12,338 research outputs found
Price dynamics in financial markets: a kinetic approach
The use of kinetic modelling based on partial differential equations for the
dynamics of stock price formation in financial markets is briefly reviewed. The
importance of behavioral aspects in market booms and crashes and the role of
agents' heterogeneity in emerging power laws for price distributions is
emphasized and discussed
Tricritical points in a Vicsek model of self-propelled particles with bounded confidence
We study the orientational ordering in systems of self-propelled particles
with selective interactions. To introduce the selectivity we augment the
standard Vicsek model with a bounded-confidence collision rule: a given
particle only aligns to neighbors who have directions quite similar to its own.
Neighbors whose directions deviate more than a fixed restriction angle
are ignored. The collective dynamics of this systems is studied by agent-based
simulations and kinetic mean field theory. We demonstrate that the reduction of
the restriction angle leads to a critical noise amplitude decreasing
monotonically with that angle, turning into a power law with exponent 3/2 for
small angles. Moreover, for small system sizes we show that upon decreasing the
restriction angle, the kind of the transition to polar collective motion
changes from continuous to discontinuous. Thus, an apparent tricritical point
is identified and calculated analytically. We also find that at very small
interaction angles the polar ordered phase becomes unstable with respect to the
apolar phase. We show that the mean-field kinetic theory permits stationary
nematic states below a restriction angle of . We calculate the
critical noise, at which the disordered state bifurcates to a nematic state,
and find that it is always smaller than the threshold noise for the transition
from disorder to polar order. The disordered-nematic transition features two
tricritical points: At low and high restriction angle the transition is
discontinuous but continuous at intermediate . We generalize our
results to systems that show fragmentation into more than two groups and obtain
scaling laws for the transition lines and the corresponding tricritical points.
A novel numerical method to evaluate the nonlinear Fredholm integral equation
for the stationary distribution function is also presented.Comment: 20 pages, 18 figure
Consensus Convergence with Stochastic Effects
We consider a stochastic, continuous state and time opinion model where each
agent's opinion locally interacts with other agents' opinions in the system,
and there is also exogenous randomness. The interaction tends to create
clusters of common opinion. By using linear stability analysis of the
associated nonlinear Fokker-Planck equation that governs the empirical density
of opinions in the limit of infinitely many agents, we can estimate the number
of clusters, the time to cluster formation and the critical strength of
randomness so as to have cluster formation. We also discuss the cluster
dynamics after their formation, the width and the effective diffusivity of the
clusters. Finally, the long term behavior of clusters is explored numerically.
Extensive numerical simulations confirm our analytical findings.Comment: Dedication to Willi J\"{a}ger's 75th Birthda
Opinion dynamics: models, extensions and external effects
Recently, social phenomena have received a lot of attention not only from
social scientists, but also from physicists, mathematicians and computer
scientists, in the emerging interdisciplinary field of complex system science.
Opinion dynamics is one of the processes studied, since opinions are the
drivers of human behaviour, and play a crucial role in many global challenges
that our complex world and societies are facing: global financial crises,
global pandemics, growth of cities, urbanisation and migration patterns, and
last but not least important, climate change and environmental sustainability
and protection. Opinion formation is a complex process affected by the
interplay of different elements, including the individual predisposition, the
influence of positive and negative peer interaction (social networks playing a
crucial role in this respect), the information each individual is exposed to,
and many others. Several models inspired from those in use in physics have been
developed to encompass many of these elements, and to allow for the
identification of the mechanisms involved in the opinion formation process and
the understanding of their role, with the practical aim of simulating opinion
formation and spreading under various conditions. These modelling schemes range
from binary simple models such as the voter model, to multi-dimensional
continuous approaches. Here, we provide a review of recent methods, focusing on
models employing both peer interaction and external information, and
emphasising the role that less studied mechanisms, such as disagreement, has in
driving the opinion dynamics. [...]Comment: 42 pages, 6 figure
Phase transitions and non-equilibrium relaxation in kinetic models of opinion formation
We review in details some recently proposed kinetic models of opinion
dynamics. We discuss the several variants including a generalised model. We
provide mean field estimates for the critical points, which are numerically
supported with reasonable accuracy. Using non-equilibrium relaxation
techniques, we also investigate the nature of phase transitions observed in
these models. We study the nature of correlations as the critical points are
approached, and comment on the universality of the phase transitions observed.Comment: 11 pages, 8 eps figures, 1 table. Contribution for proceedings of
Statphys-Kolkata-VII 26-30 November, 201
The size distribution of cities: a kinetic explanation
We present a kinetic approach to the formation of urban agglomerations which
is based on simple rules of immigration and emigration. In most cases, the
Boltzmann-type kinetic description allows to obtain, within an asymptotic
procedure, a Fokker--Planck equation with variable coefficients of diffusion
and drift, which describes the evolution in time of some probability density of
the city size. It is shown that, in dependence of the microscopic rules of
migration, the equilibrium density can follow both a power law for large values
of the size variable, which contains as particular case a Zipf's law behavior,
and a lognormal law for middle and low values of the size variable. In
particular, connections between the value of Pareto index of the power law at
equilibrium and the disposal of the population to emigration are outlined. The
theoretical findings are tested with recent data of the populations of Italy
and Switzerland
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
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