Wolfram Science Group
Phoenix, AZ USA
Registered: Aug 2003
Galilean invariance means the laws of physics must be the same in every constantly moving frame. That is, it doesn't matter whether you consider a system at rest or one in which every component is moving in some direction at the same velocity. Every such frame also implies the laws are the same if merely translated or rotated in space.
In mathematical terms, if every position x is replaced by a new position x' = x + dx, whatever relations constraints or equations were obeyed before have to still be obeyed - and similarly if first derivatives of space (velocities) are replaced by a new variable adding a constant dv to all velocities in the system.
Moments refers to statistical aspects of a distribution, and is being used in a statistical mechanics sense, to describe the distribution of model elements about an expected value (positions and velocities e.g.). The first moment of any distribution is its mean, the second moment is its variance, third is called skew and fourth kurtosis. They are numbered that way because the formula for all of them is the same except for the exponent - what power the difference between each value and its overall expectation is raised to.
Mean is the average value (each value to first power aka itself divided by number of elements), variance is calculated by subtracting the mean from each element and then squaring (2nd power, 2nd moment) before averaging, etc.
As a general modeling point, one would like to get correct not only the mean values for some quantity one is modeling, but also to get the higher moments right. If particles in a fluid show some distribution of velocities, one wants to get not only the mean of that distribution but its whole shape etc. Although normally we assume the velocity distribution will follow a Boltzman distribution, as the equilibrium distribution all sorts of different initial ensembles of (positions, velocities) will tend toward, under internal collisions etc. That assumption is what generally allows us to characterise all the velocities with one number, the temperature.
As for the equation, it looks like Euler's formula, which comes from what is called the action integral. A quantity called the action is defined in a phase space of a system - how much of its energy is kinetic and how much potential at each moment in time, thought of as two axes for the system, as a simple example. The mechanical principle of least action says the equations governing the motion will be such that the action is minimized. That involves finding the minimum (or extrema, more generally) of a functional. And Euler's theorem (and equation) tells us what that extrema will be - or more strictly, tells us a necessary condition for an extrema of the action functional.
That typically gives a second order differential equation which one can solve for the equations of motion, with the various constants of integration determined from known boundary conditions etc.
But much of that standard statistical mechanics machinery is unnecessary for a typical CA fluid model, because it can be shown (at least for typical assumptions about incompressibility etc) that it is sufficient that the neighbor interaction rules conserve particle number, momentum, energy. The basic paper showing this is Cellular Automaton Fluids: Basic Theory, 1986. Another useful treatment can be found in Ilachinski's Cellular Automaton, chapter 9 "CA models of fluid dynamics", pp 463-506.
Which among other things gives a derivation of the fluid system Euler equation and the well known Navier-Stokes equation just from a micro-assumption of the Boltzman equation plus the conservation laws - to first order at least. If one is thinking of non-equilibrium velocity distributions, then admittedly things can get more complex and one may want to go back to the equations to get more exact requirements for the local interaction rules to obey.
I hope this helps. Needless to say, it helps to know classical and statistical mechanics thoroughly if one is going to rejigger the standard lattice-Boltzman models to tailor them to a particular problem.
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