Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userSpatial First-passage Statistics of Al/Si(111)-(root3 x root3) Step Fluctuations
289
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
Spatial step edge fluctuations on a multi-component surface of Al/Si(111)-(root3 x root3) were measured via scanning tunneling microscopy over a temperature range of 720K-1070K, for step lengths of L = 65-160 nm. Even though the time scale of fluctuations of steps on this surface varies by orders of magnitude over the indicated temperature ranges, measured first-passage spatial persistence and survival probabilities are temperature independent. The power law functional form for spatial persistence probabilities is confirmed and the symmetric spatial persistence exponent is measured to be theta = 0.498 +/- 0.062 in agreement with the theoretical prediction theta = 1/2. The survival probability is found to scale directly with y/L, where y is the distance along the step edge. The form of the survival probabilities agree quantitatively with the theoretical prediction, which yields exponential decay in the limit of small y/L. The decay constant is found experimentally to be ys/L= 0.076 +/- 0.033 for y/L <= 0.2.
rate research
Read More
We study the distribution of first-passage functionals ${cal A}= int_0^{t_f} x^n(t), dt$, where $x(t)$ is a Brownian motion (with or without drift) with diffusion constant $D$, starting at $x_0>0$, and $t_f$ is the first-passage time to the origin. In the driftless case, we compute exactly, for all $n>-2$, the probability density $P_n(A|x_0)=text{Prob}.(mathcal{A}=A)$. This probability density has an essential singular tail as $Ato 0$ and a power-law tail $sim A^{-(n+3)/(n+2)}$ as $Ato infty$. The former is reproduced by the optimal fluctuation method (OFM), which also predicts the optimal paths of the conditioned process for small $A$. For the case with a drift toward the origin, where no exact solution is known for general $n>-1$, the OFM predicts the distribution tails. For $Ato 0$ it predicts the same essential singular tail as in the driftless case. For $Ato infty$ it predicts a stretched exponential tail $-ln P_n(A|x_0)sim A^{1/(n+1)}$ for all $n>0$. In the limit of large Peclet number $text{Pe}= mu x_0/(2D)gg 1$, where $mu$ is the drift velocity, the OFM predicts a large-deviation scaling for all $A$: $-ln P_n(A|x_0)simeqtext{Pe}, Phi_nleft(z= A/bar{A}right)$, where $bar{A}=x_0^{n+1}/{mu(n+1)}$ is the mean value of $mathcal{A}$. We compute the rate function $Phi_n(z)$ analytically for all $n>-1$. For $n>0$ $Phi_n(z)$ is analytic for all $z$, but for $-1<n<0$ it is non-analytic at $z=1$, implying a dynamical phase transition. The order of this transition is $2$ for $-1/2<n<0$, while for $-1<n<-1/2$ the order of transition changes continuously with $n$. Finally, we apply the OFM to the case of $mu<0$ (drift away from the origin). We show that, when the process is conditioned on reaching the origin, the distribution of $mathcal{A}$ coincides with the distribution of $mathcal{A}$ for $mu>0$ with the same $|mu|$.
A well known connection between first-passage probability of random walk and distribution of electrical potential described by Laplace equation is studied. We simulate random walk in the plane numerically as a discrete time process with fixed step length. We measure first-passage probability to touch the absorbing sphere of radius $R$ in 2D. We found a regular deviation of the first-passage probability from the exact function, which we attribute to the finiteness of the random walk step.
Levy walks (LWs) define a fundamental class of finite velocity stochastic processes that can be introduced as a special case of continuous time random walks. Alternatively, there is a hyperbolic representation of them in terms of partial probability density waves. Using the latter framework we explore the impact of aging on LWs, which can be viewed as a specific initial preparation of the particle ensemble with respect to an age distribution. We show that the hyperbolic age formulation is suitable for a simple integral representation in terms of linear Volterra equations for any initial preparation. On this basis relaxation properties and first passage time statistics in bounded domains are studied by connecting the latter problem with solute release kinetics. We find that even normal diffusive LWs may display anomalous relaxation properties such as stretched exponential decay. We then discuss the impact of aging on the first passage time statistics of LWs by developing the corresponding Volterra integral representation. As a further natural generalization the concept of LWs with wearing is introduced to account for mobility losses.
We report the results of analytic and numerical investigations of the time scale of survival or non-zero-crossing probability $S(t)$ in equilibrium step fluctuations described by Langevin equations appropriate for attachment/detachment and edge-diffusion limited kinetics. An exact relation between long-time behaviors of the survival probability and the autocorrelation function is established and numerically verified. $S(t)$ is shown to exhibit simple scaling behavior as a function of system size and sampling time. Our theoretical results are in agreement with those obtained from an analysis of experimental dynamical STM data on step fluctuations on Al/Si(111) and Ag(111) surfaces.
We investigate classic diffusion with the added feature that a diffusing particle is reset to its starting point each time the particle reaches a specified threshold. In an infinite domain, this process is non-stationary and its probability distribution exhibits rich features. In a finite domain, we define a non-trivial optimization in which a cost is incurred whenever the particle is reset and a reward is obtained while the particle stays near the reset point. We derive the condition to optimize the net gain in this system, namely, the reward minus the cost.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
