An experiment design for schools to demonstrate the Maxwell distribution

The article describes a new version of a demonstration experiment for the Maxwell distribution. In the first part students analyse the applicability of the Gaussian distribution to the projection of the particle velocities in the suggested experiment. Further, students observe two-dimensional distribution of particles by the modulus of velocity in a mechanical demonstration model and compare the results with theoretical provisions. Demonstration of the two-dimensional version of the Maxwell distribution for particle interaction allows students to independently derive formulas for the three-dimensional Maxwell distribution for particles in an ideal gas. The use of the suggested demonstration ensures active engagement in fundamentally important physical content.

Revisiting a low-dimensional model with short range interactions and mean field critical behavior

In all local low-dimensional models, scaling at critical points deviates from mean field behavior – with one possible exception. This exceptional model with “ordinary” behavior is an inherently non-equilibrium model studied some time ago by H.-M. Bröker and myself. In simulations, its 2-dimensional version suggested that two critical exponents were mean-field, while a third one showed very small deviations. Moreover, the numerics agreed almost perfectly with an explicit mean field model. In the present paper we present simulations with much higher statistics, both for 2d and 3d. In both cases we find that the deviations of all critical exponents from their mean field values are non-leading corrections, and that the scaling is precisely of mean field type. As in the original paper, we propose that the mechanism for this is “confusion”, a strong randomization of the phases of feed-backs that can occur in non-equilibrium systems.

A study of atmospheric boundary layer characteristics with ground snow cover - numerical method

ABSTRACT. A one-dimensional version of a mesoscale model was used to simulate the atmospheric variables over ground snow cover after incorporating suitable modifications. Modifications to include the effect of cloud on shortwave and long wave radiation were also made in the model. The model takes into account both the heat balance at the snow surface and at various layers of the snow pack and calculates the melt rate in situ. Srinagar (Jammu & Kashmir) winter data was used for the simulations. The diurnal variation of snowmelt rate and other atmospheric variables were simulated simultaneously by the model. Melt rate values were verified with the values obtained from standard empirical formula. The model-simulated profiles of potential temperature, specific humidity and wind speed were found to be in reasonable agreement with available observations. The results were found to be insensitive to changes in surface drag coefficients.

Numerical simulation of high-current pulsed arc discharge in air

Computational model of high-current pulsed arcdischarge in air is proposed. This is, in general, two-dimensional model with taking into account gas dynamics of the discharge channel, real air thermodynamics in a wide range of pressure and temperature, electrodynamics of the discharge including pinch effect, and radiation. One-dimensional version of the model is tested and verified on several numerical and experimental works reported recently. It is concluded that low and moderate current discharges are satisfactorily described with the developed model. Then, developed model was applied to simulate the electric discharge in air for the currents of 1 - 250 kA and characteristic rise times in 13 - 25 µs, and results of calculations were compared with experimental ones. It was concluded that most of characteristics of the discharge are predicted well. Namely, arc column radius and shock wave position agree well with experimental data for all current amplitudes and rise times considered. Radial distributions of temperature and electron density also satisfactorily agree with experimental data. It was found that pinch effect should be considered for currents higher than 100 kA.

Optimization of the dynamic transition in the continuous coloring problem

Random constraint satisfaction problems (CSPs) can exhibit a phase where the number of constraints per variable α makes the system solvable in theory on the one hand, but also makes the search for a solution hard, meaning that common algorithms such as Monte Carlo (MC) method fail to find a solution. The onset of this hardness is deeply linked to the appearance of a dynamical phase transition where the phase space of the problem breaks into an exponential number of clusters. The exact position of this dynamical phase transition is not universal with respect to the details of the Hamiltonian one chooses to represent a given problem. In this paper, we develop some theoretical tools in order to find a systematic way to build a Hamiltonian that maximizes the dynamic α d threshold. To illustrate our techniques, we will concentrate on the problem of continuous coloring, where one tries to set an angle x i ∈ [0; 2π] on each node of a network in such a way that no adjacent nodes are closer than some threshold angle θ, that is cos(x i − x j )⩽ cos θ. This problem can be both seen as a continuous version of the discrete graph coloring problem or as a one-dimensional version of the Mari–Krzakala–Kurchan model. The relevance of this model stems from the fact that continuous CSPs on sparse random graphs remain largely unexplored in statistical physics. We show that for sufficiently small angle θ this model presents a random first order transition and compute the dynamical, condensation and Kesten–Stigum transitions; we also compare the analytical predictions with MC simulations for values of θ = 2π/q, q ∈ N . Choosing such values of q allows us to easily compare our results with the renowned problem of discrete coloring.

PLANAR IMMERSIONS WITH PRESCRIBED CURL AND JACOBIAN DETERMINANT ARE UNIQUE

We prove that immersions of planar domains are uniquely specified by their Jacobian determinant, curl function and boundary values. This settles the two-dimensional version of an outstanding conjecture related to a particular grid generation method in computer graphics.

Dirac Integral Equations for Dielectric and Plasmonic Scattering

A new integral equation formulation is presented for the Maxwell transmission problem in Lipschitz domains. It builds on the Cauchy integral for the Dirac equation, is free from false eigenwavenumbers for a wider range of permittivities than other known formulations, can be used for magnetic materials, is applicable in both two and three dimensions, and does not suffer from any low-frequency breakdown. Numerical results for the two-dimensional version of the formulation, including examples featuring surface plasmon waves, demonstrate competitiveness relative to state-of-the-art integral formulations that are constrained to two dimensions. However, our Dirac integral equation performs equally well in three dimensions, as demonstrated in a companion paper.

A hybrid $$ H ^1\times H (\mathrm {curl})$$ finite element formulation for a relaxed micromorphic continuum model of antiplane shear

One approach for the simulation of metamaterials is to extend an associated continuum theory concerning its kinematic equations, and the relaxed micromorphic continuum represents such a model. It incorporates the Curl of the nonsymmetric microdistortion in the free energy function. This suggests the existence of solutions not belonging to $$ H ^1$$ H 1 , such that standard nodal $$ H ^1$$ H 1 -finite elements yield unsatisfactory convergence rates and might be incapable of finding the exact solution. Our approach is to use base functions stemming from both Hilbert spaces $$ H ^1$$ H 1 and $$ H (\mathrm {curl})$$ H ( curl ) , demonstrating the central role of such combinations for this class of problems. For simplicity, a reduced two-dimensional relaxed micromorphic continuum describing antiplane shear is introduced, preserving the main computational traits of the three-dimensional version. This model is then used for the formulation and a multi step investigation of a viable finite element solution, encompassing examinations of existence and uniqueness of both standard and mixed formulations and their respective convergence rates.