44,167 research outputs found
Incorporating spatial correlations into multispecies mean-field models
In biology, we frequently observe different species existing within the same environment. For example, there are many cell types in a tumour, or different animal species may occupy a given habitat. In modeling interactions between such species, we often make use of the mean-field approximation, whereby spatial correlations between the locations of individuals are neglected. Whilst this approximation holds in certain situations, this is not always the case, and care must be taken to ensure the mean-field approximation is only used in appropriate settings. In circumstances where the mean-field approximation is unsuitable, we need to include information on the spatial distributions of individuals, which is not a simple task. In this paper, we provide a method that overcomes many of the failures of the mean-field approximation for an on-lattice volume-excluding birth-death-movement process with multiple species. We explicitly take into account spatial information on the distribution of individuals by including partial differential equation descriptions of lattice site occupancy correlations. We demonstrate how to derive these equations for the multispecies case and show results specific to a two-species problem. We compare averaged discrete results to both the mean-field approximation and our improved method, which incorporates spatial correlations. We note that the mean-field approximation fails dramatically in some cases, predicting very different behavior from that seen upon averaging multiple realizations of the discrete system. In contrast, our improved method provides excellent agreement with the averaged discrete behavior in all cases, thus providing a more reliable modeling framework. Furthermore, our method is tractable as the resulting partial differential equations can be solved efficiently using standard numerical techniques
Modeling of Nucleation Processes
Nucleation is the onset of a first-order phase transition by which a
metastable phase transforms into a more stable one. Such a phase transition
occurs when an initial system initially in equilibrium is destabilized by the
change of an external parameter like the temperature or the pressure. If the
perturbation is small enough, the system does not become unstable but rather
stays metastable. In diffusive transformations, the system then evolves through
the nucleation, the growth and the coarsening of a second phase. Such a phase
transformation is found in a lot of situations in materials science like
condensation of liquid droplets from a supersaturated vapor, solidification,
precipitation from a supersaturated solid solution, ... The initial stage of
all these different processes can be well described within the same framework.
Since its initial formulation in 1927 by Volmer, Weber and Farkas and its
modification in 1935 by Becker and D\"oring the classical nucleation theory has
been a suitable tool to model the nucleation stage in phase transformations. In
this article, we first describe this theory. A kinetic approach, the cluster
dynamics, can also be used to describe nucleation. This constitutes the second
part of this article. The links as well as the difference between both
descriptions are emphasized. Since its initial formulation, the classical
nucleation theory has been enriched, so as to take into account the fact that
clusters other than monomers can migrate and react. It has been also extended
to multi-component systems. These generalizations of the initial formalism are
also presented
Time-dependent Circulation Flows: Iron Enrichment in Cooling Flows with Heated Return Flows
We describe a new type of dynamical model for hot gas in galaxy groups and
clusters in which gas moves simultaneously in both radial directions.
Circulation flows are consistent with (1) the failure to observe cooling gas in
X-ray spectra, (2) multiphase gas observed near the centers of these flows and
(3) the accumulation of iron in the hot gas from Type Ia supernovae in the
central galaxy. Dense inflowing gas cools, producing a positive central
temperature gradient, as in normal cooling flows. Bubbles of hot, buoyant gas
flow outward. Circulation flows eventually cool catastrophically if the outward
flowing gas transports mass but no heat; to maintain the circulation both mass
and energy must be supplied to the inflowing gas over a large volume, extending
to the cooling radius. The rapid radial recirculation of gas produces a flat
central core in the gas iron abundance, similar to many observations. We
believe the circulation flows described here are the first gasdynamic,
long-term evolutionary models that are in good agreement with all essential
features observed in the hot gas: little or no gas cools as required by XMM
spectra, the gas temperature increases outward near the center, and the gaseous
iron abundance is about solar near the center and decreases outward.Comment: 17 pages (emulateapj5) with 6 figures; accepted by The Astrophysical
Journa
Self-Consistent Projection Operator Theory for Quantum Many-Body Systems
We derive an exact equation of motion for the reduced density matrices of
individual subsystems of quantum many-body systems of any lattice dimension and
arbitrary system size. Our projection operator based theory yields a highly
efficient analytical and numerical approach. Besides its practical use it
provides a novel interpretation and systematic extension of mean-field
approaches and an adaption of open quantum systems theory to settings where a
dynamically evolving environment has to be taken into account. We show its high
accuracy for two significant classes of complex quantum many-body dynamics,
unitary evolutions of non-equilibrium states in closed and stationary states in
driven-dissipative systems.Comment: 13 pages, 4 figure
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