Heat flux of a granular gas with homogeneous temperature

A steady state of a granular gas with homogeneous granular temperature, no mass flow, and nonzero heat flux is studied. The state is created by applying an external position--dependent force or by enclosing the grains inside a curved two--dimensional silo. At a macroscopic level, the state is identified with one solution to the inelastic Navier--Stokes equations, due to the coupling between the heat flux induced by the density gradient and the external force. On the contrary, at the mesoscopic level, by exactly solving a BGK model or the inelastic Boltzmann equation in an approximate way, a one--parametric family of solutions is found. Molecular dynamics simulations of the system in the quasi--elastic limit are in agreement with the theoretical results

Generalized time evolution of the homogeneous cooling state of a granular gas with positive and negative coefficient of normal restitution

The homogeneous cooling state (HCS) of a granular gas described by the inelastic Boltzmann equation is reconsidered. As usual, particles are taken as inelastic hard disks or spheres, but now the coefficient of normal restitution $\alpha$ is allowed to take negative values $\alpha\in[-1,1]$, a simple way of modeling more complicated inelastic interactions. The distribution function of the HCS is studied at the long-time limit, as well as for intermediate times. At the long-time limit, the relevant information of the HCS is given by a scaling distribution function $\phi_s(c)$, where the time dependence occurs through a dimensionless velocity $c$. For $\alpha\gtrsim -0.75$, $\phi_s$ remains close to the gaussian distribution in the thermal region, its cumulants and exponential tails being well described by the first Sonine approximation. On the contrary, for $\alpha\lesssim -0.75$, the distribution function becomes multimodal, its maxima located at $c\ne 0$, and its observable tails algebraic. The latter is a consequence of an unbalanced relaxation-dissipation competition, and is analytically demonstrated for $\alpha\simeq -1$ thanks to a reduction of the Boltzmann equation to a Fokker-Planck-like equation. Finally, a generalized scaling solution to the Boltzmann equation is also found $\phi(c,\beta)$. Apart from the time dependence occurring through the dimensionless velocity, $\phi(c,\beta)$ depends on time through a new parameter $\beta$ measuring the departure of the HCS from its long-time limit. It is shown that $\phi(c,\beta)$ describes the time evolution of the HCS for almost all times. The relevance of the new scaling is also discussed.Comment: 19 pages, 7 figure

Mass transport of driven inelastic Maxwell mixtures

Mass transport of a driven granular binary mixture is analyzed from the inelastic Boltzmann kinetic equation for inelastic Maxwell models (IMM). The mixture is driven by a thermostat constituted by two terms: a stochastic force and a drag force proportional to the particle velocity. The combined action of both forces attempts to mimic the interaction of solid particles with the interstitial surrounding gas. As with ordinary gases, the use of IMM allows us to exactly evaluate the velocity moments of the Boltzmann collision operator and so, it opens up the possibility of obtaining the exact forms of the Navier--Stokes transport coefficients of the granular mixture. In this work, the diffusion coefficients associated with the mass flux are explicitly determined in terms of the parameters of the mixture. As a first step, the steady homogeneous state reached by the system when the energy lost by collisions is compensated for by the energy injected by the thermostat is addressed. In this steady state, the ratio of kinetic temperatures are determined and compared against molecular dynamics simulations for inelastic hard spheres (IHS). The comparison shows an excellent agreement, even for strong inelasticity and/or disparity in masses and diameters. As a second step, the set of kinetic equations for the mixture is solved by means of the Chapman-Enskog method for states near homogeneous steady states. In the first-order approximation, the mass flux is obtained and the corresponding diffusion transport coefficients identified. The results show that the predictions for IMM obtained in this work coincide with those previously derived for IHS in the first-Sonine approximation when the non-Gaussian corrections to the zeroth-order approximation are neglected.Comment: 10 pages, 2 figures; paper submitted for its publication in AIP Conference Proceedings (RGD31

The noisy voter model under the influence of contrarians

The influence of contrarians on the noisy voter model is studied at the mean-field level. The noisy voter model is a variant of the voter model where agents can adopt two opinions, optimistic or pessimistic, and can change them by means of an imitation (herding) and an intrinsic (noise) mechanisms. An ensemble of noisy voters undergoes a finite-size phase transition, upon increasing the relative importance of the noise to the herding, form a bimodal phase where most of the agents shear the same opinion to a unimodal phase where almost the same fraction of agent are in opposite states. By the inclusion of contrarians we allow for some voters to adopt the opposite opinion of other agents (anti-herding). We first consider the case of only contrarians and show that the only possible steady state is the unimodal one. More generally, when voters and contrarians are present, we show that the bimodal-unimodal transition of the noisy voter model prevails only if the number of contrarians in the system is smaller than four, and their characteristic rates are small enough. For the number of contrarians bigger or equal to four, the voters and the contrarians can be seen only in the unimodal phase. Moreover, if the number of voters and contrarians, as well as the noise and herding rates, are of the same order, then the probability functions of the steady state are very well approximated by the Gaussian distribution

Critical behavior of two freely evolving granular gases separated by an adiabatic piston

Two granular gases separated by an adiabatic piston and initially in the same macroscopic state are considered. It is found that a phase transition with an spontaneous symmetry breaking occurs. When the mass of the piston is increased beyond a critical value, the piston moves to a stationary position different from the middle of the system. The transition is accurately described by a simple kinetic model that takes into account the velocity fluctuations of the piston. Interestingly, the final state is not characterized by the equality of the temperatures of the subsystems but by the cooling rates being the same. Some relevant consequences of this feature are discussed.Comment: 6 figure

Anomalous transport of impurities in inelastic Maxwell gases

A mixture of dissipative hard grains generically exhibits a breakdown of kinetic energy equipartition. The undriven and thus freely cooling binary problem, in the tracer limit where the density of one species becomes minute, may exhibit an extreme form of this breakdown, with the minority species carrying a finite fraction of the total kinetic energy of the system. We investigate the fingerprint of this non-equilibrium phase transition, akin to an ordering process, on transport properties. The analysis, performed by solving the Boltzmann kinetic equation from a combination of analytical and Monte Carlo techniques, hints at the possible failure of hydrodynamics in the ordered region. As a relevant byproduct of the study, the behaviour of the second and fourth-degree velocity moments is also worked out.Comment: The title has been changed. The paper has been enlarged with respect to our first version. 13 pages, 9 figures. To be published in EPJ

Zealots in the mean-field noisy voter model

The influence of zealots on the noisy voter model is studied theoretically and numerically at the mean-field level. The noisy voter model is a modification of the voter model that includes a second mechanism for transitions between states: apart from the original herding processes, voters may change their states because of an intrinsic, noisy in origin source. By increasing the importance of the noise with respect to the herding, the system exhibits a finite-size phase transition from a quasi-consensus state, where most of the voters share the same opinion, to a one with coexistence. Upon introducing some zealots, or voters with fixed opinion, the latter scenario may change significantly. We unveil the new situations by carrying out a systematic numerical and analytical study of a fully connected network for voters, but allowing different voters to be directly influenced by different zealots. We show that this general system is equivalent to a system of voters without zealots, but with heterogeneous values of their parameters characterizing herding and noisy dynamics. We find excellent agreement between our analytical and numerical results. Noise and herding/zealotry acting together in the voter model yields not a trivial mixture of the scenarios with the two mechanisms acting alone: it represents a situation where the global-local (noise-herding) competitions is coupled to a symmetry breaking (zealots). In general, the zealotry enhances the effective noise of the system, which may destroy the original quasi--consensus state, and can introduce a bias towards the opinion of the majority of zealots, hence breaking the symmetry of the system and giving rise to new phases ...Comment: 13 pages, 15 figure