Gamma Limit for Transition Paths of Maximal Probability

Chemical reactions can be modelled via diffusion processes conditioned to make a transition between specified molecular configurations representing the state of the system before and after the chemical reaction. In particular the model of Brownian dynamics - gradient flow subject to additive noise - is frequently used. If the chemical reaction is specified to take place on a given time interval, then the most likely path taken by the system is a minimizer of the Onsager-Machlup functional. The Gamma limit of this functional is determined in the case where the temperature is small and the transition time scales as the inverse temperatur

Kullback--Leibler approximation for probability measures on infinite dimensional spaces

In a variety of applications it is important to extract information from a probability measure $\mu$ on an infinite dimensional space. Examples include the Bayesian approach to inverse problems and (possibly conditioned) continuous time Markov processes. It may then be of interest to find a measure $\nu$, from within a simple class of measures, which approximates $\mu$. This problem is studied in the case where the Kullback--Leibler divergence is employed to measure the quality of the approximation. A calculus of variations viewpoint is adopted, and the particular case where $\nu$ is chosen from the set of Gaussian measures is studied in detail. Basic existence and uniqueness theorems are established, together with properties of minimizing sequences. Furthermore, parameterization of the class of Gaussians through the mean and inverse covariance is introduced, the need for regularization is explained, and a regularized minimization is studied in detail. The calculus of variations framework resulting from this work provides the appropriate underpinning for computational algorithms

Kullback--Leibler approximation for probability measures on infinite dimensional spaces

In a variety of applications it is important to extract information from a probability measure $\mu$ on an infinite dimensional space. Examples include the Bayesian approach to inverse problems and (possibly conditioned) continuous time Markov processes. It may then be of interest to find a measure $\nu$, from within a simple class of measures, which approximates $\mu$. This problem is studied in the case where the Kullback--Leibler divergence is employed to measure the quality of the approximation. A calculus of variations viewpoint is adopted, and the particular case where $\nu$ is chosen from the set of Gaussian measures is studied in detail. Basic existence and uniqueness theorems are established, together with properties of minimizing sequences. Furthermore, parameterization of the class of Gaussians through the mean and inverse covariance is introduced, the need for regularization is explained, and a regularized minimization is studied in detail. The calculus of variations framework resulting from this work provides the appropriate underpinning for computational algorithms

Localization-delocalization transition for disordered cubic harmonic lattices

We study numerically the disorder-induced localization-delocalization phase transitions that occur for mass and spring constant disorder in a three-dimensional cubic lattice with harmonic couplings. We show that, while the phase diagrams exhibit regions of stable and unstable waves, the universality of the transitions is the same for mass and spring constant disorder throughout all the phase boundaries. The combined value for the critical exponent of the localization lengths of $\nu = 1.550^{+0.020}_{-0.017}$ confirms the agreement with the universality class of the standard electronic Anderson model of localization. We further support our investigation with studies of the density of states, the participation numbers and wave function statistics.Comment: 12 pages, 8 figures, 1 tabl

Screened Coulomb interactions in metallic alloys: I. Universal screening in the atomic sphere approximation

We have used the locally self-consistent Green's function (LSGF) method in supercell calculations to establish the distribution of the net charges assigned to the atomic spheres of the alloy components in metallic alloys with different compositions and degrees of order. This allows us to determine the Madelung potential energy of a random alloy in the single-site mean field approximation which makes the conventional single-site density-functional- theory coherent potential approximation (SS-DFT-CPA) method practically identical to the supercell LSGF method with a single-site local interaction zone that yields an exact solution of the DFT problem. We demonstrate that the basic mechanism which governs the charge distribution is the screening of the net charges of the alloy components that makes the direct Coulomb interactions short-ranged. In the atomic sphere approximation, this screening appears to be almost independent of the alloy composition, lattice spacing, and crystal structure. A formalism which allows a consistent treatment of the screened Coulomb interactions within the single-site mean-filed approximation is outlined. We also derive the contribution of the screened Coulomb interactions to the S2 formalism and the generalized perturbation method.Comment: 28 pages, 8 figure

Electronic structure and x-ray magnetic dichroism in random substitutional alloys of f-electron elements

The Koringa-Kohn-Rostoker —coherent-potential-approximation method combines multiple-scattering theory and the coherent-potential approximation to calculate the electronic structure of random substitutional alloys of transition metals. In this paper we describe the generalization of this theory to describe f-electron alloys. The theory is illustrated with a calculation of the electronic structure and magnetic dichroism curves for a random substitutional alloy containing rare-earth or actinide elements from first principles