# The Fermi-Pasta-Ulam Problem

The Fermi-Pasta-Ulam (FPU) problem is named after the numerical experiments first performed by Enrico Fermi, John Pasta, and Stanislaw Ulam in the summer of 1953 on the Los Alamos MANIAC computer, one of the first electronic computers. They wanted to understand how a one-dimensional crystal evolves toward thermal equilibrium by simulating a chain of particles coupled by spring-like forces that included quadratic and cubic interactions. It was assumed that these nonlinear terms would cause the system to "thermalize" by redistributing energy from an initial mode into many other modes. In other words, an initial mode of vibration would quickly become more or less random.

When Fermi, Pasta, and Ulam did the simulation, they found to their great surprise that energy from a single initial mode was shared by only a few other modes; the remaining modes were hardly excited. Moreover, after a long time the initial state was almost completely recovered. This result, which is known as the FPU problem or the FPU paradox, shows that nonlinearity is not sufficient to guarantee the equipartition of energy. It turns out that thermal (ergodic) behavior is only observed when the magnitude of the nonlinear term is more than a certain critical value.

### References

- E. Fermi, J. Pasta, and S. Ulam. "Studies in Nonlinear Problems, I." Los Alamos report LA 1940, (1955).
- T. Dauxois, M. Peyrard, and S. Ruffo, "The Fermi-Pasta-Ulam 'numerical experiment': History and pedagogical perspectives," Eur. J. Phys. 26, S3â€“S11 (2005).
- M. Tabor, "The FUP Experiment." Â§7.1.b in Chaos and Integrability in Nonlinear Dynamics: An Introduction. New York: Wiley, p. 280 (1989).
- T. P. Weissert, The Genesis of Simulation in Dynamics: Pursuing the Fermi-Pasta-Ulam Problem, Springer (1997).
- Gould and Tobochnik Statistic and Thermal Physics to be published by Princeton University Press (2010). Draft chapters are available online at <http://stp.clarku.edu/>.

### Note:

This simulation was created by Wolfgang Christian using the Easy Java Simulations (Ejs) modeling tool. It is based on a Java program from the Statistical and Thermal Physics Project. You can modify this simulation if you have Ejs installed by right-clicking within a plot and selecting "Open Ejs Model" from the pop-up menu. Information about Ejs is available at: <http://www.um.es/fem/Ejs/>.