Athens, Ga. – There are probably more molecules in
your den than there are stars in the universe. When studying numbers so vast,
researchers had to find a way to make large-scale predictions based on the
study of microscopic properties. That field of inquiry is called statistical
mechanics, and it is an important tool in explaining how the world works.
A new research paper, just published in the online version
of the journal Physical Review Letters
by M. Howard Lee, Regents Professor of Physics at the University of Georgia,
however, may lead to a reassessment of some foundations of statistical
mechanics, according to its author.
“Reassessing old problems with new tools is always a
challenge,” said Lee. “But it is a challenge that has been rewarding.”
At the heart of Lee’s new research is the work of two giants
of physics and mathematics, Ludwig Boltzmann and George David Birkhoff and a
hypothesis one proposed and the other proved. It is the story of a difficult
and intricate theorem that remains important in using microscopic pictures to
understand large-scale systems.
Boltzmann was a 19th century Austrian physicist
and one of the founders of statistical mechanics. He proposed what came to be
called the Ergodic Hypothesis: A time average is equal to an ensemble average.
This elegant idea allowed scientists to compute accurate thermodynamic
functions without having to examine how particles act and change over time.
It became one of the foundations of statistical mechanics,
but actually proving Bolzmann’s hypothesis turned out to be a classically
intractable problem, until Birkhoff, an American mathematician, came along. But
while his proof seemed to work in the field of mathematics, it never satisfied
physicists, who considered it far too abstract.
Lee’s paper in Physical
Review Letters proposes a new solution to the problem that has perplexed
researchers since Birkhoff’s solution some 70 years ago.
“Proving Bolzmann’s hypothesis is extremely difficult,
because one must first solve the equation of motion, which is a daunting task
in itself,” said Lee. “As a result, most people have come to accept the
hypothesis despite occasional evidence to the contrary.”
In 2001, Lee laid the groundwork for testing the hypothesis
by using a technique he had developed to help solve another problem in 1982
when he found an exact, general and practical solution to one of the most
important problems in statistical physics. The problem was how to solve the
so-called "Heisenberg equation of motion," which yields the response
of a system to an external probe.
While another scientist solved the problem first, Lee went
about it in different way, one that provided for the first time a theory from
which one could actually calculate. Lee’s work on that problem has had a
tremendous impact on statistical mechanics, as evidenced by nearly 600
citations since its publication 25 years ago.
When a colleague suggested that Lee use this mathematical
tool, which he calls an “ergometer,” to probe Birkhoff’s solution to Bolzmann’s
hypothesis, a light bulb went off. This might be a way to take Birkhoff from
mathematics into the very different realm of physics.
“To make sense of Birkhoff’s Theorem, let’s say that being
Ergodic means being able to walk on land,” said Lee. “In this analogy, Birkhoff
says that there is an island, but he doesn’t say how large or small the island
is. It could be as small as an islet or as large as a continent. To physicists,
it’s critical to know how large that island is.”
Lee then used his ergometer to help determine the boundaries
and therefore the size of the “island.” In the Physical Review Letters paper, Lee examined where Birkhoff’s
Theorem is violated and extracted from it the underlying physical basis for it.
“Establishing this connection puts Birkhoff’s Theorem on a
physical terrain, enabling us to begin the mapping process of that island, and
this paper is the start of that work,” said Lee.
It will also allow physicists to understand how widely valid
Boltzmann’s Hypothesis actually is and help researchers in assessing the entire
foundations of statistical mechanics.
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Note to editors: Contact Lee at 706/542-3539 or mhlee@uga.edu for a copy of the article.
