EPJ B Colloquium - Ballistic annihilation in one dimension: a critical review

Details

Published on 09 February 2022

In a new Colloquium published in EPJB, S. Biswas (Universidad de Guadalajara, Mexico) and F. Leyvraz (Universidad Nacional Autónoma de México, Mexico) review several related systems. In the simplest, all particles move in a straight line at constant velocity in one dimension, and upon meeting, irreversibly react to an inert species. The simplest approach to such systems involves the “law of mass action” which leads, for large times, to a concentration decay of 1/t. The model described above for which all particles move with two possible distinct velocities only, has been solved exactly. In this case, it is shown that the concentration decay goes as t^-1/2, so that the law of mass action is strongly violated.

If the particles diffuse on the line, again the model can be solved. The law for the concentration decay at large times is also t^-1/2. However if these two features are combined, novel effects arise. Let the particles diffuses with a fixed bias, which can be positive or negative with probability 1/2. Then the concentration decay rate is t^-3/4, faster than in either of the pure cases. Even more intricate behavior arises if we allow asymmetric distributions for the drifts : if we allow the possibility of a small imbalance between the number of particles with positive and with negative drift, we obtain three different decay laws: initially we have t^-3/4, then we have a period in which the surviving particles have the same drift, but the concentration decays according to a new exponent, t^-1/4. Finally, once all structure is eliminated, we have the decay of t^-1/2 as for a diffusing system.