Synopsis

Prime Numbers in Frozen Territory

Physics 5, s63
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


Subject Areas

Interdisciplinary PhysicsStatistical Physics

Related Articles

The Unimportance of Accurate Financial Knowledge
Statistical Physics

The Unimportance of Accurate Financial Knowledge

Simulations of the behavior of individual financial traders show that imperfect market knowledge increases risk but not overall losses. Read More »

The Two Faces of Supercooled Water
Chemical Physics

The Two Faces of Supercooled Water

Computations support the 30-year-old idea that supercooled liquid water can undergo a transition between high- and low-density states. Read More »

Microsphere Pair Converts Microwaves to Light
Interdisciplinary Physics

Microsphere Pair Converts Microwaves to Light

A pair of microspheres can convert microwave signals over a wide frequency range into optical signals, which will be essential for future quantum technologies. Read More »

More Articles