On the joint residence time of N independent two-dimensional Brownian motions

We study the behavior of several joint residence times of N independent Brownian particles in a disc of radius $R$ in two dimensions. We consider: (i) the time T_N(t) spent by all N particles simultaneously in the disc within the time interval [0,t]; (ii) the time T_N^{(m)}(t) which at least m out of N particles spend together in the disc within the time interval [0,t]; and (iii) the time {\tilde T}_N^{(m)}(t) which exactly m out of N particles spend together in the disc within the time interval [0,t]. We obtain very simple exact expressions for the expectations of these three residence times in the limit t\to\infty.Comment: 8 page

Order statistics of the trapping problem

When a large number N of independent diffusing particles are placed upon a site of a d-dimensional Euclidean lattice randomly occupied by a concentration c of traps, what is the m-th moment of the time t_{j,N} elapsed until the first j are trapped? An exact answer is given in terms of the probability Phi_M(t) that no particle of an initial set of M=N, N-1,..., N-j particles is trapped by time t. The Rosenstock approximation is used to evaluate Phi_M(t), and it is found that for a large range of trap concentracions the m-th moment of t_{j,N} goes as x^{-m} and its variance as x^{-2}, x being ln^{2/d} (1-c) ln N. A rigorous asymptotic expression (dominant and two corrective terms) is given for for the one-dimensional lattice.Comment: 11 pages, 7 figures, to be published in Phys. Rev.

LiCaFeF6 A zero strain cathode material for use in Li ion batteries

A new zero strain LiCaFeF6 cathode material for reversible insertion and extraction of lithium ions is presented. LiCaFeF6 is synthesized by a solid state reaction and processed to a conductive electrode composite via high energy ball milling. In the first cycle, a discharge capacity of 112 mAh g amp; 8315; is achieved in the voltage range from 2.0 V to 4.5 V. The electrochemically active redox couple is Fe3 amp; 8314; Fe2 amp; 8314; as confirmed by Mössbauer spectroscopy and X ray absorption spectroscopy. The compound has a trigonal colquiriite type crystal structure space group . By means of in situ and ex situ XRD as well as X ray absorption fine structure spectroscopy a reversible response to Li uptake release is found. For an uptake of 0.8 mol Li per formula unit only minimal changes occur in the lattice parameters causing a total change in unit cell volume of less than 0.5 . The spatial distribution of cations in the crystal structure as well as the linkage between their corresponding fluorine octahedra is responsible for this very small structural response. With its zero strain behaviour this material is expected to exhibit only negligible mechanical degradation. It may be used as a cathode material in future lithium ion batteries with strongly improved safety and cycle lif

Critical dimensions for random walks on random-walk chains

The probability distribution of random walks on linear structures generated by random walks in $d$-dimensional space, $P_d(r,t)$, is analytically studied for the case $\xi\equiv r/t^{1/4}\ll1$. It is shown to obey the scaling form $P_d(r,t)=\rho(r) t^{-1/2} \xi^{-2} f_d(\xi)$, where $\rho(r)\sim r^{2-d}$ is the density of the chain. Expanding $f_d(\xi)$ in powers of $\xi$, we find that there exists an infinite hierarchy of critical dimensions, $d_c=2,6,10,\ldots$, each one characterized by a logarithmic correction in $f_d(\xi)$. Namely, for $d=2$, $f_2(\xi)\simeq a_2\xi^2\ln\xi+b_2\xi^2$; for $3\le d\le 5$, $f_d(\xi)\simeq a_d\xi^2+b_d\xi^d$; for $d=6$, $f_6(\xi)\simeq a_6\xi^2+b_6\xi^6\ln\xi$; for $7\le d\le 9$, $f_d(\xi)\simeq a_d\xi^2+b_d\xi^6+c_d\xi^d$; for $d=10$, $f_{10}(\xi)\simeq a_{10}\xi^2+b_{10}\xi^6+c_{10}\xi^{10}\ln\xi$, {\it etc.\/} In particular, for $d=2$, this implies that the temporal dependence of the probability density of being close to the origin $Q_2(r,t)\equiv P_2(r,t)/\rho(r)\simeq t^{-1/2}\ln t$.Comment: LATeX, 10 pages, no figures submitted for publication in PR

Control of eye movements while recording from single units in the pigmented rat

Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/23073/1/0000647.pd

Order statistics for d-dimensional diffusion processes

We present results for the ordered sequence of first passage times of arrival of N random walkers at a boundary in Euclidean spaces of d dimensions

Retarding Sub- and Accelerating Super-Diffusion Governed by Distributed Order Fractional Diffusion Equations

We propose diffusion-like equations with time and space fractional derivatives of the distributed order for the kinetic description of anomalous diffusion and relaxation phenomena, whose diffusion exponent varies with time and which, correspondingly, can not be viewed as self-affine random processes possessing a unique Hurst exponent. We prove the positivity of the solutions of the proposed equations and establish the relation to the Continuous Time Random Walk theory. We show that the distributed order time fractional diffusion equation describes the sub-diffusion random process which is subordinated to the Wiener process and whose diffusion exponent diminishes in time (retarding sub-diffusion) leading to superslow diffusion, for which the square displacement grows logarithmically in time. We also demonstrate that the distributed order space fractional diffusion equation describes super-diffusion phenomena when the diffusion exponent grows in time (accelerating super-diffusion).Comment: 11 pages, LaTe

Early microstructural white matter changes in patients with HIV: A diffusion tensor imaging study

Background: Previous studies have reported white matter (WM) brain alterations in asymptomatic patients with human immunodeficiency virus (HIV). Methods: We compared diffusion tensor imaging (DTI) derived WM fractional anisotropy (FA) between HIV-patients with and without mild macroscopic brain lesions determined using standard magnetic resonance imaging (MRI). We furthermore investigated whether WM alterations co-occurred with neurocognitive deficits and depression. We performed structural MRI and DTI for 19 patients and 19 age-matched healthy controls. Regionally-specific WM integrity was investigated using voxel-based statistics of whole-brain FA maps and region-of-interest analysis. Each patient underwent laboratory and neuropsychological tests. Results: Structural MRI revealed no lesions in twelve (HIV-MRN) and unspecific mild macrostructural lesions in seven patients (HIV-MRL). Both analyses revealed widespread FA-alterations in all patients. Patients with HIV-MRL had FA-alterations primarily adjacent to the observed lesions and, whilst reduced in extent, patients with HIV-MRN also exhibited FA-alterations in similar regions. Patients with evidence of depression showed FA-increase in the ventral tegmental area, pallidum and nucleus accumbens in both hemispheres, and patients with evidence of HIV-associated neurocognitive disorder showed widespread FA-reduction. Conclusion: These results show that patients with HIV-MRN have evidence of FA-alterations in similar regions that are lesioned in HIV-MRL patients, suggesting common neuropathological processes. Furthermore, they suggest a biological rather than a reactive origin of depression in HIV-patients