Solving the Boltzmann equation in N log N

In [C. Mouhot and L. Pareschi, "Fast algorithms for computing the Boltzmann collision operator," Math. Comp., to appear; C. Mouhot and L. Pareschi, C. R. Math. Acad. Sci. Paris, 339 (2004), pp. 71-76], fast deterministic algorithms based on spectral methods were derived for the Boltzmann collision operator for a class of interactions including the hard spheres model in dimension three. These algorithms are implemented for the solution of the Boltzmann equation in two and three dimension, first for homogeneous solutions, then for general non homogeneous solutions. The results are compared to explicit solutions, when available, and to Monte-Carlo methods. In particular, the computational cost and accuracy are compared to those of Monte-Carlo methods as well as to those of previous spectral methods. Finally, for inhomogeneous solutions, we take advantage of the great computational efficiency of the method to show an oscillation phenomenon of the entropy functional in the trend to equilibrium, which was suggested in the work [L. Desvillettes and C. Villani, Invent. Math., 159 (2005), pp. 245-316].Comment: 32 page

Prediction of strong shock structure using the bimodal distribution function

A modified Mott-Smith method for predicting the one-dimensional shock wave solution at very high Mach numbers is constructed by developing a system of fluid dynamic equations. The predicted shock solutions in a gas of Maxwell molecules, a hard sphere gas and in argon using the newly proposed formalism are compared with the experimental data, direct-simulation Monte Carlo (DSMC) solution and other solutions computed from some existing theories for Mach numbers M<50. In the limit of an infinitely large Mach number, the predicted shock profiles are also compared with the DSMC solution. The density, temperature and heat flux profiles calculated at different Mach numbers have been shown to have good agreement with the experimental and DSMC solutionsComment: 22 pages, 9 figures, Accepted for publication in Physical Review

Investigation of initiation of gigantic jets connecting thunderclouds to the ionosphere

The initiation of giant electrical discharges called as "gigantic jets" connecting thunderclouds to the ionosphere is investigated by numerical simulation method in this paper. Using similarity relations, the triggering conditions of streamer formation in laboratory situations are extended to form a criterion of initiation of gigantic jets. The energy source causing a gigantic jet is considered due to the quasi-electrostatic field generated by thunderclouds. The electron dynamics from ionization threshold to streamer initiation are simulated by the Monte Carlo technique. It is found that gigantic jets are initiated at a height of ~18-24 km. This is in agreement with the observations. The method presented in this paper could be also applied to the analysis of the initiation of other discharges such as blue jets and red sprites.Comment: 12th International Congress on Plasma Physics, 25-29 October 2004, Nice (France

The Moment Guided Monte Carlo method for the Boltzmann equation

In this work we propose a generalization of the Moment Guided Monte Carlo method developed in [11]. This approach permits to reduce the variance of the particle methods through a matching with a set of suitable macroscopic moment equations. In order to guarantee that the moment equations provide the correct solutions, they are coupled to the kinetic equation through a non equilibrium term. Here, at the contrary to the previous work in which we considered the simplified BGK operator, we deal with the full Boltzmann operator. Moreover, we introduce an hybrid setting which permits to entirely remove the resolution of the kinetic equation in the limit of infinite number of collisions and to consider only the solution of the compressible Euler equation. This modification additionally reduce the statistical error with respect to our previous work and permits to perform simulations of non equilibrium gases using only a few number of particles. We show at the end of the paper several numerical tests which prove the efficiency and the low level of numerical noise of the method.Comment: arXiv admin note: text overlap with arXiv:0908.026

A Monte Carlo simulation of ion transport at finite temperatures

We have developed a Monte Carlo simulation for ion transport in hot background gases, which is an alternative way of solving the corresponding Boltzmann equation that determines the distribution function of ions. We consider the limit of low ion densities when the distribution function of the background gas remains unchanged due to collision with ions. A special attention has been paid to properly treat the thermal motion of the host gas particles and their influence on ions, which is very important at low electric fields, when the mean ion energy is comparable to the thermal energy of the host gas. We found the conditional probability distribution of gas velocities that correspond to an ion of specific velocity which collides with a gas particle. Also, we have derived exact analytical formulas for piecewise calculation of the collision frequency integrals. We address the cases when the background gas is monocomponent and when it is a mixture of different gases. The developed techniques described here are required for Monte Carlo simulations of ion transport and for hybrid models of non-equilibrium plasmas. The range of energies where it is necessary to apply the technique has been defined. The results we obtained are in excellent agreement with the existing ones obtained by complementary methods. Having verified our algorithm, we were able to produce calculations for Ar$^+$ ions in Ar and propose them as a new benchmark for thermal effects. The developed method is widely applicable for solving the Boltzmann equation that appears in many different contexts in physics.Comment: 14 page

Modelling opinion formation by means of kinetic equations

In this chapter, we review some mechanisms of opinion dynamics that can be modelled by kinetic equations. Beside the sociological phenomenon of compromise, naturally linked to collisional operators of Boltzmann kind, many other aspects, already mentioned in the sociophysical literature or no, can enter in this framework. While describing some contributions appeared in the literature, we enlighten some mathematical tools of kinetic theory that can be useful in the context of sociophysics

Heat Shock Protein-27, -60 and -90 expression in gastric cancer: association with clinicopathological variables and patient survival

<p>Abstract</p> <p>Background</p> <p>Heat shock proteins (HSPs) are ubiquitous, highly conserved proteins across all the species and play essential roles in maintaining protein stability within the cells under normal conditions, while preventing stress-induced cellular damage. HSPs were also overexpressed in various types of cancer, being associated with tumor cell proliferation, differentiation and apoptosis. The aim of the present study was to evaluate the clinical significance of HSP -27, -60, and -90 expression in gastric carcinoma.</p> <p>Methods</p> <p>HSP -27, -60, and -90 proteins expression was assessed immunohistochemically in tumoral samples of 66 gastric adenocarcinoma patients and was statistically analyzed in relation to various clinicopathological characteristics, tumor proliferative capacity and patients' survival.</p> <p>Results</p> <p>HSP-27, -60, -90 proteins were abundantly expressed in gastric adenocarcinoma cases examined. HSP-27 expression was significantly associated with tumor size (pT, P = 0.026), the presence of organ metastases (pM, P = 0.046) and pStage (P = 0.041), while HSP-27 staining intensity with nodal status (pN, P = 0.042). HSP-60 expression was significantly associated with patients' sex (P = 0.011), while HSP-60 staining intensity with patients' age (P = 0.027) and tumor histopathological grade (P = 0.031). HSP-90 expression was not associated with any of the clinicopathological parameters examined; however, HSP-90 staining intensity was significantly associated with tumor size (pT, P = 0.020). High HSP-90 expression was significantly associated with longer overall survival times in univariate analysis (log-rank test, P = 0.033), being also identified as an independent prognostic factor in multivariate analysis (P = 0.026).</p> <p>Conclusion</p> <p>HSP-27, -60, and -90 were associated with certain clinicopathological parameters which are crucial for the management of gastric adenocarcinoma patient. HSP-90 expression may also be an independent prognostic indicator in gastric adenocarcinoma patients.</p