A recently introduced model describing—on a 1d lattice—the velocity field of a granular fluid is discussed in detail. The dynamics of the velocity field occurs through next-neighbours inelastic collisions which conserve momentum but dissipate energy. The dynamics is described through the corresponding Master Equation for the time evolution of the probability distribution. In the continuum limit, equations for the average velocity and temperature fields with fluctuating currents are derived, which are analogous to hydrodynamic equations of granular fluids when restricted to the shear modes. Therefore, the homogeneous cooling state, with its linear instability, and other relevant regimes such as the uniform shear flow and the Couette flow states are described. The evolution in time and space of the single particle probability distribution, in all those regimes, is also discussed, showing that the local equilibrium is not valid in general. The noise for the momentum and energy currents, which are correlated, are white and Gaussian. The same is true for the noise of the energy sink, which is usually negligible
We introduce a model described in terms of a scalar velocity field on a 1D lattice, evolving through collisions that conserve momentum but do not conserve energy. Such a system possesses some of the main ingredients of fluidized granular media and naturally models them. We deduce non-linear fluctuating hydrodynamics equations for the macroscopic velocity and temperature fields, which replicate the hydrodynamics of shear modes in a granular fluid. Moreover, this Landau-like fluctuating hydrodynamics predicts an essential part of the peculiar behaviour of granular fluids, like the instability of homogeneous cooling state at large size or inelasticity. We also compute the exact shape of long range spatial correlations which, even far from the instability, have the physical consequence of noticeably modifying the cooling rate. This effect, which stems from momentum conservation, has not been previously reported in the realm of granular fluids.〈 〉= ± ± , that is, we assume that velocities at adjacent sites are uncorrelated. This hypothesis is somehow similar to the molecular chaos assumption when writing the Boltzmann equation for a low-density fluid.
We derive the hydrodynamic equations with fluctuating currents for the density, momentum, and energy fields for an active system in the dilute limit. In our model, nonoverdamped self-propelled particles (such as grains or birds) move on a lattice, interacting by means of aligning dissipative forces and excluded volume repulsion. Our macroscopic equations, in a specific case, reproduce a transition line from a disordered phase to a swarming phase and a linear dispersion law accounting for underdamped wave propagation. Numerical simulations up to a packing fraction ∼10% are in fair agreement with the theory, including the macroscopic noise amplitudes. At a higher packing fraction, a dense-diluted coexistence emerges. We underline the analogies with the granular kinetic theories, elucidating the relation between the active swarming phase and granular shear instability.
Solving the dynamics of equilibrium dense liquids is a notoriously difficult problem [1]. In the low-density limit, or at short times, the solution is obtained via kinetic theory, while at any density, hydrodynamics provides a description of the dynamics at large length and time scales. However, at high densities or low temperatures, kinetic theory breaks down, and the hydrodynamic regime is pushed to scales that are much larger than any experimentally relevant scale. In this strongly interacting "supercooled liquid" regime, dynamics is slow, viscosity is high, and both correlation functions and transport coefficients display non-trivial behavior that is neither captured by kinetic theory nor by hydrodynamics. The only microscopic description of this regime is obtained by the Mode-Coupling Theory (MCT) [2,3], which is a set of closed equations derived from a series of poorly controlled approximations of the true dynamics. Despite its non-systematic nature [4][5][6][7], MCT accurately describes the initial slowing down of liquid dynamics upon supercooling, including the wavevector dependence of correlation functions [8].Interestingly, in the formal limit in which the spatial dimension d goes to infinity, liquid thermodynamics reduces to the calculation of the second virial coefficient [9][10][11][12]. Based on this observation, a first attempt to solve exactly liquid dynamics for d → ∞ was presented in [13] (see also [14]), but the full dynamical mean field theory (DMFT) that describes exactly the equilibrium dynamics in d → ∞ was only derived recently, via a second-order virial expansion on trajectories [15] or via a dynamic cavity method [16,17]. The DMFT provides a set of closed one-dimensional integro-differential equations, which exactly describe the many-body liquid dynamics in the thermodynamic limit, and are similar in structure to those obtained for quantum systems in the same limit [18]. We refer the reader to [17,[19][20][21] for a detailed review of the solution of liquid dynamics in infinite dimensions, including its extension to the out-of-equilibrium setting.Unfortunately, the analytical solution of the DMFT equations is out of reach. In this work, we present their numerical solution, obtained through an iterative method and a straightforward discretization of time. This strategy, however, only works for differentiable interaction potentials. We thus discuss how to derive the DMFT of hard spheres via a non-trivial limit of a soft sphere interaction. We present numerical results for soft and hard spheres, supported by analytical computations at low densities, in the short time limit, and at long times in the glass phase.The paper is organized as follows. In section II we review the basic equations of the DMFT of liquids. In section III, we discuss the Hard Sphere limit of DMFT. In section IV, we discuss the discretization and convergence algorithms used in this work. In section V, we present the numerical solution for Soft and Hard Spheres. Finally, in section VI we draw our conclusions and present s...
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