Brownian Agents and Active Particles: Collective Dynamics in the Natural and Social Sciences

426 Pages·2003·14.44 MB·English

by  Frank Schweitzer

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P´eterE´rdi
CenterforComplexSystemsStudies,KalamazooCollege,USA,
andHungarianAcademyofSciences,Budapest,Hungary
KarlJ.Friston
InstituteofCognitiveNeuroscience,UniversityCollegeLonden,London,UK
HermannHaken
CenterofSynergetics,UniversityofStuttgart,Stuttgart,Germany
JanuszKacprzyk
SystemResearch,PolishAcademyofSciences,Warsaw,Poland
ScottKelso
CenterforComplexSystemsandBrainSciences,
FloridaAtlanticUniversity,BocaRaton,USA
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NonlinearDynamicsGroup,UniversityofPotsdam,
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CenterforComplexQuantumSystems,UniversityofTexas,Austin,USA
PeterSchuster
TheoreticalChemistryandStructuralBiology,UniversityofVienna,
Vienna,Austria
FrankSchweitzer
SystemsDesign,ETHZurich,Zurich,Switzerland
DidierSornette
EntrepreneurialRisk,ETHZurich,Zurich,Switzerland
Frank Schweitzer
Browning Agents
and Active Particles
Collective Dynamics
in the Natural and Social Sciences
WithaForewordbyJ.DoyneFarmer
With192Figuresand3Tables
123
FrankSchweitzer
ETHZürich
ProfessurfürSystemgestaltung
Kreuzplatz5
8032Zürich
Switzerland
E-mail:[email protected]
1sted.2003,2ndPrinting
LibraryofCongressControlNumber:2007932745
ISSN 0172-7389
ISBN 978-3-540-73844-2 SpringerBerlinHeidelbergNewYork
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Foreword
Whenwecontemplatephenomenaasdiverseaselectrochemicaldepositionor
the spatial patterns of urban development, it is natural to assume that they
havenothingincommon.Afterall,therearemanylevelsinthehierarchythat
builds up from atoms to human society, and the rules that govern atoms are
quite different from those that govern the geographical emergence of a city.
The common view among many, if not most, biologists and social scientists
is that the devil is entirely in the details. This school of thought asserts that
socialscienceandbiologyhavelittleornothingincommon,andindeedmany
biologists claim that even different fields of biology have little in common. If
theyareright,thensciencecanonlyproceedbyrecordingvastlistsofdetails
that no common principles will ever link together.
Physics, in contrast, has achieved a parsimonious description for a broad
range of phenomena based on only a few general principles. The phenomena
thatphysicsaddressesareunquestionablymuchsimplerthanthoseofbiology
or social science, and on the surface appear entirely dissimilar. A cell is far
more complicated than a pendulum or an atom, and human society, being
builtoutofagreatmanycells,isfarmorecomplicatedstill.Cellsandsocieties
have many layers of hierarchical organization, with complex functional and
computationalproperties;theyhaveidentities,idiosyncraciesstemmingfrom
an accumulation of historical contingency that makes them impossible to
characterize in simple mathematical terms. Their complexity is far beyond
that of the simple systems usually studied in physics. So, how can methods
from physics conceivably be of any use?
Theanswer,asdemonstratedbyFrankSchweitzerinthisbook,liesinthe
factthattheessenceofmanyphenomenadonotdependonalloftheirdetails.
From the study of complex systems we have now accumulated a wealth of
examples that demonstrate how simple components with simple interaction
rules can give rise to complex emergent behaviors, even when, as this book
illustrates,thecomponentsarethemselvesquitecomplicated.Thisisbecause,
forsomepurposes,onlyafewoftheirfeaturesarerelevantandthecomplexity
of the collective behavior emerges from the interactions of these few simple
featuresalone.Soalthoughindividualpeopleareverycomplicated,andtheir
decisionsaboutwheretolivemaybebasedoncomplex,idiosyncraticfactors,
it may nonetheless be possible to understand certain statistical properties of
VI Foreword
the geographic layout of a city in terms of simple models based on a few
simple rules. Furthermore, with only minor modifications of these rules, the
same explanatory framework can be used to understand the dendritic pat-
terns for zinc deposits in an electricfield. It is particularly striking that such
disparate phenomena can be explained using the same theoretical tools. We
havelongknowninphysicsthatmanydifferentphenomenacanbeexplained
with similar mathematics. For example, the equations that describe simple
electriccircuitsconsistingofresistors,inductors,andcapacitorsareprecisely
thesameasthosedescribingasystemofmassesandsprings.Thisworkshows
that such mathematical analogies apply even more broadly than one might
have suspected.
In the middle of the 20th century, John von Neumann said that “sci-
ence and technology will shift from a past emphasis on motion, force, and
energy to communication, organization, programming and control”. This is
already happening, but as we enter the 21st century, the scientific program
for understanding complex systems is still in its infancy as we continue ex-
perimenting to find the right theoretical tools. One of the obvious starting
pointcandidatesisstatisticalmechanics.Thisisanaturalsuggestionbecause
statisticalmechanicsisthebranchofphysicsthatdealswithorganizationand
disorganization.Oneexamplewherethishasalreadysucceededisinformation
theory. Claude Shannon showedhowentropy, whichwasoriginally conceived
for understanding the relationship between heat, work, and temperature,
could be generalized to measure information in an abstract setting and used
for practical purposes such as the construction of an efficient communica-
tion channel. So perhaps there are other extensions of statistical mechanics
that can be used to understand the remarkable range of emergent behaviors
exhibited by many diverse and complex systems.
But the reality is that classical statistical mechanics is mostly about
disorganization. A typical model in statistical mechanics treats atoms as
structureless Ping-Pong balls. A classic example is Brownian motion: When
a particle is suspended in a fluid, it is randomly kicked by the atoms of
the fluid and makes a random walk. This example played a pivotal role in
proving that the world was made of atoms and helped make it possible to
quantitatively understand the relationship between macroscopic properties
such as friction and microscopic properties such as molecular collisions. It
led to the development of the theory of random processes, which has proved
to be extremely useful in many other settings.
The framework Schweitzer develops here goes beyond that of Brownian
motionbymakingtheparticlessuspendedinthefluidjustalittlemorecom-
plicated. The particles become Brownian agents with internal states. They
can store energy and information and they can sense their environment and
respond to it. They can change their internal states contingently depending
on their environment or based on their interactions with each other. These
extra features endow them with simple computational capabilities. They are
Foreword VII
smarter than Ping-Pong balls, but no smarter than they need to be. By
adjusting parameters, the behavior can vary from purely stochastic at one
extreme to purely deterministic at the other. Schweitzer shows that even
whentheBrownianagentsarequitesimple,throughtheirdirectandindirect
interactions with each other, they can exhibit quite complex behaviors.
Brownian agents can be used in many different contexts, ranging from
atomic physics to macroeconomics. In this book, Schweitzer systematically
develops the power of the Brownian agent model and shows how it can be
applied to problems ranging from molecule to mind. At the lowest level they
can be simple atoms or molecules with internal states, such as excitation,
and simple rules of interaction corresponding to chemical reactions. They
can be used to describe the properties of molecular motors and ratchets. Or
they can be cells or single-celled organisms responding to stimuli, such as
electric fields, light, or chemical gradients. They can be used to study the
group feeding properties of bark beetle larvae, or the trail formation of ants
creating and responding to pheromone signals. With just a few changes in
the model, they can be pedestrians forming trails based on visual queues, or
automobile drivers stuck in traffic. Or they can be voters forming opinions
bytalkingtotheirneighbors,orworkersdecidingwheretoworkinafactory.
Agent-basedmodelingisoneofthebasictoolsthathasemergedinrecent
years for the study of complex systems. The basic idea is to encapsulate the
behavior of the interacting units of a complex system in simple programs
that constitute self-contained interacting modules. Unfortunately, however,
agent-based modelers often lack self-restraint, and create agents that are
excessively complicated. This results in models whose behavior can be as
difficulttounderstandasthesystemstheyareintendedtostudy.Oneendsup
not knowing what properties are generic and which properties are unwanted
side-effects.
This work takes just the opposite approach by including only features
that are absolutely necessary. It demonstrates that agent-based modeling is
not just for computer simulation. By keeping the agents sufficiently simple,
it is also possible to develop a theoretical framework that sometimes gives
rise to analytic results and provides a mental framework for modeling and
interpretingtheresultsofsimulationswhenanalyticmethodsfail.Byinsisting
on parsimony, it adheres to the modeling style that has been the key to the
success of physics (and that originally motivated the rational expectations
equilibrium model in economics).
This book lays out a vision for a coherent framework for understanding
complexsystems.However,itshouldbeviewedasabeginningratherthanan
end.Thereisstillagreatdealtobedoneinmakingmoredetailedconnections
to real problems and in making quantitative, falsifiable predictions. Despite
the wide range of problems discussed here, I suspect that this is only a small
subset of the possibilities where the Brownian agent method can be applied.
WhileIdon’tthinkthatBrownianagentswillgainamonopolyontheoretical
VIII Foreword
modeling in complex systems, this book does a major service by introducing
this new tool and demonstrating its generality and power in a wide range
of diverse applications. This work will likely have an important influence on
complex systems modeling in the future. And perhaps most importantly, it
addstotheunityofknowledgebyshowinghowphenomenainwidelydifferent
areas can be addressed within a common mathematical framework and, for
some purposes, most of the details can be ignored.
Santa Fe, NM, USA
February 2003 J. Doyne Farmer
Preface
The emergence of complex behavior in a system consisting of interacting
elements is among the most fascinating phenomena of our world. Examples
can be found in almost every field of today’s scientific interest, ranging from
coherentpatternformationinphysicalandchemicalsystemstothemotionof
swarmsofanimalsinbiologyandthebehaviorofsocialgroups.Thequestion
of how system properties on the macroscopic level depend on microscopic
interactions is one of the major challenges in complex systems and, despite
a number of different attempts, is still far from being solved.
To gain insight into the interplay between microscopic interactions and
macroscopic features, it is important to find a level of description that, on
the one hand, considers specific features of the system and is suitable for
reflecting the origination of new qualities, but, on the other hand, is not
flooded with microscopic details. In this respect, agent models have become
a very promising tool mainly for simulating complex systems. A commonly
accepted theory of agent systems that also allows analytical investigation is,
however,stillpendingbecauseofthediversityofthevariousmodelsinvented
forparticularapplications.Itwillbeamultidisciplinarychallengetoimprove
this situation, in which statistical physics also needs to play its part by
contributing concepts and formal methods.
This book wants to contribute to this development. First, we introduce
a particular class of agent models denoted as Brownian agents and show its
applicability to a variety of problems ranging from physicochemistry to biol-
ogy, economy, and the social sciences. As we will demonstrate, the Brownian
agent approach provides a stable and efficient method for computer simula-
tions of large ensembles of agents. Second, we do not want just to present
simulation results but also want to use the methods of statistical physics to
analyze the dynamics and the properties of systems with large numbers of
Brownian agents, in this way contributing pieces for a formal approach to
multiagent systems.
SimilartoBrownianparticles,Brownianagents aresubjecttobothdeter-
ministic andstochasticinfluences, whichwill allowustoderivea generalized
Langevindynamicsfortheiractivities.Differentfromphysicalparticles,how-
ever, Brownian agents have individual degrees of freedom that allow them
to respond differently to external signals, to interact with other agents, to