Noise-induced transitions for random versions of Verhulst model

According to the Verhulst model the rate of increase/decrease of a biological population with size x(t) at time t is equal to the sum of $\rho x\left(t\right)$ and $-x{(t)}^2$, where $\rho \in \mathbb{R}$ is a constant. The constant ρ is positive and negative for favorable and hostile environments, respectively. The limitation of resources is quantified by the term $-x{(t)}^2$. We examine random versions of the Verhulst model obtained by replacing ρ with $(\rho +\mathrm{white}\mathrm{noise})$. Gaussian (GWN) and Poisson (PWN) white noise processes are considered. The state X(t) of the random Verhulst model satisfies stochastic differential equations driven by Gaussian and Poisson white noises. Our objective is to identify noise-induced transitions, that is, noise levels at which the stationary density of X(t) exhibits qualitative changes. It is shown that noise levels causing transitions under Poisson white noise approach those under Gaussian white noise as the frequency of Poisson jumps increases indefinitely while their size approaches zero.     

Exact explicit time integration of hyperbolic partial differential equations with mesh free radial basis functions

This study is a progress report that examines the numerical solution of inviscid hyperbolic partial differential equations (PDEs) without the need for upwind differencing and other numerical artifacts. The fixed frame PDEs are locally transformed by rotating and translating the coordinate system at each local discretization point. These transformations yield a simpler PDE system that is effectively linearized. It is assumed that in this transformed local frame within a time interval, $\mathrm{\Delta }t$, the dependent variables are products of the spatial dependent radial basis functions (RBFs), and the time dependent expansion coefficients, $\mathbf{\chi }(t)$. This linearization is exploited by transforming the PDEs into systems of linear ordinary differential equations (ODEs) in terms of the expansion coefficients. The affine space decomposition is used to obtain an ODE system of $N_i$ ODEs in $N_i$ unknowns that can be integrated exactly in time. Then the entire set of N expansion coefficients is found. Numerical results show that hyperbolic PDEs can be integrated in time without upwinding and the root mean square errors between the exact and numerical solutions are indeed very small.     

Experimental and computational investigations of LaNi5-xAlx (x = 0, 0.25, 0.5, 0.75 and 1.0) tritium-storage alloys

Although already scientists in recent years have reported some experimental and theoretical results of LaNi-Al series of tritium-storage alloys, several key aspects remain the subject of considerable debate. In an effort to interpret some of these unknowns, we have performed experimental and theoretical investigations for LaNi_(5-x)Al_x(x = 0, 0.25, 0.5, 0.75 and 1.0) tritium-storage alloys. Firstly, the XRD characterization indicates that the unit cell volumes of LaNi_(5-x)Al_x increase with Al content in alloys. Secondly, the PCisotherm measurement of LaNi_(5-x)Al_xalloys shows that their hydrogen absorption/desorption plateau pressures reduce with the increase of Al content while their plateau widths narrow simultaneously. The deuterium absorption/desorption plateaus have a similar trend to hydrogen's except for their plateaus being higher than hydrogen's. To explain the above experimental findings, a series of calculations based on density functional theory(DFT) and frozen phonon approach have been performed. The results manifest that:(1) the partial substitutions of Al for Ni reduce the hydrogen formation energies of LaNi_(5-x)Al_xH and the number of available interstitial sites, and therefore lead to the absorption/desorption plateau pressures being reduced and the plateau widths being narrowed down at the same experimental temperatures;(2) the covalent interaction between H and Ni is an important factor for estimating the stability of LaNi_(5-x)Al_x-H system;(3) since the calculated enthalpy change H is generally more accurate than the calculated entropy change S with respect to the corresponding experimental value for each LaNi_(5-x)Al_xH(or D), the curves of H vs. hydrogen storage capacity instead of Van't Hoff relation, can be used to predict the experimental plateau pressures of LaNi_(5-x)Al_x-H(D or T) at a given temperature;(4) the hydrogen isotope effect of LaNi_(5-x)Al_x-H(D or T) system can be quantitatively described as a linearity relation between ⊿ZPE + ⊿H~(vib) and 1/√mQ(Q = H, D, T). From the good agreement between the predicted and experimental ln(P_H/P_0) and ln(P_D/P_0), it is deduced that predicting ln(P_T/P_0) of LaNi_(5-x)Al_x T is feasible. The procedure of pre-computing and comparing curves of H vs. hydrogen storage capacity proposed in this paper provided an attractive tool to increase the efficiency of experimental alloying design of hydrogen(deuterium or tritium) storage materials.