Search for universality in one-dimensional ballistic annihilation kinetics

Abstract

We study the kinetics of ballistic annihilation for a one-dimensional ideal gas with continuous velocity distribution. A dynamical scaling theory for the long time behavior of the system is derived. Its validity is supported by extensive numerical simulations for several velocity distributions. This leads us to the conjecture that all the continuous velocity distributions φ(v) which are symmetric, regular and such that φ(0) = 0 are attracted in the long time regime towards the same Gaussian distribution and thus belong to the same universality class. Moreover, it is found that the particle density decays as n(t) ∼ t −α , with α ≃ 0.785 ± 0.005.

Dynamics of ballistic annihilation

“…This scaling relation may be simply obtained by elementary dimensional analysis [4,5,6,9,10], and may be considered as the compatibility condition of the hierarchy with the self-similar scaling solutions [12]. It is moreover identically fulfilled by the expressions (36) and (37).…”

“…In one dimension, Ben-Naim et al [4,5] have shown that the exponent ξ could depend on the behavior near the origin of the initial velocity distribution. This problem has been revisited by Rey et al [6]. Based on the exact theoretical approach [2,3], a dynamical scaling theory was derived and its validity supported by numerical simulations for several velocity distributions.…”

Hydrodynamics of probabilistic ballistic annihilation

“…When p = 1, we recover the annihilation model originally defined by Elskens and Frisch [13], that has attracted some attention since [14,15,16,17,18,19,20,21]. In one dimension (again for p = 1), the problem is well understood for discrete initial velocity distributions [15,16]. On the contrary, higher dimensions introduce complications that make the problem much more difficult to treat [19,20].…”

“…The model of probabilistic ballistic annihilation in one dimension for bimodal discrete initial velocity distributions was introduced in [11], whereas for higher dimensions and arbitrary continuous initial velocity distributions it was considered in [12]. When p = 1, we recover the annihilation model originally defined by Elskens and Frisch [13], that has attracted some attention since [14,15,16,17,18,19,20,21]. In one dimension (again for p = 1), the problem is well understood for discrete initial velocity distributions [15,16].…”

“…Numerical evidences show that in the long-time limit, a non-Maxwellian scaling solution for the homogeneous system appears (homogeneous cooling state (HCS) [19,20], which also exists for inelastic hard spheres [25]). Nothing is known about the stability of the latter solution, the only existing result being that in one dimension, with a bimodal initial velocity distribution, clusters of particles are spontaneously formed by the dynamics [16]. In view of this situation we develop a hydrodynamic description for probabilistic ballistic annihilation.…”