# Joel Lebowitz and statistical mechanics

**Joel Lebowitz was the Foundation Sciences Mathématiques de Paris's guest during 3 weeks in November 2008 to attend the IHP Trimester entitled Interacting particle systems, statistical mechanics and probability theory (September 5 ^{th} – December 19^{th}, 2008).**

Joel Lebowitz is the **George William Hill Professor of Mathematics and Physics at Rutgers University** (State University of New Jersey). All along his rich carrier, this outstanding researcher has received many prizes and honors : the Boltzmann Medal (1992), the Henri Poincaré Prize (2000), the Nicholson Medal (1994) awarded by the American Physical Society, the Volterra Award (2001), Max-Planck Medal (2007), and others.

Joel Lebowitz has made very important contributions to **statistical mechanics**, which is a field related to probability theory, dynamical systems and other branches of mathematics. He even founded one of the most important journals in this field, the *Journal of Statistical Physics*, in 1975.

Statistical mechanics

The subject itself is traditionally defined as the analysis of how the behaviour of macroscopic systems depends on the properties of the atoms and molecules which are the microscopic constituents of macroscopic systems (for example, how does the temperature at which water boils depend on the pressure). There are many technological devices which depend on understanding such behaviour, e.g. why some materials become superconductors at low temperatures and others do not.

Put another way, statistical mechanics studies the collective behaviour of a large number of individual entities which are relatively independent but interact with each other. Many aspects of the collective phenomena do not depend on the details. For example, the flow of water and the flow of oil are essentially the same even if a molecule of oil is very different from a molecule of water. Such properties only depend on general conservative laws.

Joel Lebowitz explains: “We therefore try to understand these phenomena using very simple models. Often the results obtained from simple models are applicable to real systems.”

For example, boiling (freezing as well) is a manifestation of a collective phenomenon. The coexistence curve of vapor and liquid depends on the temperature. There is a critical temperature T_{c} above which there is no difference between liquid and vapor. When the temperature is close to this critical temperature, the density of liquid and the density of vapor are close to each other and the difference between them depends on how far below this temperature T_{c} you are, according to the formula below:

?*?* = *? _{l}* –

*?*? (T

_{v}_{c}– T)

^{?}The temperature T

_{c}depends on the liquid but

*?*is universal: it only depends on the dimension and the symmetry.

Statistical mechanics is applied not only to phase transitions in materials (liquid to vapor, solid to liquid…) but also to many other fields : biological phenomena, social phenomena, financial markets, traffic flows (transition between smooth and congested traffic), etc. The

*individual entities*can be atoms, bacteria or cars, but their collective behaviour is determined by their interactions and has many universal features.