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 $D$ 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 $D$, the zero steady state is stable. Patterns occur for small values of $D$ because the zero steady state is Turing unstable in this range; for $D=0$ 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 $L^1$ 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