Exact Solution of Two-Species Ballistic Annihilation with General Pair-Reaction Probability
Abstract
The reaction process
A+B−>C
is modelled for ballistic reactants on an infinite line with particle velocities
v
A
=c
and
v
B
=−c
and initially segregated conditions, i.e. all A particles to the left and all B particles to the right of the origin. Previous, models of ballistic annihilation have particles that always react on contact, i.e. pair-reaction probability
p=1
. The evolution of such systems are wholly determined by the initial distribution of particles and therefore do not have a stochastic dynamics. However, in this paper the generalisation is made to
p<1
, allowing particles to pass through each other without necessarily reacting. In this way, the A and B particle domains overlap to form a fluctuating, finite-sized reaction zone where the product C is created. Fluctuations are also included in the currents of A and B particles entering the overlap region, thereby inducing a stochastic motion of the reaction zone as a whole. These two types of fluctuations, in the reactions and particle currents, are characterised by the `intrinsic reaction rate', seen in a single system, and the `extrinsic reaction rate', seen in an average over many systems. The intrinsic and extrinsic behaviours are examined and compared to the case of isotropically diffusing reactants.