708 research outputs found
Well posedness and Maximum Entropy Approximation for the Dynamics of Quantitative Traits
We study the Fokker-Planck equation derived in the large system limit of the
Markovian process describing the dynamics of quantitative traits. The
Fokker-Planck equation is posed on a bounded domain and its transport and
diffusion coefficients vanish on the domain's boundary. We first argue that,
despite this degeneracy, the standard no-flux boundary condition is valid. We
derive the weak formulation of the problem and prove the existence and
uniqueness of its solutions by constructing the corresponding contraction
semigroup on a suitable function space. Then, we prove that for the parameter
regime with high enough mutation rate the problem exhibits a positive spectral
gap, which implies exponential convergence to equilibrium.
Next, we provide a simple derivation of the so-called Dynamic Maximum Entropy
(DynMaxEnt) method for approximation of moments of the Fokker-Planck solution,
which can be interpreted as a nonlinear Galerkin approximation. The limited
applicability of the DynMaxEnt method inspires us to introduce its modified
version that is valid for the whole range of admissible parameters. Finally, we
present several numerical experiments to demonstrate the performance of both
the original and modified DynMaxEnt methods. We observe that in the parameter
regimes where both methods are valid, the modified one exhibits slightly better
approximation properties compared to the original one.Comment: 28 pages, 4 tables, 5 figure
Mathematical Analysis of a System for Biological Network Formation
Motivated by recent physics papers describing rules for natural network
formation, we study an elliptic-parabolic system of partial differential
equations proposed by Hu and Cai. The model describes the pressure field thanks
to Darcy's type equation and the dynamics of the conductance network under
pressure force effects with a diffusion rate representing randomness in the
material structure. We prove the existence of global weak solutions and of
local mild solutions and study their long term behaviour. It turns out that, by
energy dissipation, steady states play a central role to understand the pattern
capacity of the system. We show that for a large diffusion coefficient , the
zero steady state is stable. Patterns occur for small values of because the
zero steady state is Turing unstable in this range; for we can exhibit a
large class of dynamically stable (in the linearized sense) steady states
Decay to equilibrium for energy-reaction-diffusion systems
We derive thermodynamically consistent models of reaction-diffusion equations
coupled to a heat equation. While the total energy is conserved, the total
entropy serves as a driving functional such that the full coupled system is a
gradient flow. The novelty of the approach is the Onsager structure, which is
the dual form of a gradient system, and the formulation in terms of the
densities and the internal energy. In these variables it is possible to assume
that the entropy density is strictly concave such that there is a unique
maximizer (thermodynamical equilibrium) given linear constraints on the total
energy and suitable density constraints.
We consider two particular systems of this type, namely, a diffusion-reaction
bipolar energy transport system, and a drift-diffusion-reaction energy
transport system with confining potential. We prove corresponding
entropy-entropy production inequalities with explicitely calculable constants
and establish the convergence to thermodynamical equilibrium, at first in
entropy and further in using Cziszar-Kullback-Pinsker type inequalities.Comment: 40 page
Sensitivity Analysis of Tech 1 - A Systems Dynamics Model for Technological Shift
This paper deals with the sensitivity analysis of TECH1 -- a system dynamics model, which describes the technological shift from an old technology to a new one, within a specific scenario. However, its goal is not to describe the model, which was done by Robinson (1979), in this case the paper's goal is threefold:
1. To show with mathematical tools which factors are important for an invention to become an innovation, by interpreting in an economic sense the results of the performed analysis.
2. To make it possible for a broader range of people to understand system dynamics models -- especially TECH1 and consequently to improve them.
3. To show what kind of mathematical analysis is useful for a class of economic models represented by differential equations.
Although TECH1 has not yet been applied to the real world, the author hopes that this paper will help to produce a better understanding of the innovation process in the real world, as well as of system dynamics models and their limits
Low Momentum Classical Mechanics with Effective Quantum Potentials
A recently introduced effective quantum potential theory is studied in a low
momentum region of phase space. This low momentum approximation is used to show
that the new effective quantum potential induces a space-dependent mass and a
smoothed potential both of them constructed from the classical potential. The
exact solution of the approximated theory in one spatial dimension is found.
The concept of effective transmission and reflection coefficients for effective
quantum potentials is proposed and discussed in comparison with an analogous
quantum statistical mixture problem. The results are applied to the case of a
square barrier.Comment: 4 figure
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