42 research outputs found
Generalised fractional diffusion equations for subdiffusion on arbitrarily growing domains
Many physical phenomena occur on domains that grow in time. When the
timescales of the phenomena and domain growth are comparable, models must
include the dynamics of the domain. A widespread intrinsically slow transport
process is subdiffusion. Many models of subdiffusion include a history
dependence. This greatly confounds efforts to incorporate domain growth. Here
we derive the fractional partial differential equations that govern
subdiffusion on a growing domain, based on a Continuous Time Random Walk. This
requires the introduction of a new, comoving, fractional derivative.Comment: 12 pages, 1 figur
Precision isotope shift measurements in Ca using highly sensitive detection schemes
We demonstrate an efficient high-precision optical spectroscopy technique for
single trapped ions with non-closed transitions. In a double-shelving
technique, the absorption of a single photon is first amplified to several
phonons of a normal motional mode shared with a co-trapped cooling ion of a
different species, before being further amplified to thousands of fluorescence
photons emitted by the cooling ion using the standard electron shelving
technique. We employ this extension of the photon recoil spectroscopy technique
to perform the first high precision absolute frequency measurement of the
D P transition in Ca,
resulting in a transition frequency of kHz.
Furthermore, we determine the isotope shift of this transition and the
S P transition for Ca,
Ca and Ca ions relative to Ca with an
accuracy below 100 kHz. Improved field and mass shift constants of these
transitions as well as changes in mean square nuclear charge radii are
extracted from this high resolution data
Reaction-diffusion and reaction-subdiffusion equations on arbitrarily evolving domains
Reaction-diffusion equations are widely used as the governing evolution
equations for modeling many physical, chemical, and biological processes. Here
we derive reaction-diffusion equations to model transport with reactions on a
one-dimensional domain that is evolving. The model equations, which have been
derived from generalized continuous time random walks, can incorporate
complexities such as subdiffusive transport and inhomogeneous domain stretching
and shrinking. A method for constructing analytic expressions for short time
moments of the position of the particles is developed and moments calculated
from this approach are shown to compare favourably with results from random
walk simulations and numerical integration of the reaction transport equation.
The results show the important role played by the initial condition. In
particular, it strongly affects the time dependence of the moments in the short
time regime by introducing additional drift and diffusion terms. We also
discuss how our reaction transport equation could be applied to study the
spreading of a population on an evolving interface.Comment: 38 pages, 10 figure
Varying constants, Gravitation and Cosmology
Fundamental constants are a cornerstone of our physical laws. Any constant
varying in space and/or time would reflect the existence of an almost massless
field that couples to matter. This will induce a violation of the universality
of free fall. It is thus of utmost importance for our understanding of gravity
and of the domain of validity of general relativity to test for their
constancy. We thus detail the relations between the constants, the tests of the
local position invariance and of the universality of free fall. We then review
the main experimental and observational constraints that have been obtained
from atomic clocks, the Oklo phenomenon, Solar system observations, meteorites
dating, quasar absorption spectra, stellar physics, pulsar timing, the cosmic
microwave background and big bang nucleosynthesis. At each step we describe the
basics of each system, its dependence with respect to the constants, the known
systematic effects and the most recent constraints that have been obtained. We
then describe the main theoretical frameworks in which the low-energy constants
may actually be varying and we focus on the unification mechanisms and the
relations between the variation of different constants. To finish, we discuss
the more speculative possibility of understanding their numerical values and
the apparent fine-tuning that they confront us with.Comment: 145 pages, 10 figures, Review for Living Reviews in Relativit
Continuous-time random walks on networks with vertex- and time-dependent forcing
We have investigated the transport of particles moving as random walks on the vertices of a network, subject to vertex- and time-dependent forcing. We have derived the generalized master equations for this transport using continuous time random walks, characterized by jump and waiting time densities, as the underlying stochastic process. The forcing is incorporated through a vertex- and time-dependent bias in the jump densities governing the random walking particles. As a particular case, we consider particle forcing proportional to the concentration of particles on adjacent vertices, analogous to self-chemotactic attraction in a spatial continuum. Our algebraic and numerical studies of this system reveal an interesting pair-aggregation pattern formation in which the steady state is composed of a high concentration of particles on a small number of isolated pairs of adjacent vertices. The steady states do not exhibit this pair aggregation if the transport is random on the vertices, i.e., without forcing. The manifestation of pair aggregation on a transport network may thus be a signature of self-chemotactic-like forcing
A Mathematical Model for the Proliferation, Accumulation and Spread of Pathogenic Proteins Along Neuronal Pathways with Locally Anomalous Trapping
There is growing evidence that many neurodegenerative disease processes involve the
proliferation, accumulation and spread of pathogenic proteins. The transport of proteins
in the brain is typically hindered on small scales by micro-domain traps and binding sites
but it may be enhanced on larger scales by neuronal pathways identified as white matter
transport networks. We have introduced a mathematical network model to simulate a
pathogenic protein neurodegenerative disease in the brain taking into account the
anomalous transport. The proliferation and accumulation of pathogenic proteins is modelled
using a set of reaction kinetics equations on the nodes of a network. Transport of the
proteins on the network is modelled as a continuous time random walk with power law
distributed waiting times on the nodes. This power law waiting time distribution is shown
to be consistent with anomalously slowed diffusion on local scales but transport is
enhanced on larger scales by the jumps between nodes. The model reveals that the disease
spreads as a propagating front throughout the brain. The anomalous behaviour leads to a
lessor variation in the concentration of pathogenic proteins. The enhanced transport on
the network ensures that the approach to equilibrium is dominated by the short time
behaviour of the waiting time density, hence the effects of subdiffusion are not as
pronounced as in a spatial continuum