          Astron. Astrophys. 326, 433-441 (1997)

## 5. Newtonian interaction and expansion

### 5.1. Newtonian interaction

Before 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, N -body systems exist in nature under Newtonian gravitational forces, and to invoke a distribution of particles in such systems is not necessarily inconsistent. The possible infinities that appear in some expressions are only mathematical problems which are not present in nature and can be solved once we give our data the conditions that will accommodate our physical reality to a mathematical model (I mean to avoid the infinite proximity of particles by means of a cut-off, etc.). We can look at it from another point of view: a statistical thermodynamical system in equilibrium cannot achieve singular states with infinities, except as a set of zero measurements, because the probability of getting a singular state, with a very small distance between some particles, would take an extremely long time, as if approaching a singularity.

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: The interaction between pairs of particles is the Newtonian gravity force. This hypothesis obviously includes Hypothesis 2. With a Newtonian potential: . Then, We set . This leads to   which gives us a relationship among certain correlations of the distribution and C.

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 Universe

When we consider the galaxy distribution in the large-scale structure of the Universe, we must bear in mind the expansion, so: HYPOTHESIS 5: The system of particles is distributed over a space in expansion. We must use comoving coordinates in order to maintain the zero peculiar velocity as the most probable one (because the above formulation gives the probability as proportional to , where p is the momentum). We consider the proper motions, not the background: the kinetic energy derived from peculiar velocities and the comoving potential energy.

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 r or and the integral , Eq. ( 20 ) is modified by simply replacing by :       © European Southern Observatory (ESO) 1997

Online publication: October 15, 1997