Synopsis: Prime Numbers in Frozen Territory

The behavior of freezing transitions in glasses is related to the statistical properties of prime numbers.

If prime numbers are the elementary particles of arithmetic, then the Riemann zeta function is one of the key tools for analyzing how they behave. The zeta function, although relatively easy to write down, contains multitudes: it encodes information about the distribution of primes along the number line, and is the centerpiece of unsolved problems in number theory.

Connections between the statistics of primes and physics have been made before, but now, in a paper in Physical Review Letters, Yan Fyodorov of Queen Mary, University of London, UK, and colleagues show a surprising correspondence between freezing in disordered systems, like glasses, and the peaks and valleys of the zeta function.

The energy of a disordered system is like a traveler moving around on a random landscape of hill and valleys. As the temperature is lowered, the traveler bounces from place to place but eventually settles into a local energy minimum, which marks the freezing transition in the glass. Fyodorov et al. show both analytically, and with numerical simulations, that the statistical mechanical properties of the freezing transition correlate with the statistical properties of extrema of the zeta function. Not only might this work guide the way physicists tackle important statistical physics problems, but our understanding of freezing could help mathematicians make progress in attacking some of the grand challenges of number theory. – David Voss


Announcements

More Announcements »

Subject Areas

Interdisciplinary PhysicsStatistical Physics

Previous Synopsis

Nonlinear Dynamics

Science of Slosh

Read More »

Next Synopsis

Interdisciplinary Physics

Force Diagrams on Skis

Read More »

Related Articles

Focus: How to Compare Books or Genomes
Complex Systems

Focus: How to Compare Books or Genomes

A mathematical technique for comparing large symbol sets suggests that less frequently used words are mainly responsible for the evolution of the English language over the past two centuries. Read More »

Viewpoint: Turning Down the Volume on Granular Materials
Statistical Physics

Viewpoint: Turning Down the Volume on Granular Materials

A reformulation of the statistical mechanics of granular materials replaces the volume of the material with a function related to its structure. Read More »

Synopsis: Playing Games with Schrödinger
Interdisciplinary Physics

Synopsis: Playing Games with Schrödinger

Models that treat economic and biological behavior in terms of game-play resemble quantum mechanics. Read More »

More Articles