90 research outputs found
On a Boltzmann mean field model for knowledge growth
In this paper we analyze a Boltzmann type mean field game model for knowledge
growth, which was proposed by Lucas and Moll. We discuss the underlying
mathematical model, which consists of a coupled system of a Boltzmann type
equation for the agent density and a Hamilton-Jacobi-Bellman equation for the
optimal strategy. We study the analytic features of each equation separately
and show local in time existence and uniqueness for the fully coupled system.
Furthermore we focus on the construction and existence of special solutions,
which relate to exponential growth in time - so called balanced growth path
solutions. Finally we illustrate the behavior of solutions for the full system
and the balanced growth path equations with numerical simulations.Comment: 6 figure
Balanced growth path solutions of a Boltzmann mean field game model for knowledge growth
In this paper we study balanced growth path solutions of a Boltzmann mean
field game model proposed by Lucas et al [13] to model knowledge growth in an
economy. Agents can either increase their knowledge level by exchanging ideas
in learning events or by producing goods with the knowledge they already have.
The existence of balanced growth path solutions implies exponential growth of
the overall production in time. We proof existence of balanced growth path
solutions if the initial distribution of individuals with respect to their
knowledge level satisfies a Pareto-tail condition. Furthermore we give first
insights into the existence of such solutions if in addition to production and
knowledge exchange the knowledge level evolves by geometric Brownian motion
Cross-diffusion systems with excluded volume effects and asymptotic gradient flows
In this paper we discuss the analysis of a cross-diffusion PDE system for a
mixture of hard spheres, which was derived by Bruna and Chapman from a
stochastic system of interacting Brownian particles using the method of matched
asymptotic expansions. The resulting cross-diffusion system is valid in the
limit of small volume fraction of particles. While the system has a gradient
flow structure in the symmetric case of all particles having the same size and
diffusivity, this is not valid in general. We discuss local stability and
global existence for the symmetric case using the gradient flow structure and
entropy variable techniques. For the general case, we introduce the concept of
an asymptotic gradient flow structure and show how it can be used to study the
behavior close to equilibrium. Finally we illustrate the behavior of the model
with various numerical simulations
Lane formation by side-stepping
In this paper we study a system of nonlinear partial differential equations,
which describes the evolution of two pedestrian groups moving in opposite
direction. The pedestrian dynamics are driven by aversion and cohesion, i.e.
the tendency to follow individuals from the own group and step aside in the
case of contraflow. We start with a 2D lattice based approach, in which the
transition rates reflect the described dynamics, and derive the corresponding
PDE system by formally passing to the limit in the spatial and temporal
discretization. We discuss the existence of special stationary solutions, which
correspond to the formation of directional lanes and prove existence of global
in time bounded weak solutions. The proof is based on an approximation argument
and entropy inequalities. Furthermore we illustrate the behavior of the system
with numerical simulations
On a Boltzmann mean field model for knowledge growth
In this paper we analyze a Boltzmann type mean field game model for knowledge growth, which was proposed by Lucas and Moll [15]. We discuss the underlying mathematical model, which consists of a coupled system of a Boltzmann type equation for the agent density and a Hamilton-Jacobi-Bellman equation for the optimal strategy. We study the analytic features of each equation separately and show local in time existence and uniqueness for the fully coupled system. Furthermore we focus on the construction and existence of special solutions, which relate to exponential growth in time - so called balanced growth path solutions. Finally we illustrate the behavior of solutions for the full system and the balanced growth path equations with numerical simulations
Boltzmann mean-field game model for knowledge growth: limits to learning and general utilities
In this paper we investigate a generalisation of a Boltzmann mean field game
(BMFG) for knowledge growth, originally introduced by the economists Lucas and
Moll. In BMFG the evolution of the agent density with respect to their
knowledge level is described by a Boltzmann equation. Agents increase their
knowledge through binary interactions with others; their increase is modulated
by the interaction and learning rate: Agents with similar knowledge learn more
in encounters, while agents with very different levels benefit less from
learning interactions. The optimal fraction of time spent on learning is
calculated by a Bellman equation, resulting in a highly nonlinear
forward-backward in time PDE system.
The structure of solutions to the Boltzmann and Bellman equation depends
strongly on the learning rate in the Boltzmann collision kernel as well as the
utility function in the Bellman equation. In this paper we investigate the
monotonicity behavior of solutions for different learning and utility
functions, show existence of solutions and investigate how they impact the
existence of so-called balanced growth path solutions, that relate to
exponential growth of the overall economy. Furthermore we corroborate and
illustrate our analytical results with computational experiments
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