence probability in the case of a disordered initial state (vanishing order parameter) as a function of the codimension D = d-d' and of the critical expon…
Persistence of a Continuous Stochastic Process with Discrete-Time Sampling
我们引入了“离散时间持续性”的概念，它涉及在离散时间T = n ΔT测量的连续随机过程X（T）的零交叉。对于松弛率为μ的高斯马尔可夫过程，我们证明了对于大的n，持续（无交叉）概率随ρ（a）^n衰减，其中a = \exp（-μΔT），并且我们高精度地计算了ρ（a）.我们还定义了“交替持续性”的概念，它对应于<0. For a>1，对应于不稳定势中的运动（μ<0），在无限时间内没有零交叉的非零概率，我们展示了如何计算它。
A study of persistence dynamics is made for the first time in a quantum system by considering the dynamics of a quantum random walk. For a discrete walk on a line starting at $x=0$ at time $t=0$, the persistence probability $P(x,t)$ that a site at $x$ has not been visited till time $t$ has been calculated. $P(x,t)$ behaves as $(t/|x|-1)^{-α}$ with $α\sim 0.3$ while the global fraction ${\cal{P}}(t) = \sum_xP(x,t)/2t$ of sites remaining unvisited at time $t$ attains a constant value. $F(x,t)$, the probability that the site at $x$ is visited for the first time at $t$ behaves as $(t/|x|-1)^{-β}/|…
We present a model which displays Griffiths phase i.e. algebraic decay of density with continuously varying exponent in the absorbing phase. In active phase, the memory of initial conditions is lost with continuously varying complex exponent in this model. This is 1-D model where fraction r of sites obey rules leading to directed percolation (DP) class and the rest evolve according to rules leading to compact directed percolation (CDP) class. For infection probability $p < p_c$, the fraction of active sites $ρ(t) = 0$ asymptotically. For $p > p_c$, $ρ(infty) > 0$. At $p = p_c$, $ρ(t)$, the sur…
We study the dynamics of ordering in ferromagnets via Monte Carlo simulations of the Ising model, employing the Glauber spin-flip mechanism, in space dimensions $d=2$ and $3$. Results for the persistence probability and the domain growth are discussed for quenches to various temperatures ($T_f$) below the critical one ($T_{c}$), from different initial temperatures $T_{i} \geq T_{c}$. In long time limit, for $T_{i} > T_{c}$, the persistence probability exhibits power-law decay with exponents $θ\simeq 0.22$ and $\simeq 0.18$ in $d=2$ and $3$, respectively. For finite $T_i$, the early time behavi…
Persistence, defined as the probability that a fluctuating signal has not reached a threshold up to a given observation time, plays a crucial role in the theory of random processes. It quantifies the kinetics of processes as varied as phase ordering, reaction diffusion or interface relaxation dynamics. The fact that persistence can decay algebraically with time with non trivial exponents has triggered a number of experimental and theoretical studies. However, general analytical methods to calculate persistence exponents cannot be applied to the ubiquitous case of non-Markovian systems relaxing…
In this article, results have been presented for the two-time correlation functions for a free and a harmonically confined Brownian particle in a simple shear flow. For a free Brownian particle, the motion along the direction of shear exhibit two distinct dynamics, with the mean-square-displacement being diffusive at short times while at late times scales as $t^3$. In contrast the cross-correlation $\la x(t) y(t) \ra $ scales quadratically for all times. In the case of a harmonically trapped Brownian particle, the mean-square-displacement exhibits a plateau determined by the strength of the co…
We investigate the persistence probability $p(t)$ of the position of a Brownian particle with shape asymmetry in two dimensions. The persistence probability is defined as the probability that a stochastic variable has not changed it's sign in the given time interval. We explicitly consider two cases -- diffusion of a free particle and that of harmonically trapped particle. The later is particularly relevant in experiments which uses trapping and tracking techniques to measure the displacements. We provide analytical expressions of $p(t)$ for both the scenarios and show that in the absence of t…
We study the local persistence probability during non-stationary time evolutions in disordered contact processes with long-range interactions by a combination of the strong-disorder renormalization group (SDRG) method, a phenomenological theory of rare regions, and numerical simulations. We find that, for interactions decaying as an inverse power of the distance, the persistence probability tends to a non-zero limit not only in the inactive phase but also in the critical point. Thus, unlike in the contact process with short-range interactions, the persistence in the limit $t\to\infty$ is a dis…
Persistence Exponents and the Statistics of Crossings and Occupation Times for Gaussian Stationary Processes
我们考虑零均值离散或（等价）离散采样高斯平稳过程（GSPs）的持续概率，占用时间分布和零交叉数的分布。我们首先考虑的Ornstein-Uhlenbeck过程中，发现的平均值和方差的交叉点的数量和“部分生存”的概率的表达式。然后，我们详细阐述了相关器扩展在早期的文件[G。C. M. A. Ehrhardt和A. J. Bray，Phys. Rev. Lett. 88，070602（2001）]，以通过相关器中的级数展开来计算已知相关器的GSP的离散采样的持续指数。我们将这种方法应用于n > 2的过程d^nx/dt^n=η（t），并将该序列外推到连续采样的极限。我们扩展相关器的方法来计算占领时间和交叉数分布，以及他们的部分生存分布和占领时间和交叉数的均值和方差。我们将这些一般方法应用于d^nx/dt^n=η（t）过程（n=1（随机行走），n=2（随机加速度）和更大的n），以及1-3维随机初始条件下的扩散。在可能的情况下，将离散采样的结果外推至连续极限。
We establish an exact formula relating the survival probability for certain Lévy flights (viz. asymmetric $α$-stable processes where $α= 1/2$) with the survival probability for the order statistics of the running maxima of two independent Brownian particles. This formula allows us to show that the persistence exponent $δ$ in the latter, non Markovian case is simply related to the persistence exponent $θ$ in the former, Markovian case via: $δ=θ/2$. Thus, our formula reveals a link between two recently explored families of anomalous exponents: one exhibiting continuous deviations from Sparre-And…
The solution of the Langevin equation of an anisotropic fluid [Colmenares P. J; López F. and Olivares-Rivas W., Phys. Rev E. 2009, 80061123] allowed the evaluation of the position dependent perpendicular and parallel diffusion coefficients, using Molecular Dynamics data. However, the time scale of the Langevin Dynamics and Molecular Dynamics are different and an anzat for the persistence probability relaxation time was needed. Here we show how the solution for the average persistence probability obtained from the Smoluchowski-Fokker-Planck equation (SE), associated to the Langevin Dynamics, sc…
We study the time-dependence of the local persistence probability during a non-stationary time evolution in the disordered contact process in $d=1,2$, and $3$ dimensions. We present a method for calculating the persistence with the strong-disorder renormalization group (SDRG) technique, which we then apply in the critical point analytically for $d=1$ and numerically for $d=2,3$. According to the results, the average persistence decays at late times as an inverse power of the logarithm of time, with a universal, dimension-dependent generalized exponent. For $d=1$, the distribution of sample-dep…
We recast the persistence probability for the spin located at the origin of a half-space arbitrarily $m$-magnetized Glauber-Ising chain as a Fredholm Pfaffian gap probability generating function with a sech-kernel. This is then spelled out as a tau-function for a certain Painlevé VI transcendent, the persistence exponent $θ(m)/2$ emerging as an asymptotic decay rate. Using a known yet remarkable correspondence that relates Painlevé equations to Bonnet surfaces, the persistence probability also acquires a geometric meaning in terms of the mean curvature of the latter, and even a topological one…
We investigate the persistence probability of a Brownian particle in a harmonic potential, which decays to zero at long times -- leading to an unbounded motion of the Brownian particle. We consider two functional forms for the decay of the confinement, an exponential and an algebraic decay. Analytical calculations and numerical simulations show, that for the case of the exponential relaxation, the dynamics of Brownian particle at short and long times are independent of the parameters of the relaxation. On the contrary, for the algebraic decay of the confinement, the dynamics at long times is d…
In this short paper we derive a formula for the spatial persistence probability of the Airy_1 and the Airy_2 processes. We then determine numerically a persistence coefficient for the Airy_1 process and its dependence on the threshold.
We consider and analyze a dynamic model of random hyperbolic graphs with link persistence. In the model, both connections and disconnections can be propagated from the current to the next snapshot with probability $ω\in [0, 1)$. Otherwise, with probability $1-ω$, connections are reestablished according to the random hyperbolic graphs model. We show that while the persistence probability $ω$ affects the averages of the contact and intercontact distributions, it does not affect the tails of these distributions, which decay as power laws with exponents that do not depend on $ω$. We also consider …
We consider stochastic energy balance and entropy production (EP) in a generalized Langevin dynamics of macrospins, allowing for both amplitude and direction fluctuations, under external magnetic field. EP is calculated using Fokker-Planck equation, distinguishing between reversible and irreversible parts of probability currents. The system entropy increases due to irreversible non-equilibrium processes, and reduces as heat dissipates to surrounding environment. Using path probability distributions of time-forward trajectories and conjugate trajectories under time reversal, we obtain fluctuati…
The persistence probability P_g(t) of the global order-parameter of a simple ferromagnet undergoing phase-ordering kinetics after a quench from a fully disordered state to below the critical temperature, T<T_c, is analysed. It is argued that the persistence probability decays algebraically with time in the entire low-temperature phase. For Markov processes, the associated global persistence exponent theta_g = (2 lambda_C -d)/(2z) is related to the autocorrelation exponent lambda_C. This relationship is confirmed for phase-ordering in the exactly solved 1D Ising model and the d-dimensional sphe…
In this paper we extend the concept of persistence, well defined for classical stochastic dynamics, to the context of quantum dynamics. We demonstrate the idea via quantum random walk and a successive measurement scheme, where persistence is defined as the time during which a given site remains unvisited by the walker. We also investigated the behavior of related quantities, e.g., the first-passage time and the succession probability (newly defined), etc. The study reveals power law scaling behavior of these quantities with new exponents. Comparable features of the classical and the quantum wa…
