Generalized Poisson distributions for systems with two-particle interactions

In a cosmological context, observational best fits for galaxies’ distributions in the Universe have been tackled by recourse to different distribution functions. We provide here arguments favoring the formulation of a rather general distribution function (DF), of Poisson origin, describing galaxy clustering. The DF should be useful irrespective of distances or temperatures. We will be discussing distribution function for gravitational interactions.


Introduction
Probability distributions constitute the essential tool for dealing with thermal quantifiers [1,2]. We are interested here in Poisson binomial distributions [3] and their specialization, the binomial distribution [3], that in certain cases yields the Gaussian distribution. The latter can, in turn, be suitably approximated by the Poisson distribution [4].
The discrete probability distribution of a sum of independent and non identical Bernoulli trials that are not necessarily identically distributed (there are varied success probabilities) is what we call the Poisson binomial distribution [3]. The binomial distribution is a special case of the Poisson binomial distribution: all success probabilities are identical. Enjoying the status of being an exact probability distribution for any number of discrete trials, the Binomial distribution function yields the Gaussian distribution for large number of events (approximately continuous functions). For small probability-values the same distribution is approximated by the Poisson distribution [4] In a cosmological context, observational best fits for galaxies' distributions in the Universe have been tackled by recourse to different distribution functions, like the ones mentioned above [5].
Astrophysical fluid dynamics (AFD) is a special branch of astronomy that invokes fluid mechanics. AFD discusses, of course, the motion of fluids. These are composed of stars as constituents [6][7][8]. A gravitational quasi-equilibrium distribution (GQED) [6][7][8] has been derived for them, that also arises from a statistical mechanics of the cosmological many body problem [9,10]. These theories are based on the assumption that gravitational clustering evolves through a sequence of quasi-equilibrium states. This assumption allows for the local dynamical time scale to be faster than the global gravitational time scale, thus guaranteeing the stability of clusters and large structures during long cosmological times in an expanding background [6][7][8].
The probabilities P i of having i galaxies inside a randomly chosen spatial cell of volume V ('counting in cells'), and in particular the void probability P 0 , have received increasing attention as a tool to scrutinize 3-D redshift catalogs from data, and to compare them with simulations or theoretical predictions. For a large-scale galaxy distribution, the dependence of counts-in-cells upon the cell's shape has been determined and a concrete prediction done concerning the void distribution for scale invariant models [11,12]. This research involves the negative binomial distribution (NBD), a discrete probability one that models the success-number. This is done for a series of independent and identically distributed Bernoulli trials just before a specific (non-random) number of failures (denoted r) takes place.
The NBD was used in a cosmological stage [11] and later derived in [12]. This distribution function was used to study the probability of distributing N galaxies in a number of disconnected boxes. Another useful distribution is the log normal one (LND), a continuous probability distribution of a random variable whose logarithm is normally distributed [if the random variable Y is log-normally distributed, then = Z Y ln( ) has a normal distribution]. The LND was first employed in [13]. In [14], its authors investigated some of the effects of quadratic non-linearities on basic statistical properties of cosmological fluctuations, an effort being regarded as proposing a fully developed stochastic model for the distribution of galactic matter density.
The Gaussian distribution is successful in explaining linear and weak density perturbation fields. The results obtained for the gravitational quasi-equilibrium distribution (GQED) [6] and the LND distributions show that the latter should be preferred. Additional parameters seem to be needed for better fitting, though, which results in a modified log normal distribution. A comparison between the NBD and LNB distributions shows that the NBD is better for small radii cells, when the shot noise effect is stronger [5] [the shot noise describes the fluctuations of the number of particles detected due to their occurrence independently of each other. This is a consequence of discretization]. The results are similar for both distributions for a cell radius of 16h −1 Mpc [5,15,16]. h is dimensionless and equivalent to the Hubble constant H 0 , being a fundamental cosmological quantity [17]. The value of Hubble's constant is taken Due to its success for small radii, the NBD has been considered as producing better fittings than the LNB distribution, but the LND-satisfactory results found at large scales make LNB distributions a good choice as well. Among other techniques that one may mention are correlation functions or power spectrum ones [18,19].
The partition function and the corresponding thermodynamics for clustering of galaxies and the gravitational structure formations have also been discussed i) with modified forms for gravity and ii) by considering the effect of the cosmological constant [20][21][22][23][24][25].
In the present work we will discuss i) particular clustering parameters of the form = , where x is a variable involving a ratio between potential gravitation energy and kinetic one, and ii) general distribution functions beyond the Gaussian distribution, that use both the thermodynamic fluctuations discussed in [26] and a Poisson formula with mean square fluctuations in the number of particles [1]. Our present distribution works well for a Boltzmann gas. In Fermi or Bose gases, the probability of fluctuations becomes meaningful for small volumes (because of quantum fluctuations). These were discussed for uniform and ideal gases in [1].
This effort constitutes an attempt to incorporate the effect of interactions into the Poisson distribution (PD). This is done by introducing into the PD ad-hoc probability factors which depend on two particle correlation functions.
We use the modulus of the interaction potential, so that the distribution also works for regions of both repulsive and attractive interactions. We will be discussing also the so called void-distribution function [see the work Ronconi and Marulli [27]] that seems to work for r → 0 and ∞.

PD Generalities
For a uniform gas, the probability that a particle be found in a given volume V is V/V 0 , where V 0 and N 0 are the total volume and number of particles. Also, the probability of not encountering such particle in the volume V is The probability for N particles to be simultaneously present is As our essential present step, we bring in now ad hoc 'probability factors' f that are introduced into the above equation in the fashion We denote the interaction between the particles as U, while we call T the kinetic energy for the system of particles. We take =f 1 U T | | as our probability factor f. Thus we get and setting unity in the form N/N we write Appeal now to the well known relation which constitutes an approximate distribution function.

General distribution
Using [1] we concoct a correlation function ν(r) for the interaction energy |U| between two particles of the form V¯i s the mean number density. Where |U| is the magnitude of the two particle interaction energy. For a homogeneous isotropic medium, the correlation function depends on the distance between the particles and, as this distance tends to infinity, the fluctuations at the two positions become statistically independent and the correlation functions tends to zero [1]. This correlation can also be taken as our former probability factor f. Thus, we make the bold assumption of claiming that the probability factor the in presence of interactions is which reduces itself to the Boltzmann factor as a proof of its reasonableness. Thus our general distribution function becomes Here we introduce a new quantity B, defined below, to write The void probability can be obtained by setting N = 0. Thus we write = --P exp , 3.10 The distribution function can also be extended to multicomponent systems with N 1 , N 2 , N 3 ,K.N i number of particles of different types simultaneously in volumes Approximating the exponential up to first order we obtain an approximation b to B [6] so that (3.17) is a first approximation to (3.15) that could serve to dealing with gravitational clustering of galaxies in circumstances for which U/T < < 1, i.e., the above called GQED approach. This is a dilute fluid approximation. Therefore, (3.15) can be considered as a generalization of the GQED approach.

Box-probabilities for gravitational interactions A well known distribution function is of the form [6]
= -+ - This distribution depends on a parameter b, which is interpreted as being the ratio =b W K 2 of the gravitational energy W to the kinetic energy K [6]. The parameter b is supposed to contain all relevant information on the physical and statistical properties of this distribution [28]. As explained in [28], the dependence of b on the volume V can be empirically obtained from the variance of the number of counts in a cell of volume V, mean N and two point correlation function ν. , where x = βρT −3 , with b = 0 for the ideal gas and b = 1 for a clustered system. The same form of b has been derived in [29]. The effect of higher order corrections on the distribution function has been studied in [30]. The critical values of b for extended structures have been studied [31]. In the present case, we take = . Where β 0 = G 3 m 6 we have used the scale invariances ρ → λ −3 ρ, T → λ −1 T, and r → λr.
Thus, we cast the probability as Remind that we attempt to incorporate the effect of gravitational interactions into the Poisson distribution (PD). This is accomplished by introducing into the PD an ad-hoc probability factor (PF) which depend on two particle correlation functions. A modulus of energies' ratio entering it guarantees the adequacy of our PF.
or, in terms of x, as above,

Conclusion
This paper is an attempt to develop a Poisson distribution function for systems involving two-particle interactions. We employed the Poisson's formula and applied it to a system where the probability of finding particles in a certain volume depends on the interactions between the particles.  A rather bold guess regarding the probabilities' forms is advanced. It refers to the ratio of the interaction energy and the kinetic one, which depends only on the temperature. The ratio lies in the interval [0, 1]. On closely examining the probability factor |U|/T, one gathers that it might be regarded as an approximation to a more refined form. In view of the virial theorem, the probability factor can also be expressed in terms of the reciprocal of the virial ratio. This was not yet sufficient for us so that we proceeded towards a bolder guess. We proposed that the probability factor should be expressed in terms of the correlation function. Further, we also took the modulus of the two particle interaction potential, so as to include both the attractive and the repulsive interactions in the probability distribution function. We applied this bolder guess to the gravitational potential and evaluated the distribution function for a gravitating gas, obtaining a rather nice distribution function, that was compared to the gravitational quasi-equilibrium distribution (GQED) [6].
Our void probability was found to be exactly the same as the void probability yielded by the GQED. We also conjectured that our distribution may also work for multi-component systems.
It follows from our work that the GQED could be viewed as a reasonable approximation to that of our present approach. We introduced a general clustering parameter B, and further found that the associated GQED  The graph shows marked deviation for > N 1 .
clustering parameter constitutes a first approximation to B. Our present treatment needs still to be compared with observational and experimental data. Working is going on to that effect.

Data availability statement
All data that support the findings of this study are included within the article (and any supplementary files).