Kizhakeyil L. Sebastian

We consider the escape of a particle trapped in a metastable potential well and acted upon by two noises. One of the noises is thermal and the other is Poisson white noise, which is non-Gaussian. Using path integral techniques, we find an analytic solution to this generalization of the classic Kramers barrier crossing problem. Using the “barrier climbing” path, we calculate the activation exponent. We also derive an approximate expression for the prefactor. The calculated results are compared with the simulations, and a good agreement between the two is found. Our results show that, unlike in the case of thermal noise, the rate depends not just on the barrier height but also on the shape of the whole barrier. A comparison between the simulations and the theory also shows that a better approximation for the prefactor is needed for agreement for all values of the parameters.

The usual Kramers theory of reaction rates in condensed media predict the rate to have an inverse dependence on the viscosity of the medium, η. However, experiments on ligand binding to proteins, performed long ago, showed the rate to have η^−ν dependence, with ν in the range of 0.4–0.8. Zwanzig [J. Chem. Phys. 97, 3587 (1992)] suggested a model in which the ligand has to pass through a fluctuating opening to reach the binding site. This fluctuating gate model predicted the rate to be proportional to η^−1/2. More recently, experiments performed by Xie et al. [Phys. Rev. Lett. 93, 180603 (2004)] showed that the distance between two groups in a protein undergoes not normal diffusion, but subdiffusion. Hence, in this paper, we suggest and solve a generalization of the Zwanzig model, viz., passage through an opening, whose size undergoes subdiffusion. Our solution shows that the rate is proportional to η^−ν with ν in the range of 0.5–1, and hence, the subdiffusion model can explain the experimental observations.

Traditionally, the delocalized π system of benzene is believed to be responsible for its perfectly symmetric D_6h geometry. However, it has also been suggested that the π system prefers a distorted D_3h geometry. Arguments for this have been based on clever use of VB methods as well as through shifts in the frequency of the distortive b_2u mode. Evidence has been provided through different ways of partitioning the total electronic energy between the σ and the π systems. These methods are plagued by the fact that there is no unique way to partition the energy, leading to questions regarding the validity of the conclusions. Here we note that even though energy cannot be partitioned exactly, force acting on a nucleus depends only on the single particle density and can hence be partitioned exactly. Using good-quality wave functions that are numerically found to obey the Hellmann–Feynman theorem to good accuracy, we calculate the σ and π components of the force and provide conclusive evidence of π-distortivity at the HF level. Our approach provides an unambiguous way to approach the problem with wave functions that account for electron correlation. Our calculations suggest that the conclusion is valid at the MP2 level, too.

We consider a particle diffusing in a bounded, crowded, rearranging medium. The rearrangement happens on a time scale longer than the typical time scale of diffusion of the particle; as a result, effectively, the diffusion coefficient of the particle varies as a stochastic function of time. What is the probability that the particle will survive within the bounded region, given that it is absorbed the first time it hits the boundary of the region in which it diffuses? This question is of great interest in a variety of chemical and biological problems. If the diffusion coefficient is a constant, then analytical solutions for a variety of cases are available in the literature. However, there is no solution available for the case in which the diffusion coefficient is a random function of time. We discuss a class of models for which it is possible to find analytical solutions to the problem. We illustrate the method for a circular, two-dimensional region, but our methods are easy to apply to diffusion in arbitrary dimensions and spherical/rectangular regions. Our solution shows that if the dimension of the region is large, then only the average value of the diffusion coefficient determines the survival probability. However, for smaller-sized regions, one would be able to see the effects of the stochasticity of the diffusion coefficient. We also give generalizations of the results to N dimensions.

It has been found in many experiments that the mean square displacement of a Brownian particle x(T) diffusing in a rearranging environment is strictly Fickian, obeying ⟨(x(T))²⟩ ∝ T, but the probability distribution function for the displacement is not Gaussian. An explanation of this is that the diffusivity of the particle itself is changing as a function of time. Models for this diffusing diffusivity have been solved analytically in the limit of small time, but simulations were necessary for intermediate and large times. We show that one of the diffusing diffusivity models is equivalent to Brownian motion in the presence of a sink and introduce a class of models for which it is possible to find analytical solutions. Our solution gives ⟨(x(T))²⟩ ∝ T for all times and at short times the probability distribution function of the displacement is exponential which crosses over to a Gaussian in the limit of long times and large displacements.

DOI: http://dx.doi.org/10.1103/PhysRevE.91.019904Published 20 January 2015Received 21 November 2014©2015 American Physical Society