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, published in the online version of the journal Physical Review Letters by M. Howard Lee, Regents Professor of Physics at UGA, 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.”

**Difficult, intricate theorem**

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.

**Classically intractable problem**

It became one of the foundations of statistical mechanics, but actually proving Boltzmann’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 Boltzmann’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.”

**Laying the groundwork**

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.

**Probing Birkhoff’s solution**

When a colleague suggested that Lee use this mathematical tool, which he calls an “ergometer,” to probe Birkhoff’s solution to Boltzmann’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.”

**Establishing a connection**

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 also will allow physicists to understand how widely valid Boltzmann’s Hypothesis actually is and help researchers in assessing the entire foundations of statistical mechanics.