Mixing transformations of N particles conserving almost all classical integrals

Abstract

We have shown hot to treat consistently a set of rapidly mixing particles. It is possible to devise various dissipative transformations and then correct them to conserve the mass, centers of mass and velocity, and especially the total angular momentum. Furthermore, for simple linear friction laws it is possible to specify in advance the energy dissipation. In that case the conservation of the other integrals constraints the energy loss not exceed a maximum value.

Among all the possible corrective transformations that we have described above, only a small part may be applicable in actual physical systems. In real systems further constraints restrict the possible mixing and thus the dissipation rate, e.g., when a flow expands its internal viscosity is much smaller than when it contracts. Only studies with specific problems will allow us to choose the “right” way to model dissipation and mixing. At least the formulaiton here is general and consistent with the most fundamental constraints that fluids are though to follow. For example in a schock, another instance where the hydrodynamic description fails, the shock conditions require precisely to conserve mass and momentum. Here not only these quantities, but also the angular momentum are automatically conserved.

Presently we are experimenting with these transformations. We have implemented the correcting transformations in simple N-body systems (N ≲ 100, and in large-scale N-body simulations (N ≈ 2 − 10⁵). It is straightforward to check that the integrals are indeed conserved exactly. Applications to disc galaxies are investigated in which self-gravity is calculated with a Particle-Mesh method in a 3D polar grid (Pfenniger & Friedli 1991, 1993). This approach allows to simulate gas at large scale much more efficiently that traditional hydrodynamical codes, and is not less realistic with respect to the ISM that the hydrodynamical models since the ISM is highly inhomogeneous at small scale, and no global equation of state is known. In comparison with the SPH technique, 10 times more particles runs 10 times faster with the law discussed in Sect. 3.2, essentially because there is no need to find the neares neighbours. Instead all the particles within the same grid cells used for the gravitation calculation mix their momentum at a specified rate.