Theories that are used to extract energy-landscape information from single-molecule pulling experiments in biophysics are all invariably based on Kramers' theory of thermally-activated escape rate from a potential well. As is well known, this theory recovers the Arrhenius dependence of the rate on the barrier energy, and crucially relies on the assumption that the barrier energy is much larger than kBT (limit of comparatively low thermal fluctuations). As was already shown in Dudko, Hummer, Szabo Phys. Rev. Lett. (2006), this approach leads to the unphysical prediction of dissociation time increasing with decreasing binding energy when the latter is lowered to values comparable to kBT (limit of large thermal fluctuations). We propose a new theoretical framework (fully supported by numerical simulations) which amends Kramers' theory in this limit, and use it to extract the dissociation rate from single-molecule experiments where now predictions are physically meaningful and in agreement with simulations over the whole range of applied forces (binding energies). These results are expected to be relevant for a large number of experimental settings in single-molecule biophysics.

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Accounting for viscous damping within Fokker-Planck equations led to various improvements in the understanding and analysis of nuclear fission of heavy nuclei. Analytical expressions for the fission time are typically provided by Kramers' theory, which improves on the Bohr-Wheeler estimate by including the time-scale related to many-particle dissipative processes along the deformation coordinate. However, Kramers' formula breaks down for sufficiently high excitation energies where Kramers' assumption of a large barrier no longer holds. In the regime T > 1 MeV, Kramers' theory should be replaced by a new theory based on the Ornstein-Uhlenbeck first-passage time method that is proposed here. The theory is applied to fission time data from fusion-fission experiments on 16 O+ 208 Pb → 224 Th. The proposed model provides an internally consistent one-parameter fitting of fission data with a constant nuclear friction as the fitting parameter, whereas Kramers' fitting requires a value of friction which falls out of the allowed range. The theory provides also an analytical formula that in future work can be easily implemented in numerical codes such as CASCADE or JOANNE4.

We study the overdamped Brownian dynamics of particles moving in piecewise-defined potential energy landscapes U (x), where the height Q of each section is obtained from the exponential distribution p(Q) = aβexp(−aβQ), where β is the reciprocal thermal energy, and a > 0. The averaged effective diffusion coefficient D eff is introduced to characterize the diffusive motion: x 2 = 2 D eff t. A general expression for D eff in terms of U (x) and p(Q) is derived and then applied to three types of energy landscape: flat sections, smooth maxima, and sharp maxima. All three cases display a transition between subdiffusive and diffusive behavior at a = 1, and a reduction to free diffusion as a → ∞. The behavior of D eff around the transition is investigated and found to depend heavily upon the shape of the maxima: Energy landscapes made up of flat sections or smooth maxima display power-law behavior, while for landscapes with sharp maxima, strongly divergent behavior is observed. Two aspects of the subdiffusive regime are studied: the growth of the mean squared displacement with time and the distribution of mean first-passage times. For the former, agreement between Brownian dynamics simulations and a coarse-grained equivalent was observed, but the results deviated from the random barrier model's predictions. The discrepancy could be a finite-time effect. For the latter, agreement between the characteristic exponent calculated numerically and that predicted by the random barrier model is observed in the large-amplitude limit.

Diffusion in spatially rough, confining, one-dimensional continuous energy landscapes is treated using Zwanzig's proposal, which is based on the Smoluchowski equation. We show that Zwanzig's conjecture agrees with Brownian dynamics simulations only in the regime of small roughness. Our correction of Zwanzig's framework corroborates well with numerical results. A numerical simulation scheme based on our coarse-grained Langevin dynamics offers significant reductions in computational time. The mean first-passage time problem in the case of random roughness is treated. Finally, we address the validity of the separation of length scales assumption for the case of polynomial backgrounds and cosine-based roughness. Our results are applicable to hierarchical energy landscapes such as that of a protein's folding and transport processes in disordered media, where there is clear separation of length scale between smooth underlying potential and its rough perturbation.