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

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- 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.

Soham Biswas and Francois Leyvraz (2021), Ballistic annihilation in one dimension: a critical review,

European Physical Journal B **94**:240, DOI: 10.1140/epjb/s10051-021-00258-w