Identify the five slow fields of a monatomic gas

A collision invariant is a function whose sum over the two particles is unchanged by every allowed encounter. For elastic equal-mass particles, constants count particles, linear functions of velocity count momentum, and the squared speed counts kinetic energy. Symmetrizing the collision integral shows that its moment against any such invariant vanishes. This cancellation is local in position: an ordinary binary collision transfers velocity-space population without moving particles across a macroscopic spatial distance.

Taking these moments of the full kinetic equation therefore leaves spatial fluxes and external-force terms, but no collisional source of mass, momentum, or energy in the isolated gas. Internal exchanges between species require more careful bookkeeping; one species’ momentum can change while total momentum stays fixed. Reactions and inelastic internal excitation alter which individual moments are protected. Our five-field account assumes one nonreacting classical species with translational energy as its only internal energy.

Zero collision production requires the logarithm of a positive equilibrium distribution to be a linear combination of collision invariants. Complete the velocity square in that expression. Normalizability fixes the coefficient of squared speed to be negative, while the linear term shifts the Gaussian’s center to the local flow velocity. The multiplicative constant fixes number density. In three dimensions the remaining width is conventionally expressed through temperature and Boltzmann’s constant.

The resulting Maxwellian has equal random variances in all directions, isotropic pressure, and zero heat flux in the local rest frame. It maximizes kinetic entropy among distributions with the same five moments. The gas may have a large mean speed in a laboratory frame without changing these local random moments. Galilean translation shifts the center of the velocity distribution; it does not broaden the distribution or heat the gas by itself.

Each spatial point can have its own density, velocity, and temperature while its collision operator vanishes. Streaming between neighboring points generally makes the full kinetic left-hand side nonzero, so such a local Maxwellian is the leading approximation in a slowly varying flow. Taking its moments gives Euler-level fluxes. Viscous stress and heat conduction require departures from this zeroth-order shape; adding those fluxes without changing the closure would be inconsistent.

The heat lab illustrates slow variation after a constitutive closure has already been chosen. Reset and set spatial mode to 1 and diffusivity to 0.5, then Step once; predict amplitude exp(−0.25). Reset and set mode to 2 and diffusivity to 0.5, then Step once; predict exp(−1). These are exact periodic heat-equation modes. They display a continuum consequence of local transport, not collision invariants, a Maxwellian velocity distribution, or a derivation of the heat equation.