Simple generalization of Kramers theory to finite barrier height in spatial diffusion regime

Motivated by the escape process at low reduced barrier heights (measured in units of $k_{B}T$) is still a stationary one, the Kramers theoretical method in spatial diffusion regime should be applicable to this process. The Kramers theory is generalized to finite barrier height in a simple manner. The integration constant is redetermined by introducing metastable equilibrium state concept and continuous condition of the probability at the joint point of the potential barrier and potential well. The parabolic barrier with local frequency is replaced by a parabolic barrier with nonlocal frequency. The modified Kramers theory is confirmed by a cubic potential case. The maximal relative error in the spatial diffusion regime is less than 3% for the applied parameters.     

高温下复合核的热裂变速率

在核裂变的动力学描述中,考虑到质量随坐标的连续变化,提出了自由能有效势,导出了一个新的裂变速率公式.改进后的公式对于核温度与势能位垒高度相当的热核裂变同样适用,其结果比Kramers公式更接近于断点裂变速率的朗之万模拟值.     

Precision of approximate kramers formulas for the fission rate: Canonical and microcanonical ensembles

The precision of the Kramers formulas, which are frequently used to calculate the rate of fission of excited nuclei, is studied quantitatively bymeans of a numerical simulation. The deformation dependence of the level-density parameter is taken here into account in contrast to what was done in previous studies on this subject. It turns out that, as the excitation energy becomes lower, the error in the integral Kramers formula decreases monotonically, reaching 2% (precision of dynamical simulation) for a potential-barrier height three to four times greater than the temperature. The approximate Kramers formula obtained from the aforementioned integral formula is in substantially poorer agreement with the results of the dynamical simulation. In the same region of the controlling parameter, its typical error may change sign and reaches 5 to 15%.     

Energy diffusion controlled reaction rate in dissipative Hamiltonian systems

In this paper the energy diffusion controlled reaction rate in dissipative Hamiltonian systems is investigated by using the stochastic averaging method for quasi Hamiltonian systems. The boundary value problem of mean first-passage time (MFPT) of averaged system is formulated and the energy diffusion controlled reaction rate is obtained as the inverse of MFPT. The energy diffusion controlled reaction rate in the classical Kramers bistable potential and in a two-dimensional bistable potential with a heat bath are obtained by using the proposed approach respectively. The obtained results are then compared with those from Monte Carlo simulation of original systems and from the classical Kramers theory. It is shown that the reaction rate obtained by using the proposed approach agrees well with that from Monte Carlo simulation and is more accurate than the classical Kramers rate.     

Stochastic resonance in multidimensional periodic potential

We study the mobility and diffusion of an underdamped Brownian particle moving in a two-dimensional (2D) periodic potential which subjects to a thermal white noise and a weak external driving force. Both the signal power amplification and the diffusion rate are calculated via Langevin simulations. It is shown that the stochastic resonance (SR) can be observed in the two dimension, namely, the output quantities as functions of the temperature show a nonmonotonic behavior, however, the SR cannot be obtained in the one dimension (1D). In the 2D potential, the height of dynamical barrier is decreased effectively along the direction of transport if the curvature of the potential at the barrier is less than that at the local minima. This leads to the SR condition being obeyed, i.e., the Kramers frequency over the barrier roughly matches the frequency of external signal.     

Uniform expansion of the transition rate in Kramers' problem

Kramers' model of diffusion over potential barriers, e.g., chemical reactions, based on the noise activated escape of a particle from a potential well, is considered. Kramers derived escape rates valid for intermediate and large damping, and in a separate analysis, for small damping. In the small damping limit, Kramers' intermediate result reduces to the transition state rate which does not agree with the small damping result. A new escape rate is derived that is uniformly valid for all values of the damping coefficient. The new rate reduces to Kramers' results in the appropriate limits and, in particular, connects Kramers' intermediate and small damping results.This work was partially supported by the Air Force Office of Scientific Research under Grant No. AFOSR-83-0086, U.S. Department of Energy under Grant No. DE-AC02-78ERO-4650, and the National Science Foundation under Grant No. MCS-83-00562. One of us (BJM) gratefully acknowledges the support of a John Simon Guggenheim Memorial Foundation Fellowship.     

A Remark on the Kramers Problem

We present new point of view on the old problem, the Kramers problem. The passages from the Fokker–Planck equation to the Smoluchowski equation, including corrections to the Smoluchowski current, is treated through an asymptotic expansion of the solution of the stochastic dynamical equations. The case of an extremely weak force of friction is also discussed.     

Tunneling as a Classical Escape Rate Induced by the Vacuum Zero-point Radiation

We make a brief review of the Kramers escape rate theory for the probabilistic motion of a particle in a potential well U(x), and under the influence of classical fluctuation forces. The Kramers theory is extended in order to take into account the action of the thermal and zero-point random electromagnetic fields on a charged particle. The result is physically relevant because we get a non-null escape rate over the potential barrier at low temperatures (T → 0). It is found that, even if the mean energy is much smaller than the barrier height, the classical particle can escape from the potential well due to the action of the zero-point fluctuating fields. These stochastic effects can be used to give a classical interpretation to some quantum tunneling phenomena. Relevant experimental data are used to illustrate the theoretical results.     

基于Kramers逃逸速率的调参随机共振机理

根据Kramers逃逸速率的特性,阐明了随机共振信号的频率被限制在Kramers逃逸速率极限值一半的范围内,这种限制是制约大频率信号产生随机共振的原因.在进一步揭示二次采样随机共振频率尺度变换机理的基础上,证明了二次采样频率尺度可以把任意信号频率映射变换到随机共振频率尺度上的结论.相对于二次采样变换方法,由于双稳系统参数的调节很难使Kramers逃逸速率的一半达到实际信号的大频率,因此系统参数只能在随机共振的小参数频率范围内调节来实现随机共振. 关键词： 双稳随机共振 二次采样频率变换 系统参数调节 Kramers逃逸速率     

Brownian motion on a manifold

The question of the existence and correct form of equations describing Brownian motion on a manifold cannot be answered by mathematics alone, but requires a study of the underlying physics. As in classical mechanics, manifolds enter through the transformation of variables needed to account for the presence of constraints. The constraints are either due to a physical agency that forces the motion to remain on a manifold, or they represent conserved quantities of the equation of motion themselves. Also the Brownian motion is described either by a Smoluchowski diffusion equation or by a Kramers equation. The four cases lead to the following conclusions, (i) Smoluchowski diffusion with a conserved quantity reduces to a diffusion equation on the manifold; (ii) The same is true for diffusion with a physical constraint in three dimensions, but in more dimensions it may happen thatno autonomous equation on the manifold results; (iii) A Kramers equation with a conserved quantity reduces to an equation on the manifold, but in general not of the form of a Kramers equation; (iv) The Kramers equation with a physical constraint reduces to an autonomous Kramers equation on the manifold only for a special shape of that constraint. Throughout, only a certain type of physical constraints has been envisaged, and global questions are ignored. Finally, the customary heuristic construction of a Fokker-Planck equation for a mechanical system on a manifold is demonstrated for the case of Brownian rotation of a rigid body, and its shortcomings are emphasized.     

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.     

Brownian motion in a periodic potential: Application to dielectric relaxation

The relaxational dynamics of a planar rotator in anM-fold cosine potential subject to a random torque is investigated in detail. For the case of a periodic potential with large barrier height, the numerical results of the relaxation dynamics are in complete agreement with an approximate analytical solution. The latter, is derived on assuming a harmonic potential at the bottom of the potential minima and a large time-scale separation between the short-time libration inside each potential minima and a long-time hopping phenomenon over the potential barriers. ForM2, the hopping phenomenon is found to be the dominant feature of the orientational autocorrelation function. The average hopping time is explained satisfactorily in terms of the Kramers activation rate theory. In particular a complete agreement is found between the numerical results of the escape rate and those obtained from the modified Kramers' predictions valid for low friction coefficient. The cosine model is applied to the study of dielectric spectroscopy. The particle mobility and the complex permitivity, of a dielectric material are calculated by numerical solutions for rotational velocity and orientational auto-correlation functions, respectively. The main features of the experimental observables are determined analytically and compared to the corresponding numerical results. The applicability of the plane rotator model to dielectric spectroscopy is also discussed.From Jan. 1^st 1985: Institut für Theoretische Physik, RWTH, Aachen, FRG     

Modified fusion probability by reflection boundary

We investigate the time-dependent probability for a Brownian particle passing over the barrier to stay at a metastable potential pocket against escaping over the barrier. This is related to the whole fusion-fission dynamical process and can be called the reverse Kramers problem. By the passing probability over the saddle point of an inverse harmonic potential multiplying the exponential decay factor of a particle in the metastable potential, we present an approximate expression for the modified passing probability over the barrier, in which the effect of the reflection boundary of the potential is taken into account. Our analytical result and Langevin Monte-Carlo simulation show that the probability of passing and against escaping over the barrier is a non-monotonous function of time and its maximal value is less than the stationary result of the passing probability over the saddle point of an inverse harmonic potential.     

Prediction of ion channel transport from Grote—Hynes and Kramers theories

Generalized Langevin equation based Grote—Hynes (GH) theory and Langevin equation based Kramers theory are used to calculate the transmission coefficient for K⁺ diffusion through a model of the biological potassium ion channel IRK1, which contains a high potential barrier in the selectivity filter. The ion friction kernel is determined from a molecular dynamics (MD) simulation of the force on a stationary ion at the barrier top. The GH and Kramers estimates of the transmission coefficient are compared with those obtained from MD simulations of ion diffusion at the barrier top of the IRK1 channel. It is found that the GH estimate agrees with the value determined by rigorous MD, but the Kramers estimate is about 40% too small. The success or failure of GH and Kramers theories for various other systems is discussed and compared with these results.     

基于Kramers逃逸速率的Duffing振子广义调参随机共振研究

以二维Duffing振子的随机共振为研究对象,提出Duffing振子的广义调参随机共振. 以Kramers逃逸速率为基础,建立了Duffing振子随机共振的判别函数,阐述了Duffing振子在不同噪声强度及信号频率输入条件下的广义调参随机共振规律,并给出了Duffing振子广义调参随机共振的一般方法. 关键词： Duffing振子 随机共振 Kramers逃逸速率 广义参数调节     

Diffusion equations and the time evolution of foreign exchange rates

We investigate which type of diffusion equation is most appropriate to describe the time evolution of foreign exchange rates. We modify the geometric diffusion model assuming a non-exponential time evolution and the stochastic term is the sum of a Wiener noise and a jump process. We find the resulting diffusion equation to obey the Kramers–Moyal equation. Analytical solutions are obtained using the characteristic function formalism and compared with empirical data. The analysis focus on the first four central moments considering the returns of foreign exchange rate. It is shown that the proposed model offers a good improvement over the classical geometric diffusion model.     

Application of the theory of open quantum systems to nuclear physics problems

Quantum diffusion equations with transport coefficients explicitly depending on time are derived from the generalized non-Markovian Langevin equations. The asymptotic behavior of the friction and diffusion coefficients is investigated in the case of the FC and RWA couplings between the collective and internal subsystems. An asymptotic expression is obtained for the propagator of the density matrix of the open quantum system with the general quadratic Hamiltonian, linearly coupled (in coordinate and momentum) to internal degrees of freedom. The effect of different sets of transport coefficients on the decoherence and decay rate of the metastable state is investigated using the master equation for the reduced density matrix of open quantum systems. The developed approach is used to study the capture of the projectile nucleus by the target nucleus at energies near the Coulomb barrier. Capture cross sections in asymmetric reactions are well described with allowance for the calculated capture probabilities. Particular cases where dissipation favors penetration through the potential barrier are found. The generalized Kramers formula for the quasi-stationary decay rate of the quantum metastable systems is analytically derived.     

Solutions of the Fokker-Planck equation for a double-well potential in terms of matrix continued fractions

Solutions of the Fokker-Planck (Kramers) equation in position-velocity space for the double-well potentiald ₂x²/2+d₄x⁴/4 in terms of matrix continued fractions are derived. It is shown that the method is also applicable to a Boltzmann equation with a BGK collision operator. Results of eigenvalues and of the Fourier transform of correlation functions are presented explicitly. The lowest nonzero eigenvalue is compared with the escape rate in the weak noise limit for various damping constants and the susceptibility is compared with the zero-friction-limit result.     

Predissociation induced by ungerade-gerade symmetry breaking in 6Li7Li molecule

In the recent sub-Doppler experiment on the B1Pi(u) state of the Li2 molecule by Bouloufa et al. [J. Chem. Phys. 111, 1926 (1999)], where the dissociation of vibrational levels due to the tunneling through the potential barrier was investigated, several vibrational levels with abnormally large dissociated rates were observed in the case of the 6Li7Li isotopomer. This dynamical effect cannot be explained by tunneling as in the case of 6Li2 or 7Li2. A simple model of coupling between B1Pi(u) and 1(1)Pi(g) states involving the u-g symmetry breaking for 6Li7Li is proposed. Rates of the B1Pi(u) predissociated levels due to this coupling are calculated. A good agreement with the experiment is found.