Galilean invariance

Recall that in the continuum case we derived the Navier-Stokes equations (9.16) by allowing for the possibility of having nonzero viscous terms in the form for the momentum tensor appearing in Euler s equation (9.9). While their LG analogs may be derived in essentially the same manner, however, the lack of Galilean invariance tend to make the calculations more involved. We will outline the procedure below. 

Recovery of Galilean Invariance There is a simple trick that is often used to transform equation 9.107 into the conventional Navier-Stokes equations. 

T. Ihle and D. M. Kroll, Stochastic rotation dynamics a Galilean-invariant mesoscopic model for fluid flow, Phys. Rev. E 63, 020201(R) (2001). 

Actual nuclear forces are of a finite range. These are, for example, Gogny forces [40] representing more realistic approximation of actual nuclear forces than Skyrme approximation. Gogny interaction has no any velocity dependence and fulfills the Galilean invariance. Instead, two-body Skyrme interaction depends on relative velocities k = j2i (Vi — V2), which just simulates the finite range effects [20]. 

It should be noted that the introduction of fluid/solid interaction has no effect on the macroscopic equations since F2jt exists only at the fluid/solid interface. The relaxation time, tk, is estimated based on the viscosity and mass density representations given by Eqs. (20) and (21) of the Mi component and is detailed in Ref. [37, 43, 44], This model has been shown to satisfy Galilean invariance.44 Furthermore, in this interparticle potential model, the separation of a two-phase fluid into its components is automatic.37... 

This Hamiltonian is Galilean invariant, as is appropriate for chemical phenomena, rather than Lorentz invariant. 

Greenspan [31] outline the transformation of the Eulerian equations governing the motion of an incompressible viscous fluid from an inertial to a rotational frame. The transformation of the Navier-Stokes equations simply results in adding the artificial forces in the momentum balance. The additional equations are apparently not changed as the substantial derivative of scalar functions are Galilean invariant so the form of the terms do not change. 

Lallemand, P. Luo, L.-S. 2003 Theory of the lattice Boltzmann method dispersion, dissipation, isotropy, Galilean invariance, and stability. Physical Review E 68,036706. 

Galilean invariance (Rothman Zalesky, 1997) is a fundamental tenet of Newtonian mechanics. It is invariance under the transformation x = x - wt, where w is the constant velocity of a moving frame of reference, and embodies the concept that only the relative velocities and positions of two bodies determine their interaction. Galilean invariance is lost in lattice gas simulations because every particle has only one possible speed. This loss is an artifact that can be eliminated for incompressible fluids by re-scaling the velocity. According to Boghosian (1993), more sophisticated lattice gas models overcome this problem. Appropriate application of lattice gas models also requires certain restrictions on the mean free path of a particle (Rothman, 1988). 

J. -M. L vy-Leblond, Galilei group and galilean invariance, in E. M. Loebl (Ed.), Group Theory and Its Applications, volume II, Academic Press, New York, 1971, p. 221. 

The conservation of the total momentum is a direct result of the symmetric form of the various forces, thus DPD may be viewed as an isotropic Galilean-invariant thermostat which preserves hydrodynamics [8]. The ergodicity of DPD (i.e. that pp is the unique measure sampled by the dynamics) has only been demonstrated in the case of a high density of particles in one dimension [334]. 

Inamuro, T., N. Konishi, and F. Ogino. A Galilean Invariant Model of the Lattice Boltzmann Method for Multiphase Huid Flows Using Free-Energy Approach. Comput. Phys. Commun. 129 32-45 (2000). 

Lallemand, P. and L.-S. Luo, Theory of the Lattice Boltzmann Method Dispersion, Isotropy, Galilean Invariance, and Stability. Phys. Rev. E 61 6546-6562 (2000). 

Speziale, C.G. (1985). Galilean invariance of subgrid scale models in the large-eddy simulation of turbulence. Journal of Fluid Mechanics 156 55-62. 

Since the partide collisions are localized, different regions of configuration space can be advanced concurrently, so that the LGA-based method lends itself readily to parallelization. Although this approach makes simulations much more effi-dent than continuum methods, it was found that LGA suffers horn several native defects lattice artifacts, lack of Galilean invariance, and so on. 

The quasi-two-body t-matrices of either the Watson or KMT optical potentials are to be evaluated in the proton-nucleus COM system. For nonrelativistic theories the r-matrix element is Galilean invariant and can be immediately obtained from the r-matrix in the proton-nucleon COM system [see eq. (3.9)]. For relativistic momenta, however, Lorentz invariance of flux requires that the quantity... 

A very important condition comes from Galilean invariance. Let us look at a system from the point of view of a moving reference frame whose origin is given by x t). The density seen from this moving frame is simply the density of the reference frame, but shifted by x t)... 

The function ITxc is a pressure-like scalar memory function of two variables. In practice, ITxc is fully determined by requiring it to reproduce the scalar hnear response of the homogeneous electron gas. Expression (4.47) is clearly non-local in the time-domain but still local in the spatial coordinates. From the previous considerations it is clear that it must violate Galilean invariance. To correct this problem we use a concept borrowed from hydrodynamics. It is assumed that, in the electron liquid, memory resides not with each fixed point r, but rather within each separate fluid element . Thus the element which arrives at location r at time t remembers what happened to it at earlier times t when it was at locations different from its present... 

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