## 5. Newtonian interaction and expansion## 5.1. Newtonian interactionBefore proceeding, I will comment on some aspects regarding the Newtonian case in particular. Certain problems are associated with the application of principles such as thermodynamics and statistical mechanics in groups of particles with -type force (Taff 1985): some divergences are found, and there is non-saturation of gravitational forces (Levy-Leblond 1969). Some authors take the view that there can be no rigorous basis for applying statistical mechanics in such a system (Fisher & Ruelle 1966). This result is not a consequence of the -type force but rather of its unshielded character (see Dyson & Lenard 1967 for a discussion of the electrostatic case). In any event, one avoids the non-self-consistency of the problem by
truncating the integral limits at a finite radius, or by cancelling
the correlation functions that fall under a given lower unit, assuming
points with negligible volume as is the case in real physical
problems. In my opinion, Expression (
19 ) has been obtained regardless of the force
type, so it possesses a general validity. Now, when we introduce a
Newtonian gravitational force the expression continues to be valid. If
an infinity appears in the next expression it is only a question of
truncating the integrals or selecting the best correlation function
that does not produce divergences. When we take the cut-off, we
neglect the probabilities near the singularity, a set with dimensions
greater than zero but small enough. The introduction of cut-offs will
make the results dependent upon the details of the regularization, so
the selection of the cut-offs must have a physical basis.
HYPOTHESIS 4: We set . This leads to which gives us a relationship among certain correlations of the
distribution and The proof that the system is valid for achieving thermodynamic equilibrium can be found in Lieb & Lebowitz (1973), where a general Coulombian system is considered. Nevertheless, this does not imply that all Newtonian gravitational systems are in equilibrium. ## 5.2. Effects of the expansion of the UniverseWhen we consider the galaxy distribution in the large-scale
structure of the Universe, we must bear in mind the expansion, so:
HYPOTHESIS 5: It is derived in Saslaw & Fang (1996) that consideration of the
expansion is equivalent to taking into account the gravitational
effects of the local fluctuating part of the density field when we use
comoving coordinates, i.e. we should substract the mean density from
the density field. Since the mean density does not depend on © European Southern Observatory (ESO) 1997 Online publication: October 15, 1997 |