Stabilisation of the lattice-Boltzmann method using the Ehrenfests' coarse-graining

The lattice-Boltzmann method (LBM) and its variants have emerged as promising, computationally efficient and increasingly popular numerical methods for modelling complex fluid flow. However, it is acknowledged that the method can demonstrate numerical instabilities, e.g., in the vicinity of shocks. We propose a simple and novel technique to stabilise the lattice-Boltzmann method by monitoring the difference between microscopic and macroscopic entropy. Populations are returned to their equilibrium states if a threshold value is exceeded. We coin the name Ehrenfests' steps for this procedure in homage to the vehicle that we use to introduce the procedure, namely, the Ehrenfests' idea of coarse-graining. The one-dimensional shock tube for a compressible isothermal fluid is a standard benchmark test for hydrodynamic codes. We observe that, of all the LBMs considered in the numerical experiment with the one-dimensional shock tube, only the method which includes Ehrenfests' steps is capable of suppressing spurious post-shock oscillations.Comment: 4 pages, 9 figure

PCA Beyond The Concept of Manifolds: Principal Trees, Metro Maps, and Elastic Cubic Complexes

Multidimensional data distributions can have complex topologies and variable local dimensions. To approximate complex data, we propose a new type of low-dimensional ``principal object'': a principal cubic complex. This complex is a generalization of linear and non-linear principal manifolds and includes them as a particular case. To construct such an object, we combine a method of topological grammars with the minimization of an elastic energy defined for its embedment into multidimensional data space. The whole complex is presented as a system of nodes and springs and as a product of one-dimensional continua (represented by graphs), and the grammars describe how these continua transform during the process of optimal complex construction. The simplest case of a topological grammar (``add a node'', ``bisect an edge'') is equivalent to the construction of ``principal trees'', an object useful in many practical applications. We demonstrate how it can be applied to the analysis of bacterial genomes and for visualization of cDNA microarray data using the ``metro map'' representation. The preprint is supplemented by animation: ``How the topological grammar constructs branching principal components (AnimatedBranchingPCA.gif)''.Comment: 19 pages, 8 figure

Elastic principal manifolds and their practical applications

Principal manifolds serve as useful tool for many practical applications. These manifolds are defined as lines or surfaces passing through "the middle" of data distribution. We propose an algorithm for fast construction of grid approximations of principal manifolds with given topology. It is based on analogy of principal manifold and elastic membrane. The first advantage of this method is a form of the functional to be minimized which becomes quadratic at the step of the vertices position refinement. This makes the algorithm very effective, especially for parallel implementations. Another advantage is that the same algorithmic kernel is applied to construct principal manifolds of different dimensions and topologies. We demonstrate how flexibility of the approach allows numerous adaptive strategies like principal graph constructing, etc. The algorithm is implemented as a C++ package elmap and as a part of stand-alone data visualization tool VidaExpert, available on the web. We describe the approach and provide several examples of its application with speed performance characteristics.Comment: 26 pages, 10 figures, edited final versio

Linking the hydrodynamic and kinetic description of a dissipative relativistic conformal theory

We use the entropy production variational method to associate a one particle distribution function to the assumed known energy-momentum and entropy currents describing a relativistic conformal fluid. Assuming a simple form for the collision operator we find this one particle distribution function explicitly, and show that this method of linking the hydro and kinetic description is a non trivial generalization of Grad's ansatz. The resulting constitutive relations are the same as in the conformal dissipative type theories discussed in J. Peralta-Ramos and E. Calzetta, Phys. Rev. D {\bfseries 80}, 126002 (2009). Our results may prove useful in the description of freeze-out in ultrarelativistic heavy-ion collisions.Comment: v2: 23 pages, no figures, accepted in Phys. Rev.

Raman and Infrared-Active Phonons in Hexagonal HoMnO3_3 Single Crystals: Magnetic Ordering Effects

Polarized Raman scattering and infrared reflection spectra of hexagonal HoMnO$_3$ single crystals in the temperature range 10-300 K are reported. Group-theoretical analysis is performed and scattering selection rules for the second order scattering processes are presented. Based on the results of lattice dynamics calculations, performed within the shell model, the observed lines in the spectra are assigned to definite lattice vibrations. The magnetic ordering of Mn ions, which occurs below T$_N$=76 K, is shown to effect both Raman- and infrared-active phonons, which modulate Mn-O-Mn bonds and, consequently, Mn exchange interaction.Comment: 8 pages, 6 figure

Boltzmann equation and hydrodynamic fluctuations

We apply the method of invariant manifolds to derive equations of generalized hydrodynamics from the linearized Boltzmann equation and determine exact transport coefficients, obeying Green-Kubo formulas. Numerical calculations are performed in the special case of Maxwell molecules. We investigate, through the comparison with experimental data and former approaches, the spectrum of density fluctuations and address the regime of finite Knudsen numbers and finite frequencies hydrodynamics.Comment: This is a more detailed version of a related paper: I.V. Karlin, M. Colangeli, M. Kroger, PRL 100 (2008) 214503, arXiv:0801.2932. It contains comparison between predictions and experiment, in particular. 11 pages, 6 figures, 2 table

Effective dynamics of a nonabelian plasma out of equilibrium

Starting from kinetic theory, we obtain a nonlinear dissipative formalism describing the nonequilibrium evolution of scalar colored particles coupled selfconsistently to nonabelian classical gauge fields. The link between the one-particle distribution function of the kinetic description and the variables of the effective theory is determined by extremizing the entropy production. This method does not rely on the usual gradient expansion in fluid dynamic variables, and therefore the resulting effective theory can handle situations where these gradients (and hence the momentum-space anisotropies) are expected to be large. The formalism presented here, being computationally less demanding than kinetic theory, may be useful as a simplified model of the dynamics of color fields during the early stages of heavy ion collisions and in phenomena related to parton energy loss.Comment: 20 two-column pages, 2 figures. v3: minor changes. Accepted for publication in Phys. Rev.

Thermodynamic Tree: The Space of Admissible Paths

Is a spontaneous transition from a state x to a state y allowed by thermodynamics? Such a question arises often in chemical thermodynamics and kinetics. We ask the more formal question: is there a continuous path between these states, along which the conservation laws hold, the concentrations remain non-negative and the relevant thermodynamic potential G (Gibbs energy, for example) monotonically decreases? The obvious necessary condition, G(x)\geq G(y), is not sufficient, and we construct the necessary and sufficient conditions. For example, it is impossible to overstep the equilibrium in 1-dimensional (1D) systems (with n components and n-1 conservation laws). The system cannot come from a state x to a state y if they are on the opposite sides of the equilibrium even if G(x) > G(y). We find the general multidimensional analogue of this 1D rule and constructively solve the problem of the thermodynamically admissible transitions. We study dynamical systems, which are given in a positively invariant convex polyhedron D and have a convex Lyapunov function G. An admissible path is a continuous curve along which $G$ does not increase. For x,y from D, x\geq y (x precedes y) if there exists an admissible path from x to y and x \sim y if x\geq y and y\geq x. The tree of G in D is a quotient space D/~. We provide an algorithm for the construction of this tree. In this algorithm, the restriction of G onto the 1-skeleton of $D$ (the union of edges) is used. The problem of existence of admissible paths between states is solved constructively. The regions attainable by the admissible paths are described.Comment: Extended version, 31 page, 9 figures, 69 cited references, many minor correction

The Five Factor Model of personality and evaluation of drug consumption risk

The problem of evaluating an individual's risk of drug consumption and misuse is highly important. An online survey methodology was employed to collect data including Big Five personality traits (NEO-FFI-R), impulsivity (BIS-11), sensation seeking (ImpSS), and demographic information. The data set contained information on the consumption of 18 central nervous system psychoactive drugs. Correlation analysis demonstrated the existence of groups of drugs with strongly correlated consumption patterns. Three correlation pleiades were identified, named by the central drug in the pleiade: ecstasy, heroin, and benzodiazepines pleiades. An exhaustive search was performed to select the most effective subset of input features and data mining methods to classify users and non-users for each drug and pleiad. A number of classification methods were employed (decision tree, random forest, $k$-nearest neighbors, linear discriminant analysis, Gaussian mixture, probability density function estimation, logistic regression and na{\"i}ve Bayes) and the most effective classifier was selected for each drug. The quality of classification was surprisingly high with sensitivity and specificity (evaluated by leave-one-out cross-validation) being greater than 70\% for almost all classification tasks. The best results with sensitivity and specificity being greater than 75\% were achieved for cannabis, crack, ecstasy, legal highs, LSD, and volatile substance abuse (VSA).Comment: Significantly extended report with 67 pages, 27 tables, 21 figure

Nonlinear viscosity and velocity distribution function in a simple longitudinal flow

A compressible flow characterized by a velocity field $u_x(x,t)=ax/(1+at)$ is analyzed by means of the Boltzmann equation and the Bhatnagar-Gross-Krook kinetic model. The sign of the control parameter (the longitudinal deformation rate $a$) distinguishes between an expansion ($a>0$) and a condensation ($a<0$) phenomenon. The temperature is a decreasing function of time in the former case, while it is an increasing function in the latter. The non-Newtonian behavior of the gas is described by a dimensionless nonlinear viscosity $\eta^*(a^*)$, that depends on the dimensionless longitudinal rate $a^*$. The Chapman-Enskog expansion of $\eta^*$ in powers of $a^*$ is seen to be only asymptotic (except in the case of Maxwell molecules). The velocity distribution function is also studied. At any value of $a^*$, it exhibits an algebraic high-velocity tail that is responsible for the divergence of velocity moments. For sufficiently negative $a^*$, moments of degree four and higher may diverge, while for positive $a^*$ the divergence occurs in moments of degree equal to or larger than eight.Comment: 18 pages (Revtex), including 5 figures (eps). Analysis of the heat flux plus other minor changes added. Revised version accepted for publication in PR