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We consider diffusion over a potential barrier for n degrees of freedom. Generalizing the procedure of Kramers, we find a quasistationary solution to the associated Fokker-Planck equation. This yields an expression for the diffusion current over the barrier and, finally, a simple and elegant generalization of Kramers' formula for the diffusion rate.

We compare multifocal intraocular lenses (MFIOLs) to monofocal IOLs for visual acuity (VA), contrast sensitivity, and adverse events using data from 21 randomized controlled trials with 2951 subjects. There was no statistical difference between uncorrected distance VA and corrected distance VA. Compared with monofocal IOLs, MFIOLs showed a better performance on uncorrected intermediate VA measured at 60 cm and uncorrected near VA; the mean differences were-0.06 (95% confidence interval [

Using a Fokker-Planck equation, we consider induced nuclear fission for large values of the friction constant. Then, the equation simplifies to the Smoluchowski equation which we solve using a method of van Kampen. We find that the fission rate obeys a scaling law. From our numerical results and those obtained previously, we estimate the transient time~for fission. For small nuclear temperature,~is defined as the time delay between the onset of a nuclear reaction and the attainment of the quasistationary probability flow over the fission barrier. For nuclear temperatures large compared to the fission barrier, the entire fission process is governed by transients, and~essentially yields the fission lifetime. We speculate on the significance of our results for recent experimental observations.

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Based on the unified Hauser–Feshbach and exciton model for light nuclei, the calculations of reaction cross sections and the double-differential cross sections for n + 6Li are performed. Since all of the first-particle emissions are from the compound nucleus to the discrete levels, the angular momentum coupling effect in pre-equilibrium mechanism must be taken into account. The fitting of the measured data indicates that the three-body break-up process needs to be involved, and the pre-equilibrium reaction mechanism dominates the reaction processes. In light nucleus reactions the recoil effect must be taken into account.

BackgroundTo compare the clinical performance of bifocal and trifocal intraocular lenses (IOLs) in cataract surgery, a meta-analysis on randomized controlled trials was conducted.MethodsA comprehensive literature retrieval of PubMed, Science Direct and EMBASE was performed in this systematic review. Clinical outcomes included visual acuity (VA), contrast sensitivity (CS), spectacle independence, postoperative refraction and surgical satisfaction.ResultsThere were 8 RCTs included in this study. The difference of uncorrected near VA (UNVA) between the bifocal IOLs and trifocal IOLs had no significance [MD = 0.02, 95%CI: (− 0.03,0.06)]. There was no significant difference in the distant-corrected near VA (DCNVA) with MD of 0.04 [95%CI (− 0.02, 0.10)]. Compared with trifocal group, the uncorrected intermediate visual acuity (UIVA) [MD = 0.09,95%CI:(0.01,0.17)] was significantly worse in the bifocal group. No difference was found in distance-corrected intermediate VA (DCIVA) [MD = 0.09, 95%CI: (− 0.04, 0.23)] between two groups. Analysis on AT LISA subgroup indicated the bifocal group had worse intermediate VA than trifocal group (AT LISA tri 839 M) [MD = 0.18, 95%CI: (0.12, 0.24) for UIVA and MD = 0.19, 95%CI: (0.13, 0.25) for DCIVA]. However, there was no statistically significant difference between the two groups in the uncorrected distance VA (UDVA) and corrected distance visual acuity (CDVA) [MD = 0.01, 95%CI: (− 0.01,0.04) for UDVA; MD = 0.00, 95%CI: (− 0.01,0.01) for CDVA].The postoperative refraction of bifocal group was similar to that of trifocal group [MD = -0.08, 95% CI: (− 0.19, 0.03) for spherical equivalent; MD = -0.09, 95%CI: (− 0.21, 0.03) for cylinder; MD = -0.09, 95% CI: (− 0.27, 0.08) for sphere]. No difference was found for spectacle independence, posterior capsular opacification (PCO) incidence and patient satisfaction between bifocal IOLs and trifocal IOLs. [RR = 0.89, 95% CI: (0.71, 1.12) for spectacle independence; RR = 1.81, 95% CI: (0.50, 6.54) for PCO incidence; RR = 0.98, 5% CI: (0.86, 1.12) for patient satisfaction].ConclusionPatients receiving trifocal IOLs, especially AT LISA tri 839 M, have a better intermediate VA than those receiving bifocal IOLs. Near and distance visual performance, spectacle independence, postoperative refraction and surgical satisfaction of bifocal IOLs were similar to those of trifocal IOLs.Electronic supplementary materialThe online version of this article (10.1186/s12886-019-1078-1) contains supplementary material, which is available to authorized users.