CORE CONCEPTS
Core Concept: Ergodic theory plays a key rolein multiple
fieldsSteven AshleyScience Writer
Statistical mechanics is a powerful set ofmathematical tools
that uses probability the-ory to bridge the enormous gap between
theunknowable behaviors of individual atomsandmolecules and those
of large aggregate sys-tems of thema volume of gas, for
example.Fundamental to statistical mechanics is
ergodic theory, which offers a mathematicalmeans to study the
long-term average behaviorof complex systems, such as the behavior
ofmolecules in a gas or the interactions of vi-brating atoms in a
crystal. The landmark con-cepts and methods of ergodic theory
continueto play an important role in statistical mechan-ics,
physics, mathematics, and other fields.Ergodicity was first
introduced by the
Austrian physicist Ludwig Boltzmann in the1870s, following on
the originator of statisti-cal mechanics, physicist James Clark
Max-well. Boltzmann coined the word ergodiccombining two Greek
words: o (ergon:work) and (odos: path or way)to describe his
hypothesis. Although someformulations of his hypothesis were not
com-pletely accurate, his ideas seeded an impor-tant set of
principles and tools.Boltzmann, working with gases, suggested
that the spatial average values giving rise tomacroscopic
features also arose as averagesover time of observable quantities
that couldbe calculated from microscopic states. Herealized that
this might hold if a system passedthrough all of the possible
microscopic states.This strong form of Boltzmanns ergodic
hypothesis was, however, later seen to be toostrong in practice
on topological and measuretheoretic grounds, says Nndor Simnyi,
pro-fessor of mathematics at the University ofAlabama at
Birmingham. Ergodicity wasloosely defined. It was an assumption
madeabout the time-evolution of a dynamical sys-tem that worked,
but the idea that a systemgoes through every state, that a system
goesthrough every point in the phase space, isimpossible;
mathematically, its nonsense,he says. It was, however, satisfactory
for aphysical understanding of systems.The path toward greater
theoretical rigor,
explains University of Warwick mathematics
professor TomWard, reached a key milestonein the early 1930s
when American mathema-tician George D. Birkhoff and
Austrian-Hun-garian (and later, American) mathematicianand
physicist John von Neumann
separatelyreconsideredandreformulatedBoltzmanns er-godic
hypothesis, leading to the pointwise andmean ergodic theories,
respectively (see ref. 1).These results consider a dynamical
sys-
temwhether an ideal gas or other complexsystemsinwhich some
transformation func-tion maps the phase state of the system intoits
state one unit of time later. Given a mea-sure-preserving system, a
probability spacethat is acted on by the transformation ina way
that models physical conservation laws,what properties might it
have? asks Ward,who is managing editor of the journal ErgodicTheory
and Dynamical Systems. The measureon a mathematical space assigns
weights toparts of the space.Measure theory provides a precise
formu-
lation, saying that while a system doesnt lit-erally pass
through every point in the phasespace, it may come arbitrarily
close to everypoint, he explains. That is, theres no fa-vored part
of the space. This formulationgives rise to a notion of ergodicity
weakenough to hold for many systems and strongenough to have
significant consequences.After decades of relative quiescence,
efforts
to devise ergodicity models underwent a re-vival in the 1960s,
notably including work byYakov Sinai, a mathematician now at
Prince-ton University, Simnyi says. In 1963, Sinaiintroduced the
idea of dynamical billiards, orSinai Billiards, in which an
idealized hardparticle bounces around inside a squareboundary
without loss of energy. Inside thesquare is a circular wall, off of
which the par-ticle bounces. He proved that for most
initialtrajectories of the ball, this system is ergodicthat is,
over infinite time, the time average ofan observed quantity will
equal its averageover the space. For any part of the space,the
proportion of time the system spendsin that region will be
proportional to its size.If the system spends 3% of the time in
a
region of its phase space, that region is 3%
of the total phase-space volume; they areproportional in size,
Simnyi says. Itwas the first time that someone had rigor-ously
proved that a dynamical system thatseemed related to a real
physical situationwas ergodic.Ergodic theory continues to have
wide
impact in statistical physics, number theory,probability theory,
functional analysis, andother fields. One prominent example is
theGreenTao theorem. In 2004, Ben Green ofthe University of Oxford
and Terrence Taoof University of California, Los Angeles,building
on the work of American-Israelimathematician Hillel Furstenberg,
fusedmethods from analytic number theory andergodic theory to prove
a folklore conjec-ture dating back to the 1770s that the se-quence
of prime numbers contains arbi-trarily long sequences of numbers in
whichthe difference between consecutive terms isa constant (for
example, 3, 5, 7 is such a se-quence of length three). Tao went on
to earnthe 2006 Fields Medal.
1 Moore CC (2015) Ergodic theorem, ergodic theory,and
statistical mechanics. Proc Natl Acad Sci USA112:19071911.
Austrian physicist Ludwig Boltzmann laidthe foundation for
modern-day ergodic the-ory. Image courtesy of Goethe
UniversityFrankfurt.
1914 | PNAS | February 17, 2015 | vol. 112 | no. 7
www.pnas.org/cgi/doi/10.1073/pnas.1500429112
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