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Ballistic annihilation with superimposed diffusion in one dimension

We consider a one-dimensional system with particles having either positive or negative velocity, which annihilate on contact. To the ballistic motion of the particle, a diffusion is superimposed. The annihilation may represent a reaction in which the two particles yield an inert species. This model has been the object of previous work, in which it was shown that the particle concentration decays faster than either the purely ballistic or the purely diffusive case. We report on previously unnoticed behaviour for large times, when only one of the two species remains and also unravel the underlying fractal structure present in the system. We also consider in detail the case in which the initial concentration of right-going particles is $1/2+\varepsilon$, with $\varepsilon\neq0$. It is shown that a remarkably rich behaviour arises, in which two crossover times are observed as $\varepsilon \to 0$.