  The Mpemba effect occurs when two samples at different initial temperatures
evolve in such a way that the temperatures cross each other during the
relaxation towards equilibrium. In this paper we show the emergence of a
Mpemba-like effect in a molecular binary mixture in contact with a thermal
reservoir (bath). The interaction between the gaseous particles of the mixture
and the thermal reservoir is modeled via a viscous drag force plus a stochastic
Langevin-like term. The presence of the external bath couples the time
evolution of the total and partial temperatures of each component allowing the
appearance of the Mpemba phenomenon, even when the initial temperature
differences are of the same order of the temperatures themselves. Analytical
results are obtained by considering multitemperature Maxwellian approximations
for the velocity distribution functions of each component. The theoretical
analysis is carried out for initial states close to and far away (large
Mpemba-like effect) from equilibrium. The former situation allows us to develop
a simple theory where the time evolution equation for the temperature is
linearized around its asymptotic equilibrium solution. This linear theory
provides an expression for the crossover time. We also provide a qualitative
description of the large Mpemba effect. Our theoretical results agree very well
with computer simulations obtained by numerically solving the Enskog kinetic
equation by means of the direct simulation Monte Carlo method and by performing
molecular dynamics simulations. Finally, preliminary results for driven
granular mixtures also show the occurrence of a Mpemba-like effect for
inelastic collisions.
