Research

Under what conditions does a system thermalize? What is the role of chaos in thermalization? Is there a critical value of non-linearity in a system for chaos and does it persist in the thermodynamic limit? These are some of the questions I have been studying in the 1D mean-field Bose-Hubbard Model.

Publications

A Brief Introduction to Research

The study of thermalization in non-linear systems dates back to the numerical experiments of Fermi, Pasta, and Ulam in the 1950's 1. They modeled a non-linear string as a series of anharmonically coupled oscillators. Consider a one-dimensional chain of harmonic oscillators, with linear coupling. The dynamics of this chain can be described in terms of normal modes of the system. In the linear system, these normal modes do not interact with one another and there is no energy exchange between the modes. The distribution of energy in the modes is constant and equal to the initial distribution. What happens when a small non-linearity is added, which couples the normal modes? Will energy spread between the modes? Will the distribution of energy among the modes equal the predictions of thermodynamics?

At the time the FPU carried out their experiment, the expectation was that a system with a large number of degrees of freedom would thermalize in the presence of any non-linearity, no matter how small. For the FPU system, thermalization would be marked by equipartition of energy among the normal modes. The results of the experiment presented a paradox. Initially one of the low-energy modes was excited and while energy spread to the higher modes at first, it did not continue to spread and eventually returned to the mode that was excited initially. It was later shown that the cubic (β) FPU model is very close to the Korteweg-de-Vries equation, which is fully integrable2, offering one explanation for the absence of thermalization. Other work revealed that there is a critical threshold of the non-linearity and that FPU thermalizes above the threshold.

In our work, we study chaos and thermalization in the 1D mean-field Bose-Hubbard Model. We calculate a finite-time Lyapunov exponent, which is a measure of chaos. A non-zero Lyapunov exponent indicates that two neighboring phase-space trajectories diverge exponentialy, a signature of chaos. A suitable spectral entropy is defined, which measures how close the system is to the thermodynamic prediction. We find a threshold for chaos, which depends on the strength of the nonlinearity as well as the total energy of the system. Far above the threshold, the system thermalizes.

1E. Fermi, J. Pasta, and S. Ulam, Los Alamos Report LA-1940 (1955).
2N. J. Zabusky and M. D. Kruskal Phys. Rev. Lett. 15, 240 (1965).