Calculations for Extended Thermodynamics of dense gases up to whatever order and with all the symmetries

The 14 moments model for dense gases, introduced in the last years by Arima, Taniguchi Ruggeri, Sugiyama, is here considered. They have found the closure of the balance equations up to second order with respect to equilibrium; here the closure is found up to whatever order with respect to equilibrium, but for a more constrained system where more symmetry conditions are imposed and this in agreement with the suggestion of the kinetic theory. The results, when restricted at second order with respect to equilibrium, are the same of the previously cited model but under the further restriction of full symmetries.Comment: arXiv admin note: substantial text overlap with arXiv:1410.405

Snyder-de Sitter model from two-time physics

We show that the symplectic structure of the Snyder model on a de Sitter background can be derived from two-time physics in seven dimensions and propose a Hamiltonian for a free particle consistent with the symmetries of the model.Comment: 9 page

Big Ideas of Cryptography in Primary School

We present a learning path on cryptography for primary school students (Grade 5), which we designed and tested. The project aims to raise initial awareness of the core ideas of modern cryptography, which are fundamental concepts for becoming informed and active citizens in modern digital society. For this reason, we designed a progression of unplugged activities (sometimes integrated with task-specific, block-based programming environments) to expose students to different cryptographic techniques, where each new activity is motivated by an analysis of the criticalities encountered previously. We used unplugged activities to show that Computer Science (CS) does not necessarily imply the use of digital devices and to allow students to focus on its general scientific principles. We describe the designed learning module, discuss the main hinges in the experiment, and reflect on the lessons learned. The final evaluation showed excellent learning outcomes and high satisfaction with the activities. As cryptography uses mathematical tools (e.g., modular arithmetic, statistics), some parts of which are within reach of primary school children but rarely taught to them, it was possible to dwell on these aspects, making them experience mathematical objects in a non-standard context, and also stimulating a greater awareness of the impact of mathematics and CS in everyday life

An exact fluid model for relativistic electron beams: The many moments case

An interesting and satisfactory fluid model has been proposed in literature for the the description of relativistic electron beams. It was obtained with 14 independent variables by imposing the entropy principle and the relativity principle. Here the case is considered with an arbitrary number of independent variables, still satisfying the above mentioned two principles; these lead to conditions whose general solution is here found. We think that the results satisfy also a certain ordering with respect to a smallness parameter $\epsilon$ measuring the dispersion of the velocity about the mean; this ordering generalizes that appearing in literature for the 14 moments case

Extended Thermodynamics for Dense Gases up to Whatever Order and with Only Some Symmetries

Extended Thermodynamics of dense gases is characterized by two hierarchies of field equations, which allow one to overcome some restrictions on the generality of the previous models. This idea has been introduced by Arima, Taniguchi, Ruggeri and Sugiyama. In~the case of a 14-moment model, they have found the closure of the balance equations up to second order with respect to equilibrium. Here, the closure is obtained up to whatever order and imposing only the necessary symmetry conditions. It comes out that the first non-symmetric parts of the higher order fluxes appear only at third order with respect to equilibrium, even if Arima, Taniguchi, Ruggeri and Sugiyama found a non-symmetric part proportional to an arbitrary constant also at first order with respect to equilibrium. Consequently, this constant must be zero, as Arima, Taniguchi, Ruggeri and Sugiyama assumed in the applications and on an intuitive ground

The general exact solution for the many moments macroscopic approach to extended thermodynamics of polyatomic gases

A new model for Polyatomic and for Dense Gases has been proposed in literature in the last five years in the framework of Extended Thermodynamics. The case with an arbitrary but fixed number of moments has been recently studied, both with the kinetic approach than with the macroscopic approach; this last one is more general and includes the results of the kinetic approach only as a particular case. \\ Scope of the "closure problem" is to find the expression of some arbitrary functions which appear in the balance equations. Up to now only a recurrence procedure has been published which outlines how to find the solution of this problem with the macroscopic approach; by using this procedure, a numberable set of solutions has been found and written explicitly, while we find here the most general exact solution. It is determined except for some arbirary terms and it is interesting that these terms appear also in the 24 moments model; so we find here that they are transmitted from the model with 24 moments to those with an arbitrary number of moments, without any further arbitrary term

An Exact Solution for the Macroscopic Approach to Extended Thermodynamics of Dense Gases with Many Moments

Extended Thermodynamics of Dense Gases with an arbitrary but fixed number of moments has been recently studied in literature; the arbitrariness of the number of moments is linked to a number N and the resulting model is called an (N)−Model. As usual in Extended Thermodynamics, in the field equations some unknown functions appear; restriction on their generalities are obtained by imposing the entropy principle, the Galilean relativity principle and some symmetry conditions. The solution of these conditions is called the ”closure problem” and it has not been written explicitly because an hard notation is necessary, but it has been shown how the theory is selfgenerating in the sense that, if we know the closure of the (N) −Model, than we will be able to find that of the (N + 1) − Model. Instead of this, we find here an exact solution which holds for every number N

A kinetic type extended model for dense gases and macromolecular fluids

Extended thermodynamics is an important theory which is appreciated from mathematicians and physicists. Following its ideas and considering the macroscopic approach with suggestions from the kinetic one, we find in this paper, the solution of an interesting model: the model for dense gases and macromolecular fluids