Normalize the distribution before taking moments

Use f(x,v,t) so that f times a small position volume and a small velocity volume is the expected number of particles in that cell. Integrating over velocity gives the local number density, and integrating over both spaces gives the total particle number. A normalized probability density would instead integrate to one. These are related conventions, but their formulas for conditional averages have different prefactors. Always state which object a symbol represents before applying a familiar integral.

Velocity and momentum coordinates also use different densities. For particles of one fixed mass, momentum equals mass times velocity, so the three-dimensional measure acquires a factor m³. Equality of particle counts then requires the velocity density to be m³ times the momentum density evaluated at the corresponding momentum. A missing Jacobian silently alters every moment. Quantum occupation numbers using phase-space cells divided by Planck’s constant require another convention, which we do not adopt here.

The local mean velocity is the first velocity moment divided by number density. Subtract it from each velocity to define the peculiar velocity. Its first moment vanishes by construction. The second peculiar moment defines the kinetic pressure tensor: particles crossing an imagined surface carry fluctuating momentum relative to the local flow. Its diagonal components need not be equal away from equilibrium, so a general distribution cannot be summarized by one scalar temperature without discarding information.

For a classical monatomic gas, one may define kinetic temperature from the trace of this tensor even before full equilibrium. The resulting scalar describes total random translational energy, but isotropic pressure and a Maxwellian velocity shape are additional claims. Two counterstreaming narrow beams can have the same density and scalar kinetic temperature as a thermal gas while differing strongly in higher moments. Moment agreement alone therefore does not establish local equilibrium or justify a fluid closure.

The full momentum flux consists of bulk transport plus the pressure tensor. Likewise, total kinetic energy splits into bulk kinetic energy and random internal energy. The heat flux uses a third peculiar moment, not the total laboratory energy flux. This distinction lets conservation equations describe moving gases without treating an ordinary advected energy current as heat conduction. Closing those equations requires relating these higher moments to density, velocity, temperature, and possibly their gradients.

The probability lab gives a finite counting illustration. Reset and set success probability to 0.75 and trials to 8; predict mean count 6 and variance 1.5. Reset and set probability to 0.5 and trials to 8; predict mean 4 and variance 2. These are exact binomial probabilities for independent trials. They illustrate normalization and moments, not a measured molecular velocity distribution, collision correlations, or an anisotropic pressure tensor.