On kinetics and extreme values in systems with random interactions

Abstract

Biological environments such as the cytoplasm are comprised of many different molecules, which makes explicit modeling intractable. In the spirit of Wigner, one may be tempted to assume interactions to derive from a random distribution. Via this approximation, the system can be efficiently treated in the mean-field, and general statements about expected behavior of such systems can be made. Here, I study systems of particles interacting via random potentials, outside of mean-field approximations. These systems exhibit a phase transition temperature, under which part of the components precipitate. The nature of this transition appears to be non-universal, and to depend intimately on the underlying distribution of interactions. Above the phase transition temperature, the system can be efficiently treated using a Bethe approximation, which shows a dependence on extreme value statistics. Relaxation timescales of this system tend to be slow, but can be made arbitrarily fast by increasing the number of neighbors of each particle.