18 research outputs found
Semi-analytical solutions of boundary value problems for the stationary diffusion equation in three-dimensional canonical domains
We present the generalized method of separation of variables (GMSV) to solve boundary value problems for the stationary diffusion equation in three-dimensional canonical domains (e.g., parallelepipeds, cylinders, spheres, spheroids, ..., and also their combinations)
Universal spectral features of different classes of random diffusivity processes
Stochastic models based on random diffusivities, such as the diffusing- diffusivity approach, are popular concepts for the description of non-Gaussian diffusion in heterogeneous media. Studies of these models typically focus on the moments and the displacement probability density function. Here we develop the complementary power spectral description for a broad class of random diffusivity processes. In our approach we cater for typical single particle tracking data in which a small number of trajectories with finite duration are garnered. Apart from the diffusing-diffusivity model we study a range of previously unconsidered random diffusivity processes, for which we obtain exact forms of the probability density function. These new processes are different versions of jump processes as well as functionals of Brownian motion. The resulting behaviour subtly depends on the specific model details. Thus, the central part of the probability density function may be Gaussian or non-Gaussian, and the tails may assume Gaussian, exponential, log-normal or even power-law forms. For all these models we derive analytically the moment-generating function for the single-trajectory power spectral density. We establish the generic -scaling of the power spectral density as function of frequency in all cases. Moreover, we establish the probability density for the amplitudes of the random power spectral density of individual trajectories. The latter functions reflect the very specific properties of the different random diffusivity models considered here. Our exact results are in excellent agreement with extensive numerical simulations.Severo Ochoa.SEV-2017-0718
BERC.2018-202
Kinetics of active surface-mediated diffusion in spherically symmetric domains
We present an exact calculation of the mean first-passage time to a target on
the surface of a 2D or 3D spherical domain, for a molecule alternating phases
of surface diffusion on the domain boundary and phases of bulk diffusion. We
generalize the results of [J. Stat. Phys. {\bf 142}, 657 (2011)] and consider a
biased diffusion in a general annulus with an arbitrary number of regularly
spaced targets on a partially reflecting surface. The presented approach is
based on an integral equation which can be solved analytically. Numerically
validated approximation schemes, which provide more tractable expressions of
the mean first-passage time are also proposed. In the framework of this minimal
model of surface-mediated reactions, we show analytically that the mean
reaction time can be minimized as a function of the desorption rate from the
surface.Comment: Published online in J. Stat. Phy
Mean first-passage time of surface-mediated diffusion in spherical domains
We present an exact calculation of the mean first-passage time to a target on
the surface of a 2D or 3D spherical domain, for a molecule alternating phases
of surface diffusion on the domain boundary and phases of bulk diffusion. The
presented approach is based on an integral equation which can be solved
analytically. Numerically validated approximation schemes, which provide more
tractable expressions of the mean first-passage time are also proposed. In the
framework of this minimal model of surface-mediated reactions, we show
analytically that the mean reaction time can be minimized as a function of the
desorption rate from the surface.Comment: to appear in J. Stat. Phy
On distributions of functionals of anomalous diffusion paths
Functionals of Brownian motion have diverse applications in physics,
mathematics, and other fields. The probability density function (PDF) of
Brownian functionals satisfies the Feynman-Kac formula, which is a Schrodinger
equation in imaginary time. In recent years there is a growing interest in
particular functionals of non-Brownian motion, or anomalous diffusion, but no
equation existed for their PDF. Here, we derive a fractional generalization of
the Feynman-Kac equation for functionals of anomalous paths based on
sub-diffusive continuous-time random walk. We also derive a backward equation
and a generalization to Levy flights. Solutions are presented for a wide number
of applications including the occupation time in half space and in an interval,
the first passage time, the maximal displacement, and the hitting probability.
We briefly discuss other fractional Schrodinger equations that recently
appeared in the literature.Comment: 25 pages, 4 figure
Exponential decay of Laplacian eigenfunctions in domains with branches
The behavior of Laplacian eigenfunctions in domains with branches is
investigated. If an eigenvalue is below a threshold which is determined by the
shape of the branch, the associated eigenfunction is proved to exponentially
decay inside the branch. The decay rate is twice the square root of the
difference between the threshold and the eigenvalue. The derived exponential
estimate is applicable for arbitrary domains in any spatial dimension.
Numerical simulations illustrate and further extend the theoretical estimate
Semi-analytical solutions of boundary value problems for the stationary diffusion equation in three-dimensional canonical domains
We present the generalized method of separation of variables (GMSV) to solve boundary value problems for the stationary diffusion equation in three-dimensional canonical domains (e.g., parallelepipeds, cylinders, spheres, spheroids, ..., and also their combinations)
Semi-analytical solutions of boundary value problems for the stationary diffusion equation in three-dimensional canonical domains
We present the generalized method of separation of variables (GMSV) to solve boundary value problems for the stationary diffusion equation in three-dimensional canonical domains (e.g., parallelepipeds, cylinders, spheres, spheroids, ..., and also their combinations)
Optimal least-squares estimators of the diffusion constant from a single Brownian trajectory
Modern developments in microscopy and image processing are revolutionising areas of physics, chemistry, and biology as nanoscale objects can be tracked with unprecedented accuracy. However, the price paid for having a direct visualisation of a single particle trajectory with high temporal and spatial resolution is a consequent lack of statistics. This naturally calls for reliable analytical tools which will allow one to extract the properties specific to a statistical ensemble from just a single trajectory. In this article we briefly survey different analytical methods currently used to determine the ensemble average diffusion coefficient from single particle data and then focus specifically on weighted least-squares estimators, seeking the weight functions for which such estimators are ergodic. Finally, we address the question of the effects of disorder on such estimators
On the Problem of Diffusivity in Heterogeneous Biological Materials with Random Structure
Biological tissues are multicompartmental heterogeneous media composed of cellular and subcellular domains. Randomly walking water molecules may have different diffusion coefficients and densities (concentrations) in different domains, namely within cells and within the outer medium. Results of the proposed effective media scale-averaging iterative scheme are used to explore the effects of a large range of microstructural and compositional parameters on the apparent (effective) diffusion coefficient. A self-consistent modelling framework for predicting the steady-state effective diffusion coefficient is presented; the framework reveals the strong dependence of the apparent diffusion coefficient on the ratio of the microscopic diffusion coefficients of the comprising phases, permeability of the cells, and their volume fractions