Escape by diffusion from a square well across a square barrier

The problem of escape of a particle by diffusion from a square potential well across a square barrier is studied on the basis of the one-dimensional Smoluchowski equation for the space- and time-dependent probability distribution. For the model potential the Smoluchowski equation is solved exactly by a Laplace transform with respect to time. In the limit of a high barrier the rate of escape is given by an asymptotic result similar to that derived by Kramers for a curved well and a curved barrier. An approximate analytic formula is derived for the outward time-dependent probability current in terms of the width and depth of the well and the width and height of the barrier. A similar expression holds for the complete probability distribution.     

Diffusion in a bistable potential

The problem of diffusion of a particle in a bistable potential is studied on the basis of the one-dimensional Smoluchowski equation for the space- and time-dependent probability distribution. The potential is modeled as two parabolic wells separated by a parabolic barrier. For the model potential the Smoluchowski equation is solved exactly by a Laplace transform with respect to time for the initial condition that at time zero the probability distribution is given by a thermal equilibrium distribution in one of the wells. In the limit of a high barrier the rate of transition to the other well is given by an asymptotic result due to Kramers. For a potential barrier of moderate height there are significant corrections to the asymptotic result.     

Diffusion dynamics in external noise-activated non-equilibrium open system-reservoir coupling environment

The diffusion process in an external noise-activated non-equilibrium open system-reservoir coupling environment is studied by analytically solving the generalized Langevin equation. The dynamical property of the system near the barrier top is investigated in detail by numerically calculating the quantities such as mean diffusion path, invariance, barrier passing probability, and so on. It is found that, comparing with the unfavorable effect of internal fluctuations, the external noise activation is sometimes beneficial to the diffusion process. An optimal strength of external activation or correlation time of the internal fluctuation is expected for the diffusing particle to have a maximal probability to escape from the potential well.     

Angle and speed distributions of hydrogen desorbing thermally from metal surfaces

A quantum theoretical treatment of the angle and speed distributions of recombinatively desorbing hydrogen from metal surfaces is proposed. The desorption rate is discussed in the framework of the transition state theory. The recombinative reaction process of hydrogen due to thermal activation leads to the formation of an activated complex in the transition state. In the vicinity of a saddle point on a three-dimensional potential energy surface, the translational motion of the activated complex in the direction perpendicular to the metal surface is accompanied by its center-of-mass vibrational motion parallel to the metal surface. In order to carry out the quantum mechanical calculation, the potential surface is replaced by a simplified model potential, which provides a square potential barrier along the surface normal. It is shown that, on leaving the potential barrier, the activated complex is reflected by the boundary of the potential barrier with a certain probability and, at the same time, the center-of-mass modes of vibration with frequencies v ₁ and v ₂ are coupled with the translational motion along the surface normal. Vibrational wave functions in the momentum representation are used to calculate the transmission coefficient, which is incorporated into the conventional rate formula. The angle-dependent speed distributions of desorbing molecules are derived from the rate formula.     

Transitions between wells and escape by diffusion in a one-dimensional potential landscape

The time-dependence of the occupation probabilities of neighboring wells due to diffusion in one dimension is formulated in terms of a set of generalized rate equations describing transitions between neighboring wells and escape across a final barrier. The equations contain rate coefficients, memory coefficients, and a long-time coefficient characterizing the amplitude of long-time decay. On a more microscopic level the stochastic process is described by a Smoluchowski equation for the one-dimensional probability distribution. A numerical procedure is presented which allows calculation of the transport coefficients in the set of generalized rate equations on the basis of the Smoluchowski equation.     

Angle and speed distributions of hydrogen desorbing thermally from metal surfaces

On the basis of the theory developed in a previous paper (Part I), the angle and speed distributions of D₂ molecules desorbing thermally from the Ni(111) surface are numerically calculated in the present paper (Part II). The dynamical motion of the activated complex of the recombinative desorption process is calculated by using a model potential surface, which involves a few parameters introduced to describe the necessary features of the potential surface at the transition state. Numerical calculations are carried out, and the results show that the present dynamical theory reproduces very well a variety of the characteristic features of the experimental results, e.g., sharply focused angular distribution, mean translational energy lower than 2k _B T at grazing angle, and the non-Maxwellian profile of the TOF distribution.     

Modeling polymerization of microtubules: A semi-classical nonlinear field theory approach

In this paper, for the first time, a three-dimensional treatment of microtubules’ polymerization is presented. Starting from fundamental biochemical reactions during microtubule’s assembly and disassembly processes, we systematically derive a nonlinear system of equations that determines the dynamics of microtubules in three dimensions. We found that the dynamics of a microtubule is mathematically expressed via a cubic-quintic nonlinear Schrödinger (NLS) equation. We show that in 3D a vortex filament, a generic solution of the NLS equation, exhibits linear growth/shrinkage in time as well as temporal fluctuations about some mean value which is qualitatively similar to the dynamic instability of microtubules. By solving equations numerically, we have found spatio-temporal patterns consistent with experimental observations.     

Slowly rocking symmetric, spatially periodic Hamiltonians: The role of escape and the emergence of giant transient directed transport

The nonintegrable Hamiltonian dynamics of particles placed in a symmetric, spatially periodic potential and subjected to a periodically varying field is explored. Such systems can exhibit a rich diversity of unusual transport features. In particular, depending on the setting of the initial phase of the drive, the possibility of a giant transient directed transport in a symmetric, space-periodic potential when driven with an adiabatically varying field arises. Here, we study the escape scenario and corresponding mean escape times of particles from a trapping region with the subsequent generation of a transient directed flow of an ensemble of particles. It is shown that for adiabatically slow inclination modulations the unidirectional flow proceeds over giant distances. The direction of escape and, hence, of the flow is entirely governed whether the periodic force, modulating the inclination of the potential, starts out initially positive or negative. In the phase space, this transient directed flow is associated with a long-lasting motion taking place within ballistic channels contained in the non-uniform chaotic layer. We demonstrate that for adiabatic modulations all escaping particles move ballistically into the same direction, leading to a giant directed current.     

Computer simulation study of the Levy flight process

The random walk simulation of a Levy flight shows a linear relation between the mean square displacement 〈r²〉 and time. We have analyzed different aspects of this linearity. It is shown that the restriction of jump length to a maximum value (l_m) affects the diffusion coefficient, even though it remains constant for l_m greater than 1464. So, this factor has no effect on the linearity. In addition, it is shown that the number of samples does not affect the results. We have demonstrated that the relation between the mean square displacement and time remains linear in a continuous space, while continuous variables just reduce the diffusion coefficient. The results also imply that the movement of a Levy flight particle is similar to the case in which the particle moves in each time step with an average jump length 〈l〉. Finally, it is shown that the non-linear relation of the Levy flight will be satisfied if we use a time average instead of an ensemble average. The difference between the time average and ensemble average results shows that the Levy distribution may be a non-ergodic distribution.     

Transition from transient response to steady state for a layered medium

The transition from transient response to steady-state for a layered medium subjected to antiplane loadings is studied. The steady-state formula for a layered medium is derived and the solutions for a layered half-space are then expressed explicitly in the form of wave number integrals. The transient response solutions for a layered half-space are obtained by the convolution of time harmonic loading function with transient response formula derived analytically from an effective matrix method. Two layered half-spaces with different ratios of wave velocities in the layer and half-space are considered and investigated by means of extensive numerical results to show their quite different transition behavior. The numerical results indicate that transient responses will approach steady state after certain characteristic times when the transient effects die away. The transition phenomena and characteristic times are investigated in detail through the responses from near field to far field as well as from low frequency to high frequency.     

Model-dependent nature of blowtorch effect on the escape and equilibration rates

We analyze the relaxation behavior of a bistable system when the background temperature profile is inhomogeneous due to the presence of a localized hot region (blowtorch) on one side of the potential barrier. Since the diffusion equation for inhomogeneous medium is model-dependent, we consider two physical models to study the kinetics of such system. Using a conventional stochastic method, we obtain the escape and equilibration rates of the system for the two physical models. For both models, we find that the hot region enhances the escape rate from the well where it is placed while it retards the escape rate from the other well. However, the value of the escape rate from the well where the hot region is placed differs for the two models while that of the escape rate from the other well is identical for both. This work, for the first time, gives a detailed report of the similarities and differences of the escape rates and, hence, exposes the common and distinct features of the two known physical models in determining the way the bistable system relaxes. Received 25 September 2001     

The Uphill Turtle Race; On Short Time Nucleation Probabilities

The short time behavior of nucleation probabilities is studied by representing nucleation as a diffusion process in a potential well with escape over a barrier. If initially all growing nuclei start at the bottom of the well, the first nucleation time on average is larger than the inverse nucleation frequency. Explicit expressions are obtained for the short time probability of first nucleation. For very short times these become independent of the shape of the potential well. They agree well with numerical results from an exact enumeration scheme. For a large number N of growing nuclei the average first nucleation time scales as 1/log N in contrast to the long-time nucleation frequency, which scales as 1/N. For linear potential wells closed form expressions are given for all times.     

Escape for System with Non-Fluctuating Potential Barrier Only Driven by Three-State Noise

We study the escape for the mean first passage time （MFPT） over a potential barrier for a system with non- fluctuating potential barrier and only driven by a three-state noise. It is shown that in some circumstances, the three-state noise can induce the resonant activation for the MFPT over the potential barrier; but in other circumstances, it can not. There are three resonant activations for the MFPT over the potential barrier, which are respectively as the functions of the transition rates of the three-state noise.     

Could one single dichotomous noise cause resonant activation for exit time over potential barrier？

This paper studies the mean first passage time （or exit time, or escape time） over the non-fluctuating potential harrier for a system driven only by a dichotomous noise. It finds that the dichotomous noise can make the particles escape over the potential barrier, in some circumstances; but in other circumstances, it can not. In the case that the particles escape over the potential harrier, a resonant activation phenomenon for the mean first passage time over the potential barrier is obtained.     

Directed transient long-range transport in a slowly driven Hamiltonian system of interacting particles

We study the Hamiltonian dynamics of a one-dimensional chain of linearly coupled particles in a spatially periodic potential which is subjected to a time-periodic mono-frequency external field. The average over time and space of the related force vanishes and hence, the system is effectively without bias which excludes any ratchet effect. We pay special attention to the escape of the entire chain when initially all of its units are distributed in a potential well. Moreover for an escaping chain we explore the possibility of the successive generation of a directed flow based on large accelerations. We find that for adiabatic slope-modulations due to the ac-field transient long-range transport dynamics arises whose direction is governed by the initial phase of the modulation. Most strikingly, that for the driven many particle Hamiltonian system directed collective motion is observed provides evidence for the existence of families of transporting invariant tori confining orbits in ballistic channels in the high-dimensional phase spaces.     

Photoelectron escape from iron oxide

Iron oxide of nanometer thickness were grown in situ by step-by-step oxidation of an iron foil to measure the escape probabilities of O 1s photoelectrons as a function of depth of origin, the so-called emission depth distribution function (EDDF), and the mean escape depth (MED). To record photoelectron spectra for a wide range of emission angles, the X-ray excitation source was positioned on the opposite side of the iron foil with respect to the analyzer. To excite photoelectrons on the side of the foil surface adjacent to the analyzer, the foil was made thin enough to be semitransparent to the X-ray radiation. The O 1s spectra were recorded for a wide range of oxide thicknesses until no features of metallic iron were recognized in the photoelectron spectra. The escape probability of the O 1s photoelectrons in the iron oxide was derived from the oxide-thickness dependence of the O 1s peak areas. The resulting EDDF reached a maximum beneath the oxide surface for X-ray incidence and electron-emission angles located along the surface normal. For the same incidence angle and an emission angle of 60°, the escape probability could be well approximated by a simple exponential function. The mean escape depths were obtained for both experimental geometries and agreed well with the available theory.     

Escape probability of O 1s photoelectrons leaving copper oxide

The escape probability of O 1s photoelectrons as a function of depth of origin in CuO has been studied experimentally and by the Monte Carlo technique. It has been proved experimentally that the escape probability in the direction of X-ray propagation is no longer described by a simple exponential law but is a complex function with a maximum at a depth of 4–10 Å. The mean escape depth is derived from the experimental data and is compared with the results of theory; satisfactory agreement is obtained.     

表面势垒对梯度掺杂GaN光电阴极电子逸出几率的影响

研究表面势垒对梯度掺杂GaN光电阴极电子逸出几率的影响.计算梯度掺杂透射式GaN光电阴极的电子能量分布及逸出几率,结果显示梯度掺杂与均匀掺杂相比,可以获得更大的电子逸出几率;I势垒对电子逸出几率的影响显著,而Ⅱ势垒影响较小.利用GaN光电阴极多信息量测试系统,测试两种GaN阴极样品的光电流.实验结果表明,梯度掺杂GaN样品比均匀掺杂电子逸出几率更大;单独进行Cs激活形成的I势垒对电子逸出几率有显著影响,而Cs/O共同激活形成的Ⅱ势垒对其影响较小.     

GaN光电阴极表面势垒对电子逸出几率的影响

针对GaN光电阴极表面势垒对电子逸出几率的影响问题,应用玻尓兹曼分布和基于Airy函数的传递矩阵法计算了GaN光电阴极的电子逸出几率,发现电子逸出几率主要由I势垒决定,II势垒对电子逸出几率的影响有限.利用自行研制的GaN光电阴极激活评估实验系统,测试了透射式GaN光电阴极样品的激活光电流.实验发现,Cs单独激活引起电子逸出几率的显著增加,而Cs单独充分激活后的Cs/O交替激活对电子逸出几率的影响有限.理论计算结果与激活光电流测试结果一致,其原因是Cs单独激活对降低真空能级的贡献远大于Cs/O共同激活.