# Total gravitational force and the classification of long range interactions

In equilibrium statistical mechanics the distinction between short and long range interactions is given by the integrability or not of the pair potential. However for what concerns only the clustering dynamics of a particle distribution under the effect of an attractive pair interaction, it seems by recent works that the distinction is given by the integrability of the pair force instead of the potential.

For and integrable pair force the aggregation dynamics in a closed volume is local and does not lead in general to a global collapse, while for forces with diverging integral over the infinite volume the aggregation dynamics is collective and leads to a global collapse of the matter density (i.e. spherical gravitational collapse).

A first important information to explore the dynamics of such systems comes from the study, in an infinite and homogeneous particle stochastic system, of the probability distribution of the total force acting on one of the particle under the effect of the others particles. Not always this quantity is well defined in the thermodynamic limit. It depends on both the large scale behavior of the pair interaction and on the large scale correlations of the particle distribution, and depending on both a regularization can be required (e.g. Jean’s swindle for the gravitational force).

In general if the pair force is integrable, in order to find the probability distribution of the total force, no regularization is required for any spatially homogeneous stochastic particle distribution. If instead the pair force has a diverging integral in the infinite volume limit, a regularization in the definition of the total force is always necessary and such force is then well defined only for “sufficiently regular” stochastic particle systems.

Given an infinite homogeneous and correlated particle distribution the study of the statistics of the total force on a system particle permits then to introduce a classification of all pair interactions from a dynamical point of view.

Moreover, as for instance in the gravitational case, the knowledge of the statistics of the fluctuation of total force permits to have a prediction of the times scales involved in numerical n-body simulations of the dynamics. This information is in general very important both to fix the different parameters of the system in a simulation and to get a prediction of the clusterization dynamics at short times.

References

[1] A. Gabrielli, F. Sylos Labini, M. Joyce, and L. Pietronero, “STATISTICAL PHYSICS FOR COSMIC STRUCTURES”, Springer Verlag Inc. (New York-Berlin, 2004).

[2] A. Gabrielli, A. P. Masucci, and F. Sylos Labini, “Gravitational force in weakly correlated particle spatial distributions”, Phys. Rev. E, 69, 031110 (2004).

[3] A. Gabrielli, “Scale invariant forces in one-dimensional shuffled lattices”, Phys. Rev. E, 72, 066113 (2005).

[4] A. Gabrielli, T. Baertschiger, M. Joyce, B. Marcos, F. Sylos Labini, “Force distribution in a randomly perturbed lattice of identical particles with 1/r2 pair interaction”, Phys. Rev. E, 74, 021110 (2006).

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