Kinetic Theory of Granular Gases provides an introduction to the rapidly developing theory of dissipative gas dynamics — a theory which has mainly evolved over the last decade. The book is aimed at readers from the advanced undergraduate level upwards and leads on to the present state of research. Throughout, special emphasis is put on a microscopically consistent description of pairwise particle collisions which leads to an impact-velocity-dependent coefficient of restitution. The description of the many-particle system, based on the Boltzmann equation, starts with the derivation of the velocity distribution function, followed by the investigation of self-diffusion and Brownian motion. Using hydrodynamical methods, transport processes and self-organized structure formation are studied. An appendix gives a brief introduction to event-driven molecular dynamics. A second appendix describes a novel mathematical technique for derivation of kinetic properties, which allows for the application of computer algebra. The text is self-contained, requiring no mathematical or physical knowledge beyond that of standard physics undergraduate level. The material is adequate for a one-semester course and contains chapter summaries as well as exercises with detailed solutions. The molecular dynamics and computer-algebra programs can be downloaded from a companion web page.

We propose a model for collisions between particles of a granular material and calculate the restitution coefficients for the normal and tangential motion as functions of the impact velocity from considerations of dissipative viscoelastic collisions. Existing models of impact with dissipation as well as the classical Hertz impact theory are included in the present model as special cases. We find that the type of collision (smooth, reflecting or sticky) is determined by the impact velocity and by the surface properties of the colliding grains. We observe a rather nontrivial dependence of the tangential restitution coefficient on the impact velocity.

Saturn's moon Enceladus emits plumes of water vapour and ice particles from fractures near its south pole, suggesting the possibility of a subsurface ocean. These plume particles are the dominant source of Saturn's E ring. A previous in situ analysis of these particles concluded that the minor organic or siliceous components, identified in many ice grains, could be evidence for interaction between Enceladus' rocky core and liquid water. It was not clear, however, whether the liquid is still present today or whether it has frozen. Here we report the identification of a population of E-ring grains that are rich in sodium salts ( approximately 0.5-2% by mass), which can arise only if the plumes originate from liquid water. The abundance of various salt components in these particles, as well as the inferred basic pH, exhibit a compelling similarity to the predicted composition of a subsurface Enceladus ocean in contact with its rock core. The plume vapour is expected to be free of atomic sodium. Thus, the absence of sodium from optical spectra is in good agreement with our results. In the E ring the upper limit for spectroscopy is insufficiently sensitive to detect the concentrations we found.

We perform a dimension analysis for colliding viscoelastic spheres to show that the coefficient of normal restitution epsilon depends on the impact velocity g as epsilon=1-gamma(1)g(1/5)+gamma(2)g(2/5)-/+..., in accordance with recent findings. We develop a simple theory to find explicit expressions for coefficients gamma(1) and gamma(2). Using these and few next expansion coefficients for epsilon(g) we construct a Padé approximation for this function which may be used for a wide range of impact velocities where the concept of the viscoelastic collision is valid. The obtained expression reproduces quite accurately the existing experimental dependence epsilon(g) for ice particles.

A quantitative theory for polyelectrolytes in salt-free dilute solutions is developed. Depending on the electrostatic interaction strength, polyelectrolytes in solutions can undergo strong stretching (with polyelectrolyte dimension R g ϳ l

One of the spectacular discoveries of the Cassini spacecraft was the plume of water vapour and icy particles (dust) originating near the south pole of Saturn's moon Enceladus. The data imply considerably smaller velocities for the grains than for the vapour, which has been difficult to understand. The gas and dust are too dilute in the plume to interact, so the difference must arise below the surface. Here we report a model for grain condensation and growth in channels of variable width. We show that repeated wall collisions of grains, with re-acceleration by the gas, induce an effective friction, offering a natural explanation for the reduced grain velocity. We derive particle speed and size distributions that reproduce the observed and inferred properties of the dust plume. The gas seems to form near the triple point of water; gas densities corresponding to sublimation from ice at temperatures less than 260 K are generally too low to support the measured particle fluxes. This in turn suggests liquid water below Enceladus' south pole.

Using molecular dynamics simulations and theoretical analysis of velocity-autocorrelation functions, we study ion transport mechanisms in typical room-temperature ionic liquids. We show that ions may reside in two states: free and bound with an interstate exchange. We investigate quantitatively the exchange process and reveal new qualitative features of this process. To this end, we propose a dynamic criterion for free and bound ions based on the ion trajectory density and demonstrate that this criterion is consistent with a static one based on interionic distances. Analyzing the trajectories of individual cations and anions, we estimate the time that ions spend in bound "clustered states" and when they move quasifreely. Using this method, we evaluate the average portion of "free" ions as approximately 15%-25%, increasing with temperature in the range of 300-600 K. The ion diffusion coefficients and conductivities as a function of the temperature calculated from the velocity and electrical-current autocorrelation functions reproduce the reported experimental data very well. The experimental data for the direct-current conductivity (constant ionic current) is in good agreement with theoretical predictions of the Nernst-Einstein equation based on the concentrations and diffusion coefficients of free ions obtained in our simulations. In analogy with electronic semiconductors, we scrutinize an "ionic semiconductor" model for ionic liquids, with valence and conduction "bands" for ions separated by an energy gap. The obtained band gap for the ionic liquid is small, around 26 meV, allowing for easy interchange between the two dynamic states. Moreover, we discuss the underscreening paradox in the context of the amount of free charge carriers, showing that the obtained results do not yet approve its simplistic resolution.

Saturn's rings consist of a huge number of water ice particles, with a tiny addition of rocky material. They form a flat disk, as the result of an interplay of angular momentum conservation and the steady loss of energy in dissipative interparticle collisions. For particles in the size range from a few centimeters to a few meters, a power-law distribution of radii, ∼ r −q with q ≈ 3, has been inferred; for larger sizes, the distribution has a steep cutoff. It has been suggested that this size distribution may arise from a balance between aggregation and fragmentation of ring particles, yet neither the power-law dependence nor the upper size cutoff have been established on theoretical grounds. Here we propose a model for the particle size distribution that quantitatively explains the observations. In accordance with data, our model predicts the exponent q to be constrained to the interval 2.75 ≤ q ≤ 3.5. Also an exponential cutoff for larger particle sizes establishes naturally with the cutoff radius being set by the relative frequency of aggregating and disruptive collisions. This cutoff is much smaller than the typical scale of microstructures seen in Saturn's rings. (1-3) and the observation of rapid processes in the ring system (4) indicate that the shape of the particle size distribution is likely not primordial or a direct result of the ring-creating event. Rather, ring particles are believed to be involved in active accretion-destruction dynamics (5-13) and their sizes vary over a few orders of magnitude as a power law (14-17), with a sharp cutoff for large sizes (18-21). Moreover, tidal forces fail to explain the abrupt decay of the size distribution for house-sized particles (22). One wishes to understand the following: (i) Can the interplay between aggregation and fragmentation lead to the observed size distribution? And (ii) is this distribution peculiar for Saturn's rings, or is it universal for planetary rings in general? To answer these questions quantitatively, one needs to elaborate a detailed model of the kinetic processes in which the ring particles are involved. Here we develop a theory that quantitatively explains the observed properties of the particle size distribution and show that these properties are generic for a steady state, when a balance between aggregation and fragmentation holds. Our model is based on the hypothesis that collisions are binary and that they may be classified as aggregative, restitutive, or disruptive (including collisions with erosion); which type of collision is realized depends on the relative speed of colliding particles and their masses. We apply the kinetic theory of granular gases (23, 24) for the multicomponent system of ring particles to quantify the collision rate and the type of collision. Results and DiscussionModel. Ring particles may be treated as aggregates built up of primary grains (9) of a certain size r 1 and mass m 1 . [Observations indicate that particles below a certain radius are absent in dense rings (16).] Denote by m k = km 1 the mass of rin...