4. A force equation for homogeneous, isotropic distributions in Boltzmann equilibrium
In this section, I derive the force equation and then introduce a Newtonian form for the interaction.
The only trick is to apply the operator over expression ( 11 ) and to develop the expression
HYPOTHESIS 2: The interaction among particles is between pairs of particles, and is proportional to their masses, as a function of their distance (it is a central force). By this hypothesis, the total potential energy is the sum of the potential energies between pairs of particles ( ). Hence,
HYPOTHESIS 3: We assume homogeneity and isotropy. This means that only depends on distance and . The operator can be expanded and thus (I note instead of to avoid confusion; the same with and in the following equations)
This is the force equation. It gives us a relationship between different correlation functions in a distribution that follows our hypothesis and the force that is represented by means of v.
© European Southern Observatory (ESO) 1997
Online publication: October 15, 1997