12,092 research outputs found
Well-balanced and asymptotic preserving schemes for kinetic models
In this paper, we propose a general framework for designing numerical schemes
that have both well-balanced (WB) and asymptotic preserving (AP) properties,
for various kinds of kinetic models. We are interested in two different
parameter regimes, 1) When the ratio between the mean free path and the
characteristic macroscopic length tends to zero, the density can be
described by (advection) diffusion type (linear or nonlinear) macroscopic
models; 2) When = O(1), the models behave like hyperbolic equations
with source terms and we are interested in their steady states. We apply the
framework to three different kinetic models: neutron transport equation and its
diffusion limit, the transport equation for chemotaxis and its Keller-Segel
limit, and grey radiative transfer equation and its nonlinear diffusion limit.
Numerical examples are given to demonstrate the properties of the schemes
Full-counting statistics of charge and spin transport in the transient regime: A nonequilibrium Green's function approach
We report the investigation of full-counting statistics (FCS) of transferred
charge and spin in the transient regime where the connection between central
scattering region (quantum dot) and leads are turned on at . A general
theoretical formulation for the generating function (GF) is presented using a
nonequilibrium Green's function approach for the quantum dot system. In
particular, we give a detailed derivation on how to use the method of path
integral together with nonequilibrium Green's function technique to obtain the
GF of FCS in electron transport systems based on the two-time quantum
measurement scheme. The correct long-time limit of the formalism, the
Levitov-Lesovik's formula, is obtained. This formalism can be generalized to
account for spin transport for the system with noncollinear spin as well as
spin-orbit interaction. As an example, we have calculated the GF of
spin-polarized transferred charge, transferred spin, as well as the spin
transferred torque for a magnetic tunneling junction in the transient regime.
The GF is compactly expressed by a functional determinant represented by
Green's function and self-energy in the time domain. With this formalism, FCS
in spintronics in the transient regime can be studied. We also extend this
formalism to the quantum point contact system. For numerical results, we
calculate the GF and various cumulants of a double quantum dot system connected
by two leads in transient regime. The signature of universal oscillation of FCS
is identified. On top of the global oscillation, local oscillations are found
in various cumulants as a result of the Rabi oscillation. Finally, the
influence of the temperature is also examined
Derivation of the bacterial run-and-tumble kinetic equation from a model with biochemical pathway
Kinetic-transport equations are, by now, standard models to describe the
dynamics of populations of bacteria moving by run-and-tumble. Experimental
observations show that bacteria increase their run duration when encountering
an increasing gradient of chemotactic molecules. This led to a first class of
models which heuristically include tumbling frequencies depending on the
path-wise gradient of chemotactic signal.
More recently, the biochemical pathways regulating the flagellar motors were
uncovered. This knowledge gave rise to a second class of kinetic-transport
equations, that takes into account an intra-cellular molecular content and
which relates the tumbling frequency to this information. It turns out that the
tumbling frequency depends on the chemotactic signal, and not on its gradient.
For these two classes of models, macroscopic equations of Keller-Segel type,
have been derived using diffusion or hyperbolic rescaling. We complete this
program by showing how the first class of equations can be derived from the
second class with molecular content after appropriate rescaling. The main
difficulty is to explain why the path-wise gradient of chemotactic signal can
arise in this asymptotic process.
Randomness of receptor methylation events can be included, and our approach
can be used to compute the tumbling frequency in presence of such a noise
Traveling wave solution of the Hele-Shaw model of tumor growth with nutrient
Several mathematical models of tumor growth are now commonly used to explain
medical observations and predict cancer evolution based on images. These models
incorporate mechanical laws for tissue compression combined with rules for
nutrients availability which can differ depending on the situation under
consideration, in vivo or in vitro. Numerical solutions exhibit, as expected
from medical observations, a proliferative rim and a necrotic core. However,
their precise profiles are rather complex, both in one and two dimensions.Comment: 25 page
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