Ulam Paradox

A Computational Analysis of the Fermi, Pasta, Ulam Paradox


Author: Mary Zischkau
Date: February 12, 2015


This paper presents a computational study of the Fermi, Pasta, Ulam (FPU) paradox as it is applied to a system of nonlinear springs and thirty-two masses. The chain of springs and masses experiences motion only in one direction without the forces of friction or internal heating present to reduce the energy of the system. The program, written in Python, uses the Runge-Kutta method and Newtons second law for its calculations of motion. A Fourier Transformation is performed in the calculation of the energy for the system. The energy for different modes is compared over a long time. As Enrico Fermi, John Pasta, and Stanislaw Ulam discovered in their computational experiment, the energy of the different modes, rather than achieving equipartition over time, periodically returns to its initial state where the one mode is excited and the next two are not.


The purpose of this program was to simulate the motion of a chain of nonlinear springs and masses oscillating in one direction. The energy of the entire system was examined over time for different frequency modes. The results were compared to those obtained by Fermi, Pasta, and Ulam who discovered from their data that the system would not come to an equipartition of energy after a long period of time. While the ergodic hypothesis suggests that over long periods of times the energy of the system should tend to equipartition among the different normal modes, the calculations performed in the program showed that the energy periodically returned to its initial state where the k=1 mode was excited and the modes for k =2 and k =3 were not. The return of the energy to its initial state suggested that nonlinearity does not necessarily result in equipartition. The solution to the problem of why the system does not come to equipartition lies in studying the presence of solitons and deterministic chaos in the FPU chain as determined by the work of Zabusky and Kruskal in 1965 and of Chirikov in 1973.

Theoretical Background

The system of nonlinear springs and masses was first investigated by Fermi, Pasta, and Ulam using the most powerful computer at the time, the MANIAC, to perform their calculations. Fermi, Pasta, and Ulam believed that their system with nonlinear springs would conform to the equipartition theorem of statistical mechanics. The equipartition theorem states that among different modes of motion, energy will be shared evenly. For instance, the kinetic energy of helium molecules at equilibrium is distributed evenly among the three components of motion. Thus, there would be no more atoms moving along the z-axis than there are atoms moving along the x-axis and y-axis. Since the nonlinear springs provide different modes of oscillation, Fermi, Pasta, and Ulam proposed that the energy would equally divide between all the modes over time. Initially what they observed seemed to verify their proposition. The first mode where k = 1 slowly drifted to the other modes, and modes for k =2 and k=3 became excited. Nevertheless, when the scientists left their program running for a longer duration, they noted that the energies of each of the modes returned to their original states.  The results of their computation suggested that nonlinearity did not guarantee equipartition. This seemingly small discovery of Fermi, Pasta, and Ulam had monumental results. Scientists, in continuing the studies of the FPU chain, made significant developments and discoveries in the fields of mathematics and physics, including the study of chaos and of solitons, a phenomenon of solitary waves. This paper confirms the results of the Fermi, Pasta, and Ulam computations.

Computational Procedures

In order to simulate the motion of the spring-mass system, Newtons second law was used to derive the equation for the acceleration (See Figure 1 in the appendix). The motion of the springs follows Hookes Law but is adjusted with nonlinear terms due to deformation and the restoring force.  The equation for acceleration implemented in the program was derived by Fermi, Pasta, and Ulam for masses and springs with quadratic nonlinear terms, the system commonly known as the FPU -model.

x n = x n+1 + x n-1 -2 x n + ( x n+1 - x n ) 2 - ( x n - x n-1 ) 2

Integrating over the accelerations through the Runge-Kutta method returns the positions and velocities of each mass in the system. This was completed with for loops and functions for the derivatives and for the Runge-Kutta integration.

Next an analysis of the energy for the system was performed. To find the energy of the system, the kinetic and potential energies had to be calculated using the displacement and velocity found from the Runge-Kutta integration. The displacement for the system is given by the following equation which includes a normalizing factor and a Fourier Transformation.


The frequencies for the system are given by the equation below.


Using the above two equations, the energy for the system can easily be found as the sum of the kinetic and potential energies.


The xx term calculates the kinetic energy and cx is the potential energy. Once the energy of the system is found, it is analyzed for a long period of time.

Data and Analysis

The energy for the system of thirty-two masses and nonlinear springs in the first mode, where k = 1, starts in an excited state, descends rapidly, and eventually returns to its initial excited state after about 155 periods.  For the k =2 mode, the energy starts at zero, becomes excited and then drops back to zero after approximately 150 periods. Similarly, the energy of the k=3 mode begins at zero, peaks twice and then returns to zero after about 158 periods. As shown in Figure 2, the energy of the different modes briefly comes to equipartition, but then returns to its initial state where only the k = 1 mode is excited and the other two modes have no energy. This same result can be seen in Figure 3 from a similar study of the FPU paradox completed by Dr. Thierry Dauxois and Dr. Stefano Ruffo.

In calculating the energy of the system, the initial conditions play a very important role. The coefficient of the nonlinear term has to be set at approximately 0.25 in order to obtain the reoccurrence relationship seen in Figure 2. If the nonlinear term is too strong due to a higher coefficient or too weak due to a smaller coefficient, the energy of each mode will not return to its initial state. See Figure 4 and 5.

Moreover, the time step factors into whether or not the energy of each mode periodically reoccurs. Through experimentation, a time step of 1.00079 was found to be optimal for the reoccurrence. A smaller time step leads to chaos as can be seen in Figure 6.


The results from this project show that nonlinearity alone does not necessitate that energy come to equipartition. Indeed, the energy of the different modes of the FPU -model of 32 masses and nonlinear springs returned to its initial state in which only one mode was excited and the other two were at zero.  Sensitivity to the initial conditions and the time step was also examined and the results show the presence of chaos in the system. Overall, the algorithms implemented in the code were effective in the examination of the FPU paradox and the results obtained were very similar to the results obtained in previous studies of the problem. Future work would include looking for solitons when the number of masses is increased.


Figure 1

Figure1: Example of mass and nonlinear spring system investigated by Fermi, Pasta, and Ulam. The ends of the system would be fixed and the masses would only move in the horizontal direction.


Figure 2:  Graph of the energy [dfd]  of the system for different modes over time; asf . This graph was obtained with the program developed for this project. The graph shows periodic reoccurrence of the energy.


Figure 3: Graph of the energy [3]  of the system for different modes over time; zzz . This graph comes from a study done by Dr. Thierry Dauxois and Sr. Stefano Ruffo. It shows periodic reoccurrence of the energy of the system.


Figure 4: The coefficient of the nonlinear term is set to 0.30 and so the energy of the system for k = 1 is not enough to reach to its initial state a second time after 140 periods.

 Figure 5

Figure 5: The coefficient of the nonlinear term is set to 0.20 and so the energy of the system for k=1 increases beyond its initial state after 160 periods.


Figure 6: When the time step was increased from 1.00079 to 1.01, the FPU system is chaotic.


Works Cited

Dauxois, Thierry and Stefano Ruffo. Fermi-Pasta-Ulam nonlinear lattice oscillation. Scholarpedia, 2008. Web. 14 Dec. 2014. http://www.scholarpedia.org/article/Fermi-Pasta-Ulam_nonlinear_lattice_oscillations.

Dauxois, Thierry, Michel Peyrard and Stefano Ruffo. Ther Fermi-Pasta-Ulam numerical experiment: history and pedagogical perspectives. European Journal of Physics, Vol. 26, Issue 5, pp S3-S11, 2005. Web. 14 Dec. 2014. http://arxiv.org/pdf/nlin/0501053.pdf.

Porter, Mason A., Norman J. Zabusky, Bambi Hu, and David K. Campbell. Fermi, Pasta, Ulam and the Birth of Experimental Mathematics. American Scientist Vol. 97, No. 3, 2009. Web. 14 Dec. 2014. https://people.maths.ox.ac.uk/porterm/papers/fpupop_final.pdf.


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