Mpemba effect refers to the counterintuitive effect where a system which is initially further from the final steady state equilibrates faster than an identical system that is initially closer. The closeness to the final state is defined in terms of a distance measure. For driven granular systems, the Mpemba effect has been illustrated in terms of an ad-hoc measure of mean kinetic energy as the distance function. In this paper, by studying four different distance measures based on the mean kinetic energies as well as velocity distribution, we show that the Mpemba effect depends on the definition of the measures.

We study ballistic aggregation on a two dimensional square lattice, where particles move ballistically in between momentum and mass conserving coalescing collisions. Three models are studied based on the shapes of the aggregates: in the first the aggregates remain point particles, in the second they retain the fractal shape at the time of collision, and in the third they assume a spherical shape. The exponents describing the power law temporal decay of number of particles and energy as well as dependence of velocity correlations on mass are determined using large scale Monte Carlo simulations. It is shown that the exponents are universal only for the point particle model. In the other two cases, the exponents are dependent on the initial number density and correlations vanish at high number densities. The fractal dimension for the second model is close to 1.49.