Weak solutions of the Landau--Lifshitz--Bloch equation

The Landau--Lifshitz--Bloch (LLB) equation is a formulation of dynamic micromagnetics valid at all temperatures, treating both the transverse and longitudinal relaxation components important for high-temperature applications. We study LLB equation in case the temperature raised higher than the Curie temperature. The existence of weak solution is showed and its regularity properties are also discussed. In this way, we lay foundations for the rigorous theory of LLB equation that is currently not available

Decorrelation of total mass via energy

The main result of this small note is a quantified version of the assertion that if u and v solve two nonlinear stochastic heat equations, and if the mutual energy between the initial states of the two stochastic PDEs is small, then the total masses of the two systems are nearly uncorrelated for a very long time. One of the consequences of this fact is that a stochastic heat equation with regular coefficients is a finite system if and only if the initial state is integrable

Probing the elastic limit of DNA bending

Many structures inside the cell such as nucleosomes and protein-mediated DNA loops contain sharply bent double-stranded (ds) DNA. Therefore, the energetics of strong dsDNA bending constitutes an essential part of cellular thermodynamics. Although the thermomechanical behavior of long dsDNA is well described by the worm-like chain (WLC) model, the length limit of such elastic behavior remains controversial. To investigate the energetics of strong dsDNA bending, we measured the opening rate of small dsDNA loops with contour lengths of 40-200 bp using Fluorescence Resonance Energy Transfer (FRET). From the measured relationship of loop stability to loop size, we observed a transition between two separate bending regimes at a critical loop size below 100 bp. Above this loop size, the loop lifetime decreased with decreasing loop size in a manner consistent with an elastic bending stress. Below the critical loop size, however, the loop lifetime became less sensitive to loop size, indicative of softening of the double helix. The critical loop size was measured to be ~60 bp with sodium only and ~100 bp with 5 mM magnesium, which suggests that magnesium facilitates the softening transition. We show that our results are in quantitative agreement with the kinkable worm-like chain model. Furthermore, the model parameters constrained by our data can reproduce previously measured J factors between 50 and 200 bp. Our work provides powerful means to study dsDNA bending in the strong bending regime

Numerical solution of the time-fractional Fokker-Planck equation with general forcing

We study two schemes for a time-fractional Fokker-Planck equation with space- and time-dependent forcing in one space dimension. The first scheme is continuous in time and is discretized in space using a piecewise-linear Galerkin finite element method. The second is continuous in space and employs a time-stepping procedure similar to the classical implicit Euler method. We show that the space discretization is second-order accurate in the spatial $L_2$-norm, uniformly in time, whereas the corresponding error for the time-stepping scheme is $O(k^\alpha)$ for a uniform time step $k$, where $\alpha\in(1/2,1)$ is the fractional diffusion parameter. In numerical experiments using a combined, fully-discrete method, we observe convergence behaviour consistent with these results.Comment: 3 Figure