5.5 Macroscopic Quantities

While the description using Boltzmann’s equation is already a reduction with regard to the degrees of freedom, it is still a statistical description on a microscopic level. It is important to be able to link this modelling on a microscopic level to macroscopic quantities, which are accessible to measurements. The statistical nature already offers a means in the form of expectation values (Definition 101), which also allows to compact the statistics of the micro scale to simple quantities of the macro scale. Thus, a macroscopic quantity X on the manifold 𝔐  (Definition 35) can be obtained from the microscopic description using ρ on the phase space ∗
TM by


⟨X ⟩ =   X ρdp.                               (5.39)
This resembles a projection of the form
π : ρ ∈ 𝔽(𝔗 ∗𝔐 ) → 𝒯 rs (𝔐 ),                         (5 .40)
which again reveals a fiber bundle (Definition 40) structure on the base manifold 𝔐   (Definition 35), indicating the powerful versatility of the fiber bundle abstraction.

An important feature of this prescription is that this procedure can not only be applied to already obtained solutions of ρ as a post processing step, but can also be applied in order to derive macroscopic equations from Boltzmann’s equation. Several differing procedures are available to provide such a mapping resulting in differing macroscopic equations, ranging from convection-diffusion, also called drift-diffusion, to the Euler equations or the Navier-Stokes equations. This mapping may either be viewed as a defining derivation or as a link between two different descriptions, which have been derived independently or have even been postulated by empiric means.

A basic method uses a collection of weight functions χ(p)  to perform the contraction by integrating (Definition 94) over sub-manifold slices formed by the momentum coordinates p . It thus bears a certain resemblance to the procedure involved in the derivation of the BBGKY, except that the integration is carried out only with respect to the momentum coordinate using the distribution as measure, while preserving the spatial components.