This paper reviews applications of the lattice-Boltzmann method to simulations of particle-fluid suspensions. We first summarize the available simulation methods for colloidal suspensions together with some of the important applications of these methods, and then describe results from lattice-gas and lattice-Boltzmann simulations in more detail. The remainder of the paper is an update of previously published work,(69, 70) taking into account recent research by ourselves and other groups. We describe a lattice-Boltzmann model that can take proper account of density fluctuations in the fluid, which may be important in describing the short-time dynamics of colloidal particles. We then derive macro-dynamical equations for a collision operator with separate shear and bulk viscosities, via the usual multi-time-scale expansion. A careful examination of the second-order equations shows that inclusion of an external force, such as a pressure gradient, requires terms that depend on the eigenvalues of the collision operator. Alternatively, the momentum density must be redefined to include a contribution from the external force. Next, we summarize recent innovations and give a few numerical examples to illustrate critical issues. Finally, we derive the equations for a lattice-Boltzmann model that includes transverse and longitudinal fluctuations in momentum. The model leads to a discrete version of the Green–Kubo relations for the shear and bulk viscosity, which agree with the viscosities obtained from the macro-dynamical analysis. We believe that inclusion of longitudinal fluctuations will improve the equipartition of energy in lattice-Boltzmann simulations of colloidal suspensions.

Lattice-Boltzmann Simulations of Particle-Fluid Suspensions

Preface.- Primal and Dual Problems.- One-Dimensional Issues.- L^1 and L^infinity Theory.- Minimal Flows.- Wasserstein Spaces.- Numerical Methods.- Functionals over Probabilities.- Gradient Flows.- Exercises.- References.- Index.

Optimal Transport for Applied Mathematicians : Calculus of Variations, PDEs, and Modeling

Voyager 2 photography has complemented that of Voyager I in revealing many additional characteristics of Saturn and its satellites and rings. Saturn's atmosphere contains persistent oval cloud features reminiscent of features on Jupiter. Smaller irregular features track out a pattern of zonal winds that is symmetric about Saturn's equator and appears to extend to great depth. Winds are predominantly eastward and reach 500 meters per second at the equator. Titan has several haze layers with significantly varying optical properties and a northern polar "collar" that is dark at short wavelengths. Several satellites have been photographed at substantially improved resolution. Enceladus' surface ranges from old, densely cratered terrain to relatively young, uncratered plains crossed by grooves and faults. Tethys has a crater 400 kilometers in diameter whose floor has domed to match Tethys' surface curvature and a deep trench that extends at least 270° around Tethys' circumference. Hyperion is cratered and irregular in shape. Iapetus' bright, trailing hemisphere includes several dark-floored craters, and Phoebe has a very low albedo and rotates in the direction opposite to that of its orbital revolution with a period of 9 hours. Within Saturn's rings, the "birth" of a spoke has been observed, and surprising azimuthal and time variability is found in the ringlet structure of the outer B ring. These observations lead to speculations about Saturn's internal structure and about the collisional and thermal history of the rings and satellites.

A new look at the saturn system: the voyager 2 images.

http://repository.iucaa.in:8080/jspui/bitstream/11007/175/1/77_1990.pdf

Statistical mechanics of gravitating systems

Stellar polytropes and Tsallis' entropy

On donne un critere simple permettant de determiner si une fonction de distribution donnee est plus melangee qu'une autre. On utilise ce critere pour montrer qu'une galaxie ayant subi une relaxation violente ne ressemble aux galaxies elliptiques observees que si l'etat initial est froid ou a grumeaux

H-functions and mixing in violent relaxation

We show that the deterministic past history of the Universe can be uniquely reconstructed from knowledge of the present mass density field, the latter being inferred from the three-dimensional distribution of luminous matter, assumed to be tracing the distribution of dark matter up to a known bias. Reconstruction ceases to be unique below those scales – a few Mpc – where multistreaming becomes significant. Above 6 h−1 Mpc we propose and implement an effective Monge–Ampere–Kantorovich method of unique reconstruction. At such scales the Zel'dovich approximation is well satisfied and reconstruction becomes an instance of optimal mass transportation, a problem which goes back to Monge. After discretization into N point masses one obtains an assignment problem that can be handled by effective algorithms with not more than O(N3) time complexity and reasonable CPU time requirements. Testing against N-body cosmological simulations gives over 60 per cent of exactly reconstructed points.



We apply several interrelated tools from optimization theory that were not used in cosmological reconstruction before, such as the Monge–Ampere equation, its relation to the mass transportation problem, the Kantorovich duality and the auction algorithm for optimal assignment. A self-contained discussion of relevant notions and techniques is provided.

/pdf/reconstruction-of-the-early-universe-as-a-convex-2nnmph6rej.pdf

Reconstruction of the early Universe as a convex optimization problem

We show that the deterministic past history of the Universe can be uniquely reconstructed from the knowledge of the present mass density field, the latter being inferred from the 3D distribution of luminous matter, assumed to be tracing the distribution of dark matter up to a known bias. Reconstruction ceases to be unique below those scales -- a few Mpc -- where multi-streaming becomes significant. Above 6 Mpc/h we propose and implement an effective Monge-Ampere-Kantorovich method of unique reconstruction. At such scales the Zel'dovich approximation is well satisfied and reconstruction becomes an instance of optimal mass transportation, a problem which goes back to Monge (1781). After discretization into N point masses one obtains an assignment problem that can be handled by effective algorithms with not more than cubic time complexity in N and reasonable CPU time requirements. Testing against N-body cosmological simulations gives over 60% of exactly reconstructed points. 
We apply several interrelated tools from optimization theory that were not used in cosmological reconstruction before, such as the Monge-Ampere equation, its relation to the mass transportation problem, the Kantorovich duality and the auction algorithm for optimal assignment. Self-contained discussion of relevant notions and techniques is provided.

The distribution of particle sizes in Saturn's rings is probably continuous over many orders of magnitude; there is no ‘typical particle size’. A model based on this view accounts for a number of observed properties of the rings: apparent thickness, radar and radio observations, number and size distribution of the gaps discovered by Voyager 1, and optical thickness.

Michel Hénon

Papers

H-functions and mixing in violent relaxation

Reconstruction of the early Universe as a convex optimization problem

Reconstruction of the early Universe as a convex optimization problem

A simple model of Saturn's rings

Low-viscosity lattice gases