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Register a new userLocal Anisotropy of Fluids using Minkowski Tensors
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Statistics of the free volume available to individual particles have previously been studied for simple and complex fluids, granular matter, amorphous solids, and structural glasses. Minkowski tensors provide a set of shape measures that are based on strong mathematical theorems and easily computed for polygonal and polyhedral bodies such as free volume cells (Voronoi cells). They characterize the local structure beyond the two-point correlation function and are suitable to define indices $0leq beta_ u^{a,b}leq 1$ of local anisotropy. Here, we analyze the statistics of Minkowski tensors for configurations of simple liquid models, including the ideal gas (Poisson point process), the hard disks and hard spheres ensemble, and the Lennard-Jones fluid. We show that Minkowski tensors provide a robust characterization of local anisotropy, which ranges from $beta_ u^{a,b}approx 0.3$ for vapor phases to $beta_ u^{a,b}to 1$ for ordered solids. We find that for fluids, local anisotropy decreases monotonously with increasing free volume and randomness of particle positions. Furthermore, the local anisotropy indices $beta_ u^{a,b}$ are sensitive to structural transitions in these simple fluids, as has been previously shown in granular systems for the transition from loose to jammed bead packs.
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The local structure of disordered jammed packings of monodisperse spheres without friction, generated by the Lubachevsky-Stillinger algorithm, is studied for packing fractions above and below 64%. The structural similarity of the particle environments to fcc or hcp crystalline packings (local crystallinity) is quantified by order metrics based on rank-four Minkowski tensors. We find a critical packing fraction phi_c approx 0.649, distinctly higher than previously reported values for the contested random close packing limit. At phi_c, the probability of finding local crystalline configurations first becomes finite and, for larger packing fractions, increases by several orders of magnitude. This provides quantitative evidence of an abrupt onset of local crystallinity at phi_c. We demonstrate that the identification of local crystallinity by the frequently used local bond-orientational order metric q_6 produces false positives, and thus conceals the abrupt onset of local crystallinity. Since the critical packing fraction is significantly above results from mean-field analysis of the mechanical contacts for frictionless spheres, it is suggested that dynamic arrest due to isostaticity and the alleged geometric phase transition in the Edwards framework may be disconnected phenomena.
Experiments of the recrystallization processes in two-dimensional complex plasmas are analyzed in order to rigorously test a recently developed scale-free phase transition theory. The Fractal-Domain-Structure (FDS) theory is based on the kinetic theory of Frenkel. It assumes the formation of homogeneous domains, separated by defect lines, during crystallization and a fractal relationship between domain area and boundary length. For the defect number fraction and system energy a scale free power-law relation is predicted. The long range scaling behavior of the bond order correlation function shows clearly that the complex plasma phase transitions are not of KTHNY type. Previous preliminary results obtained by counting the number of dislocations and applying a bond order metric for structural analysis are reproduced. These findings are supplemented by extending the use of the bond order metric to measure the defect number fraction and furthermore applying state-of-the-art analysis methods, allowing a systematic testing of the FDS theory with unprecedented scrutiny: A morphological analysis of lattice structure is performed via Minkowski tensor methods. Minkowski tensors form a complete family of additive, motion covariant and continuous morphological measures that are sensitive to non-linear properties. The FDS theory is rigorously confirmed and predictions of the theory are reproduced extremely well. The predicted scale-free power law relation between defect fraction number and system energy is verified for one more order of magnitude at high energies compared to the inherently discontinuous bond order metric. Minkowski Tensor analysis turns out to be a powerful tool for investigations of crystallization processes. It is capable to reveal non-linear local topological properties, however, still provides easily interpretable results founded on a solid mathematical framework.
We apply the Minkowski tensor statistics to three dimensional Gaussian random fields. Minkowski tensors contain information regarding the orientation and shape of excursion sets, that is not present in the scalar Minkowski functionals. They can be used to quantify globally preferred directions, and additionally provide information on the mean shape of subsets of a field. This makes them ideal statistics to measure the anisotropic signal generated by redshift space distortion in the low redshift matter density field. We review the definition of the Minkowski tensor statistics in three dimensions, focusing on two coordinate invariant quantities $W^{0,2}_{1}$ and $W^{0,2}_{2}$. We calculate the ensemble average of these $3 times 3$ matrices for an isotropic Gaussian random field, finding that they are proportional to products of the identity matrix and a corresponding scalar Minkowski functional. We show how to numerically reconstruct $W^{0,2}_{1}$ and $W^{0,2}_{2}$ from discretely sampled fields and apply our algorithm to isotropic Gaussian fields generated from a linear $Lambda$CDM matter power spectrum. We then introduce anisotropy by applying a linear redshift space distortion operator to the matter density field, and find that both $W^{0,2}_{1}$ and $W^{0,2}_{2}$ exhibit a distinct signal characterised by inequality between their diagonal components. We discuss the physical origin of this signal and how it can be used to constrain the redshift space distortion parameter $Upsilon equiv f/b$.
The terminology granular matter refers to systems with a large number of hard objects (grains) of mesoscopic size ranging from millimeters to meters. Geological examples include desert sand and the rocks of a landslide. But the scope of such systems is much broader, including powders and snow, edible products such a seeds and salt, medical products like pills, and extraterrestrial systems such as the surface regolith of Mars and the rings of Saturn. The importance of a fundamental understanding for granular matter properties can hardly be overestimated. Practical issues of current concern range from disaster mitigation of avalanches and explosions of grain silos to immense economic consequences within the pharmaceutical industry. In addition, they are of academic and conceptual importance as well as examples of systems far from equilibrium. Under many conditions of interest, granular matter flows like a normal fluid. In the latter case such flows are accurately described by the equations of hydrodynamics. Attention is focused here on the possibility for a corresponding hydrodynamic description of granular flows. The tools of nonequilibrium statistical mechanics, developed over the past fifty years for fluids composed of atoms and molecules, are applied here to a system of grains for a fundamental approach to both qualitative questions and practical quantitative predictions. The nonlinear Navier-Stokes equations and expressions for the associated transport coefficients are obtained.
We derive a hierarchy of equations which allow a general $n$-body distribution function to be measured by test-particle insertion of between $1$ and $n$ particles, and successfully apply it to measure the pair and three-body distribution functions in a simple fluid. The insertion-based methods overcome the drawbacks of the conventional distance-histogram approach, offering enhanced structural resolution and a more straightforward normalisation. They will be especially useful in characterising the structure of inhomogeneous fluids and investigating closure approximations in liquid state theory.
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