Statistical Properties of the Gravitational Force in a Random Homogeneous Medium Constantin Payerne

Statistical properties of the gravitational force in a random homogeneous medium Constantin Payerne

To cite this version:

Constantin Payerne. Statistical properties of the gravitational force in a random homogeneous medium. [Research Report] Université Grenoble-alpes; Laboratoire de Physique et de Modélisation des Milieux Condensés. 2020. hal-02551263

HAL Id: hal-02551263 https://hal.archives-ouvertes.fr/hal-02551263 Submitted on 22 Apr 2020

HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Statistical properties of the gravitational force in a random homogeneous medium

Constantin PAYERNE, UniversitéGrenoble-Alpes, 2nd year of Master’s degree in Subatomic Physics and Cosmology Supervisor : Vincent ROSSETTO, Laboratoire de Physique et de Modélisationdes Milieux Condensés

April 22, 2020

Abstract

We discuss the statistical distribution of the gravitational (Newtonian) force exerted on a test particle in a infinite random and homogeneous gas of non-correlated particles (stars, galaxies, ...) where the first configuration of particles in space is a Poisson distribution. The exact solution is known as the Holtsmark distribution at the limit of infinite system corresponding to the number of particle N within the volume and the volume go to infinity. The statistical behavior of the gravitational force for scale comparable to the inter-distance particle can be analyzed through the combination of the n-th nearest neighbor particle contribution to the total gravitational force, which can be derived from the joint probability density of location for a set of N particles. We investigate two independent approaches to derive the joint probability density of location for a set of N neighbors using integral forms and order statistics to give a general expression of such probability distribution with generalized dimension of space d.

I. INTRODUCTION corresponding distribution of force can be simply derived.

The knowledge of the evolution and the statistical proper- II. DENSITY AS A STOCHASTIC VARIABLE ties of the gravitational field in a infinite self-gravitating system with Poisson-like spatial distribution of masses We consider a d-volume vd filled with N particles. The (stars, galaxies, ...) presents an interesting and widely local density can be assimilated to a random variable with investigated subject of research. The basic mechanism for specific probability distribution. We note the density at the emergence of coherent structure in gravitational N- location x as the random stochastic variableρ ˆ(x) given body system with long-range interacting sources from an by the number of particle within the elementary volume initial spatial and mass distribution, and subject to small dV = dx1 × ... × dxd. When each particle with index i is fluctuations of density still presents open problems in stel- at location xi, the stochastic density can be expressed by lar dynamics, in cosmological N-body simulations. the sum: N X ρˆ(x) = δ(d)(x − x ) (1) We recall the main exact result giving the probability i i=1 density function of the gravitational force exerted on an (d) arbitrary particle within a homogeneous isotropic gas of Where δ is the Dirac delta distribution in dimension d. non-correlated particles, known as the Holtsmark distri- The local average density ρ(x) is simply given by the en- bution, first introduced by Chandrasekhar [2]. Additional semble average h·i over all the configurations of particles in information can be obtained by considering order statistics the volume vd: such as the contribution of a single n-th nearest neighbor N X (d) particle to the total gravitational force exerted on a test ρ(x) = hδ (x − xi)i (2) particle. The nearest neighbor statistical properties is re- i=1 lated to the two point auto-correlation function, which has Giving f(x ) the probability density function for finding an important status in astrophysics and cosmology. To i the particle denoted by i within the elementary volume dV evaluate each neighbor’s contribution, it requires to get a around x , the ensemble average is given by: general expression of the joint probability density function i to find the set of N nearest neighbors at respective loca- Z hδ(d)(x − x )i = ddx f(x )δ(d)(x − x ) (3) tions with general dimension of space d, from which the i i i i vd

1 When particles are non-correlated, the probability for oc- to the probability density function W (F) to find the grav- cupying the volume dV does not depend on position. Then itational force exerted on the test particle comprised in a 3 f(xi) is a constant over space, and the ensemble average small volume d F around F due to all the N particles in (d) of the Dirac distribution results in hδ (x − xi)i = 1/vd. a sphere centered on it. Such probability can be expressed The local average density is deduced by the sum over the as: N indexes i giving ρ0(x) = ρ0 = N/vd. Z Z N ! 3N X W (F) = ... dx f(x1, ..., xN )δ F − Fi (5) II.a. The Poisson point process V V i The probability q for a given particle for being in a elemen- where f(x , ..., x ) is the joint probability density function tary volume dV within the total volume v is q = dV/v . 1 N d d to find the N particles at location r , ..., r . We consider the random variable N as the number of par- 1 N ticles within the elementary volume dV . When particles are non-correlated (independent), N follows a binomial law III.a. The Holtsmark distribution with parameter q: We present the steps to reach the expression of the proba- N! bility density function of force F for non-correlated parti- P(N = n) = qn(1 − q)N−n n!(N − n)! cles with equal mass m. In this case, the joint probability function f is the product of the constant probability den- The Poisson limit Theorem is valid when the total number sity function for each particle given by 1/V . The proba- of particles N n and the probability q 1. We consider bility that the n particle occupies the volume dx3N is: : the number of particle N and the volume vd going to infinity, as the density ρ keeps constant. Noting the average 3N 0 3N dx f(x1, ..., xN )dx = (6) of the binomial law as λ = Nq = ρ0dV , the Poisson limit V N theorem gives the distribution, when λ 1: The distribution of force is then given by the integral over λn λn N volumes: lim P(N = n) = exp[−λ] ≈ (1 − λ) N,v →∞ d n! n! N ! Z Z dx3N X W (F) = ... δ F − F (7) giving P(0) and P(1) much higher than P(n > 1) which is H V N i n V V i proportional to (ρ0dV ) . The stochastic local densityρ ˆ(x) verifies the probability distribution: By giving the expression of WH as a inverse Fourier trans- 1 form, it gives: with probability ρ0dV ρˆ(x) = dV 0 with probability 1 − ρ0dV Z 1 3 WH (F) = 3 d kA(k) exp(−ik · F) (8) Either the volume is occupied by a particle with probabil- (2π) Ω(k) ity ρ0dV or it is empty with the complementary probability 1 − ρ0dV . The average gives the density hρˆ(x)i = ρ0. where A(k) is given by the expression:

Z 3 N III. THE PROBABILITY DISTRIBUTION OF d x A(k) = exp(ik · Fi(x)) (9) FORCE V V Z ρ0V ρ0 3 In dimension d = 3, we note v3 = V . We consider a test = 1 − d x[1 − exp(ik · Fi(x))] (10) particle of mass m located at x in a gas of identical parti- ρ0V V cles of mass m denoted by i. The total gravitational force By using the definition of the exponential function, for infi- F exerted on the test particle by its neighborhood can be nite volume v , infinite number of particle N. and constant obtained by the vectorial sum over all the particle contri- d density ρ0, we get: butions at respective location xi: N N 2 N ρ0C(k) X Gm X A(k) = lim 1 − = exp(−ρ0C(k)) (11) F(x) = (x − x) = F (4) N→∞ N |x − x|3 i i i=1 i i=1 with: Z with the gravitational constant G = 6, 674 × 3 10−11 m3.kg−1.s−2. We can give a general expression C(k) = d x[1 − exp(ik · Fi(x))] (12) V

2 By considering an arbitrary axis given by the vector k, The first two term of the sum are reported in the case of integrating over all the directions gives: the regime β → 0: Z ∞ 4 2 10 2 3/2 du 2 4 C(k) = 2π(kGm ) [u − sin(u)] (13) WH (F ) ≈ β − Γ β (23) 5/2 3πF 9πF 3 0 u 0 0 4 = (2πkGm2)3/2 (14) For F → ∞, we now use the Taylor expansion of the expo- 15 nential function (142) giving: By using the equation (143): 2 X (−1)n Z ∞ W (F ) = β−(3n/2+1) x3n/2+1 sin xdx √ H πF n! Z ∞ du Z ∞ sin(u) 4 2π 0 n=0 0 [u − sin(u)] = du = (15) 5/2 3/2 (24) 0 u 0 u 15 giving the explicit formula for WH (F ) using (143): The Holtsmark distribution is given by the integral: 2 X (−1)n+1 3πn W (F ) = sin × (25) Z H πF n! 4 1 3 4ρ0 2 3/2 0 n=0 WH (F) = 3 d k exp −ik · F − (2πkGm ) (2π) Ω(k) 15 3n Γ + 2 β−(3n/2+1) (26) (16) 2 By considering, as before, an arbitrary axis oriented along F giving k · F = kF cos θ, integrating over the directions reporting the two first non-zero terms for large F : and radial variable k gives the expression: 1/2 15 2 24 W (F ) ≈ β−5/2 − β−4 (27) Z ∞ H 1 h 3/2i 8F0 π πF0 WH (F) = 2 3 dx(x sin x) exp −(x/β) (17) 2π F 0 IV. FIRST NEAREST NEIGHBOR The Holtsmark distribution [1],[2] for the modulus F = |F| (represented in Figure 5) is given by integrating overall the We investigate the statistical properties of the first nearest directions (multiplying by 4πF 2): neighbor of a random particle in a gas with Poisson-like Z ∞ distribution of particles in general dimension d. We first 2 h 3/2i WH (F ) = dx(x sin x) exp −(x/β) (18) consider a homogeneous and isotropic gas of particle, and πF 0 then we treat the Poisson-like case. where β = F/F0 and the typical force F0 defined as: IV.a. Auto-correlation function 2/3 2 4 Gm we define the reduced correlation function ξ as the ensem- F0 = 2π (19) 15 λ2 ble average [1]:

−1/3 ξ(r) = hδ(x1)δ(x2)i (28) with λ = ρ0 the length measure associated to the average inter-particle separation. The average force is given by where r = |x1 − x2| and δ(x) = (ˆρ(x) − ρ0)/ρ0 is the the integral: reduced density perturbation over space. the radial depen- Z ∞ dence ξ(x) = ξ(r) is due to the assumption of homogeneous 2 4F0 1 hF i = F βWH (β)dβ = Γ ≈ 3.412F0 (20) isotropic gas of particles. By using the definition of the lo- 0 π 3 0 cal density in (1) and the joint probability density in (6), The moment of order 2 is not finite. We now evaluates two the ensemble average hρˆ(x1)ˆρ(x2)i can be expressed after specific regimes; for F → 0, it is required to develop W in development: a series by using the the Taylor expansion of sin(x): N X hρˆ(x )ˆρ(x )i = hδ(d)(x − x )δ(d)(x − x )i (29) 2 X (−1)n Z ∞ h i 1 2 1 i 2 j W (F ) = x2n+1 exp −(x/β)3/2 dx i,j=1 H πβF (2n + 1)! 0 n=0 0 N N (21) X X = fi,i(x1, x2) + fi,j(x1, x2) (30) Recognizing the Gamma-Euler function in (138), the sum i=1 i6=j=1 becomes: where fi,j(x1, x2) is the joint probability density function 4 X (−1)n 2n + 3 W (F ) = Γ β2n+2 (22) to find respectively the i−th and the j−th particle at lo- H 3πF (2n + 1)! 3/2 0 n=0 cation x1 and x2. In the case of a random homogeneous

3 media, the marginal density fi(x) = 1/vd. Then, without IV.b. Density of location for the nearest neighbor loss of generality we can write: The probability density function w1(r) to find the nearest 1 neighbor particle at a distance comprises between r and fi,j(x1, x2) = wi,j(x1|x2) (31) vd r + dr from an arbitrary particle is given by the ensemble average [2]: where wi,j(x1|x2) is the conditional probability to find the i−th particle at location x knowing that the j−th particle * N N + 1 X Y Z r is at location x2. For homogeneous and isotropic gas, prob- w1(r) = δ(r − rj) 1 − δ(x − rk)dx 0 ability density function fi,j does not depend on indexes i j=2 k6=(i,j) and j when i 6= j, then we have: (38) where rj is the distance of the j-th particle around the cen- (d) δ (|x1 − x2|) if i = j ter particle. The nearest neighbor density function takes wi,j(x1|x2) = w(|x1 − x2|) if i 6= j the form: Z r Then we can derive: d−1 w1(r) = 1 − w1(u)du G(r)αdr (39) (d) N(N − 1) 0 hρ(x1)ρ(x2)i = ρ0δ (r) + w(r) (32) vd The general solution of function defined such that: ≈ ρ δ(d)(r) + Nρ w(r) (33) 0 0 Z ∞ The function w is the probability density to find a given g(x) = f(x) g(u)du (40) x particle within an elementary volume at a distance r and verifies the normalization: is given by, with K an appropriated constant: Z Z x ddxw(|x|) = 1 (34) g(x) = Kf(x) exp − f(u)du (41) vd 0

d d The correlation function ξ(r) expresses as: giving N(r) = ρ0αdr /d = (r/λd) the average number of particles within the d-volume or radius r with the typical δ(d)(r) N ξ(r) = + w(r) − 1 (35) measure of length λd: ρ0 ρ0 1/d Then, the w function reaches the expression in term of ξ: d λd = (42) ρ0αd 1 w(r) = [1 + ξ(r)] (36) vd we can write the density w1(r) after normalization:

The function ξ appears as a probability excess. It charac- 0 −N(r)+R r N 0(x)ξ(x)dx w1(r) = N (r)(1 + ξ(r))e 0 (43) terises a deviation to the probability density function 1/vd and following this convention, we have three possible cases: The assumption of a Poisson distribution of particle means particles are uncorrelated, such that the auto-correlation  > 0 particles tend to attract each other  function gets ξ(r) = 0 if r 6= 0. we can write the density ξ(r): = 0 particles are uncorrelated w (r) as:  < 0 particles tend to repel each other 1 0 0 and the ξ function verifies the integration: w1(r) = N (r) exp[−N(r)] = (1 − exp[−N(r)]) (44) 1 Z In dimension d = 3, we get naturally: ξ(r)ddx = 0 (37) vd vd 3 2 4πρ0r w1(r) = 4πρ0r exp − (45) We now get the probability w(r) to find a given particle 3 at distance r knowing a particle is at location r = 0. The probability density function G to find a particle at location Giving the average distance hri1,d of the nearest neighbor r regardless is index is the sum over all the particles in the particle as the integral: gas of w (except the center particle), such that: Z ∞ d + 1 hri1,d = w1(r)rdr = λdΓ (46) N−1 d X 0 G(r) = w(r) ≈ ρ0[1 + ξ(r)] −1/3 ≈ 0.554ρ0 (for d = 3) (47) i=1

4 2 and the average square distance hr i1,d: with a = 1 + 1/d > 1. If x is smaller enough, we can consider the first term of the sum: 2 2 d + 2 d + 1 d hr i1,d = [λd] Γ (48) hri? ≈ λ Γ − u1+1/d exp[u ] (59) d 1,d d d d + 1 ? ? −2/3 ≈ 0.347ρ0 (for d = 3) (49) By developing the exponential term at first order t, it re- mains: The dispersion σ is obtained by combining previous re- 1,d d + 1 d d + 1 sults: hri? ≈ λ Γ − u1+1/d + u Γ 1,d d d d + 1 ? ? d 1/2 (60) d + 2 2 d + 1 σ1,d = λd Γ − Γ (50) 2 d d 1+1/d ≈ hri1,d + u?hri1,d − λd u? (61) −1/3 d + 1 ≈ 0.202ρ0 (for d = 3) (51) We introduce the volume fraction Φ = ρ0v(r?/2) which is the N-sum of each volume occupied by sphere of radius IV.c. Density of location for a particle with finite size a = r?/2 divided by the volume V , it gives: We now discuss the statistical properties of the nearest ? d d d+1 1+1/d neighbor particle when it has a finite size. For particle hri = hri1,d 1 + 2 Φ − 2 λdΦ (62) 1,d d + 1 with radius a, the probability density function to find the nearest neighbour at a distance r has to include a prohib- We note that the modified average distance in the presence of finite size particles is proportional to the average ited area of size r? = 2a, then: distance calculated with point-like particle with an addi- Z ∞ tional term when the volume fraction Φ 1. d−1 w1,?(r) = 1 − w1,?(u)du G(r)αdr (52) We can note that the influence of a finite volume for the r n-th nearest neighbor for n > 1 can be neglected when the with: average distance increases with n, because the probability 1 to the n-th neighbor particle center within the restricted w(r) = Θ(r − r?) (53) vd area defined by the presence of th n − 1-th neighbor be- The auto correlation function is negative, particle tend to comes negligible when n is large enough (the approxima- repel each other due to volume exclusion, and expresses as: tion is valid when the volume fraction Φ 1).

ξ(r) = −Θ(r? − r) (54) V. The n > 1 nearest neighbor particles. where Θ is the Heaviside function. Giving the same for- V.a. Asymptotic behaviour of average position and disper- mula as before (43), we get the density: sion

0 w1,?(r) = Θ(r − r?)N (r) exp[−N(r) + N(r?)] (55) We investigate the asymptotic behaviour of statistical properties of location for the n-th nearest neighbor in a ? The calculus of the average distance hri1,d is similar to the mean-filed approximation. For large n, we can first esti- previous one, considering the upper incomplete Gamma- mate roughly the average position of the n > 1 nearest Euler function Γ(u, p) (139): neighbor particle hrin,d being the radius of a d-sphere con- d taining the average number of particle n = ρ0αdr /d, such Z ∞ ? d + 1 that: hri1,d = w1,?(r)rdr = λdΓ ,N(r?) exp[N(r?)] 1/d 0 d dn 1/d (56) hrin,d ≈ ∝ hri1,dn (63) ρ0αd We note u? = N(r?). Giving the expression of Γ(a, x): V.b. Joint density of location for the N nearest neighbors Γ(a, x) = Γ(a) − γ(a, x) (57) We now consider a set of N particles denoted by respective where the lower Gamma-Euler function (140) admits the distance ri from a particle at r = 0. The joint probabil- Taylor extension for a > 1: ity density function of finding the N neighbors at distance r , ..., r must fulfill the condition: n Z x n n+a 1 N X (−1) n−1+a X (−1) x γ(a, x) = u du = N n! 0 n! n + a Y n=0 n=0 w(r1, ..., rN ) = Θ(ri − rj)w(r1, ..., rN ) (64) (58) i>j

5 where the normalization can be written as: V.c. φ as a function of the variable r1 Z ∞ Z ∞ Z ∞ 1 = dr1... drN−1 drN w(r1, ..., rN ) (65) This expression leads to the following system; knowing that 0 rN−2 rN−1 f verifies f(0) = 0 and admits first derivative it is possible We give the final result before developing the mathemati- to find another function ν verifying: cal aspects: Z y 1 0 Y 0 −N(rN ) f(y) = f (x)dx (76) w(r1, ..., rN ) = N (ri)e (66) 0 i=N Z y = ν(x, y)dx (77) First, we try to determine the joint probability density 0 function w(r1, r2) for finding the first neighbor at a distance r1 and the second nearest neighbor at r2. It is neces- We note that this situation happens when we express the sary that the density verifies w(r1, r2) = 0 if r1 > r2. The marginal density w2 in the two different ways: joint probability density function w(r1, r2) can be written in general terms using conditional probability (Bayes for- Z r2 0 mula): w2(r2) = w2(x)dx (78) 0 r w(r1, r2) = w1(r1)g2(r2|r1) (67) Z 2 = w(x, r2)dx (79) = w2(r2)g1(r1|r2) (68) 0 where the g(a|b) are the conditional probability density Now, we assume that the φ function only depends on the function of the occurrence of event a assuming event b location of the first nearest neighbor r . I(r ) diverges to to occur. We can write an equivalent expression, for the 1 1 ∞ and we get the general form of the joint density w(r1, r2) conditional probability g2(r2|r1) to find the second nearest in terms of φ(r1): neighbor at r2 assuming the first nearest neighbor is at r1, verifying the integral equation of the same form as equation 0 −φ(r1)[N(r2)−N(r1)] (40): w(r1, r2) = w1(r1)N (r2)φ(r1)e (80) Z ∞ g2(r2|r1) = ϕ(r1, r2) g(x|r1)dx (69) r2 It is crucial to note that the marginal density w1 for the verifying the general solution given by (41): first nearest neighbor doesn’t depend on the specific form of the φ function, because the normalization implies in (67): R r2 − ϕ(r1,x)dx g2(r2|r1) = µ(r1)ϕ(r1, r2)e r1 (70) Z ∞ where µ is a function of the variable r1. The normalization w1(r1) = w1(r1) g2(x|r1)dx (81) for the conditional probability imposes that: r1 ∞ Z R y ∞ h − ϕ(r1,x)dxi 1 = g2(y|r1)dy = −µ(r1) e r1 (71) However, the w2 density reaches the integral form, with a r1 r1 suitable change of variable (r → N(r)): = µ(r1)(1 − exp[−I(r1)]) (72) Z N(r2) where I(r1) is the integral: 0 −φ(x)[N(r2)−x]−x w2(r2) = N (r2) e φ(x)dx (82) Z ∞ 0 I(r1) = ϕ(r1, x)dx (73) r1 We can investigate the case φ(x) = 1, which doesn’t pre- We can rewrite the joint probability density function cise particular condition for the first nearest neighbor to w(r1, r2) in terms of the function ϕ: be full-filled in the expression of the conditional probabil-

R r2 ity g2(r2|r1) in equation (69). We get the marginal density w1(r1) − ϕ(r1,x)dx w(r , r ) = ϕ(r , r )e r1 (74) 1 2 1 2 w2(r2) given by, with appropriated normalization: 1 − exp[−I(r1)] We note that ϕ has the dimension of a density for a sin- w (r ) = N 0(r )N(r )e−N(r2) (83) gle variable (∼ L−1). We can express the ϕ function in 2 2 2 2 terms of the dimensionless quantity φ(r1, r2) > 0 by using 0 leading to the joint density w(r , r ) given by: the probability N (r2) to ﬁnd a particle at r2 regardless its 1 2 index: 0 ϕ(r1, r2) = N (r2)φ(r1, r2) (75) w(r1, r2) = w1(r1)w1(r2) exp[N(r1)] (84)

6 We can now discussed the case for a set of N particles. In the conﬁgurations of the particles with index from 1 to general, a joint probability density function takes the form: n − 1:

n−1 N Z r Z r2 Y Y w (r) = N 0(r)e−N(r) dr ... dr N 0(r ) (94) w(r1, ..., rN ) = w1(r1) w(ri|ri−1, ..., r1) (85) n n−1 1 i 0 0 i=2 i=1 1 where the term w(r |r , ..., r ) means the probability den- = N 0(r)N(r)n−1e−N(r) (95) i i−1 1 (n − 1)! sity function of finding the i-th nearest neighbor at a distance ri assuming the respective locations of the n = using the Cauchy formula for repeated integration: 1, ...i − 1 nearest neighbors. It verifies w(ri|ri−1, ..., r1) = 0 Z x Z xn−1 Z x2 if r < r , ..., r . We can use the same property φ = 1 for i i−1 1 In(x) = dxn−1 dxn−2... dx1f(x1) (96) the conditional probabilities as expressed in equation (69) a a a which was valid for two nearest neighbors. We can write 1 Z x = (y − x)n−2f(y)dy (97) the conditional probability w(ri|ri−1, ..., r1) as follow: (n − 2)! a Z ∞ 0 w(ri|ri−1, ..., r1) = N (ri) w(x|ri−1, ..., r1)dx (86) ri which verifies the solution:

0 w(ri|ri−1, ..., r1) = N (ri) exp[−N(ri) + N(ri−1)] (87)

Then, the joint probability can be written as:

N N P −N(r )+N(r ) Y 0 i i−1 w(r1, ..., rN ) = w1(r1) N (ri)ei=2 (88) i=2 1 Y 0 −N(rN ) = N (ri)e (89) i=N with Heaviside explicit development, we get the general form:

N−1  i−1  Y 0 Y w(r1, ..., rN ) = w1(rN ) N (ri) Θ(ri − rj) (90) Figure 1: Probability density function wn(x) for finding the n- i=1 j=1 th nearest neighbor at distance r from arbitrary center as a function of the dimensionless variable x = r/λd V.d. Marginal density of location for the n-th nearest for dimension d = 3. neighbor For a given index n ≤ N, we evaluate the density We can verify that the probability to find a particle at w(r , ..., r ) by integrating the joint density over the con- r regardless its index n given by G(r) corresponds to the 1 n 0 figurations of the particles with index from n + 1 to N: function N (r) by the sum: ∞ ∞ n n Z ∞ Z ∞ X 0 −N(r) X N(r) 0 Y 0 G(r) = wn(r) = N (r)e = N (r) w(r1, ..., rn) = N (ri) drn+1... drN × (91) n! r r n=1 n=0 i=1 n N−1 (98) N The cumulative probability distribution is given by: Y 0 −N(rN ) N (rj)e (92) r j=n+1 Z γ(n, N(r)) Ω (r) = w (x)dx = (99) n n n (n − 1)! Y 0 0 = N (ri) exp[−N(rn)] (93) k i=1 The moment of order k hr in,d is given by the integral:

Z ∞ k The marginal density of index n noted wn(r) (represented k k [λd] k in Figure 1) is simply given by integrating w(r , ..., r ) over hr in,d = x wn(x)dx = Γ + n (100) 1 n 0 (n − 1)! d

7 with the limit for large n of the Gamma-Euler function with the conditional probability g(rn|rn+1) given by the given by: ratio: Γ(n + α) lim = 1 (101) n−1 n→∞ Γ(n)nα w(rn, rn+1) N(rn) 0 g(rn|rn+1) = = n n N (rn) (107) k k k/d wn+1(rn+1) N(rn+1) we get hr in,d ≈ [λd] n , which is the main ﬁeld approximation in (63). We can evaluate the dispersion σn,d: The average hrnrn+1i becomes:

1/2 2 λ 2 Γ2(1/d + n) λd 1 2 d hrnrn+1i = Γ n + 1 + (108) σn,d = Γ + n − (102) (n − 1)! n + 1/d d p(n − 1)! d (n − 1)! where we can generalize to hr r i with i < j: We note that√ for dimension d = 1 we get the dispersion: i j σn,1 = λd n. The dispersion diverges when n increases contrary for dimensions d > 1 (Figure 2). As the relative j−i !−1 λ2 Y 1 2 error σ /hri converges to 0 when n increases (Figure hr r i = d + j − k Γ + j (109) n,d n,d i j (i − 1)! d d 2). k=1

We plot the correlation coeﬃcient ρ as a function of the index n of the ﬁrst particle within the pair (n, n + 1) in Figure 3.

Figure 2: Relative error σn,d/hrin,d and dispersion σn,d/λd as a function of the index n.

V.e. Joint density of location for two successive nearest neighbors The joint probability to ﬁnd two successive nearest neighbors with index n and n + 1 at respective distances rn and rn+1 can be evaluated by the n − 1 integral of w(r1, ..., rn+1):

Z rn Z r2 w(rn, rn+1) = drn−1... dr1w(r1, ..., rn+1) (103) 0 0 n−1 N(rn) = N 0(r )N 0(r )e−N(rn+1) (104) Figure 3: Correlation coefficient ρ(n) of the random variable r (n − 1)! n n+1 n and rn+1 in several dimensions of space. The correlation coefficient ρ(n) for the two random variables rn and rn+1 follows the expression: We first note that correlation is not strongly affected by the dimension d of the space. Besides, the radial coor- hrnrn+1i − hrnihrn+1i ρ(n) = (105) dinates r and r are strongly correlated when the index σ σ n n+1 n,d n+1,d of the pair n increases. It describes the increasing accumu- It requires to calculate the average hrnrn+1i by using the lation of particles within the layers of arbitrary thickness 1/d conditional probability g(rn|rn+1) to find the n-th nearest at distance Rn ∝ n λd surrounding the d-sphere centered neighbor at rn assuming the n + 1-th nearest neighbor is of the test particle, due to geometrical effects. Let’s note at rn+1: that the correlation between the vectorial positions xn and xn+1 necessary decreases with n in dimension d > 1. We Z ∞ Z rn+1 can now proceed to numerical simulations and validate our hrnrn+1i = dxwn+1(x)x dygn(y|x)y (106) 0 0 approach by considering order statistics applications.

8 VI. Numerical simulations

VI.a. Density of location of a single particle in a d-sphere of radius R

For numerical applications, generating random variables from 0 to infinity can simplified by considering suitable re- scaling. We now consider a finite spherical d-volume of d radius R given by vd(R) = αdR /d. Let’s note r the radial coordinate of a particle within the sphere and s = r/R the reduced variable comprised between 0 and 1. Considering a free particle in the system, the probability p(s)ds that it is located at s is given by the ratio between the volume of the layer comprised between r and r + dr and the total volume vd. We can write:

d−1 dr αdr(s) d−1 d 0 p(s) = = ds = (s ) (110) ds vd Figure 4: Probability density functions fn(x) for the n-th min- The volume is now ﬁlled with N particles with average imum of a list of ordered realization of random vari- density ρ = N/v . For each realization of N particles, we ables {si}1≤i≤N following p(s) (and normalized his- 0 d tograms for N = 20 particles and for 105 simulations) order the positions {s } in a list. Let’s note that the i 1≤i≤N for dimension d = 3. formalism given by the variable s = r/R generalizes the equivalent radius of the system to 1 and makes statistical properties only depending on the total number of particles Now we investigate the change of variable x → r to get the probability density function wn(r) that must verify: N within the volume vd.

fn(x)dx = wn(r)dr (115) VI.b. Order statistics We can express a general form for wn such as:

It is possible to deduce an evaluation of wn by considering drdn−1 w (r) = KCn n [1 − (r/R)d]N−n (116) order statistics results. We consider the new random vari- n N Rdn able X which corresponds to the n−th value of a set of N elements {si}1≤i≤N distributed following p(s) and ordered where K is a normalization constant associated to the in a list. We define the probability density function fn that change of space (change from a interval with finite borders verifies the equality: x ∈ [0, 1] to an interval with an infinite border r ∈ [0, ∞[). We note that the following expression depends on the two extensive properties of the system R and N. The limit of Pn(X ∈ [x, x + dx[) = fn(x)dx (111) infinite system N,R → ∞ and N/vd = ρ0 can be consid- We can write the general expression for f (x): ered by developing 1 − x ≈ exp[−x] when x = r/R 1, n then: Z x n−1 Z 1 N−n drdn−1 w (r) ≈ KCn n exp−(N − n)(r/R)d (117) fn(x) = Kp(x) p(s)ds p(s)ds (112) n N Rdn 0 x = Kdxdn−1[1 − xd]N−n (113) For n small enough compared to N, the difference N −n ≈ N 1, then we get the exponent: With K an appropriated normalization constant. By using h r id R d h r id the integral in (144), we derive the final result [4] (repre- (N − n) ≈ = N(r) (118) sented in Figure 4, with numerical applications): R λd R

n dn−1 d N−n The wn density function can be written as: fn(x) = [nCN ]dx (1 − x ) (114) Kd drdn−1 w (r) = e−N(r) (119) n (n − 1)! Rdn

9 The normalization constant K is given by the integral of 5): dn wn from 0 to ∞, giving K = (R/λd) . The wn function √ n 5β−3/2 [5/2 2π] 3 −(1+3n/2) − √ is given by the expression: Wn(β) = β e 2 2π (122) F0(n − 1)! 2 dn d r 1 −N(r) where β = F/F0 and F0 the typical force deﬁned in the wn(r) = e (120) (n − 1)! λd r case of the Holtsmark distribution of force in (19). 1 = N(r)n−1N 0(r)e−N(r) (121) (n − 1)!

We retrieve the same result as before by considering statistical properties of ordered list elements distributed following the density p(s). We discussed the general expression of the density wn for ﬁnding any nearest neighbor with index n at location r. Such distribution can be used to evaluate the respective contribution of each n-th nearest neighbor to the gravitational force exerted on the center particle at r = 0 due to its neighborhood.

VII. Nearest neighbor contribution to the total gravitational force

VII.a. Density for the gravitational force of the n-th nearest neighbor

The probability density Wn(F ) of the force modulus Fn = Figure 6: Average force hF in/F0 and dispersion σW,n/F0 as a |Fn| Wn(F ) for the n-th nearest neighbor can be simply de- function of n. rived from its density wn(r) by the equality Wn(F )dF = wn(r)dr. The average force modulus hF in (represented in Figure 6) is given by: 2/3 F0 5 2 hF in = √ Γ n − (123) (n − 1)! 2 2π 3

The average force modulus hF in decreases to 0 when n increases. The moment of order 2 reaches the following expression for n > 1 (not ﬁnite for n = 1): 4/3 2 F0 5 4 hF in = √ Γ n − (124) (n − 1)! 2 2π 3 More generally, we can show that for k > 2, the moment k of order k is proportional to hF in ∝ Γ(n − 2k/3) which is ﬁnite and positive for n > 2k/3. The moment of order k converges for Wn(F ) with n > 2k/3. The dispersion σW,n (represented in Figure 6) is given as follow: 2/3 1/2 F0 5 4 Γ(n − 2/3) σW,n = √ Γ n − − p(n − 1)! 2 2π 3 (n − 1)! (125) Figure 5: Probability density functions Wn(β) as a function of We can investigate the case n = 1 where the distribution the normalized force β (with y-axis log-scale). of force takes the form: 15β−5/2 5β−3/2 W (F ) is given by the expression (represented in Figure W1(F ) = √ exp − √ (126) n 4 2πF0 2 2π

10 with the average force modulus hF i1: where ui = cos(θi) a random variable distributed uni- formly between 1 and −1. We note that βnun is the radial 2/3 5 1 coordinate of the n-th vectorial force contribution Fn to hF i1 = √ Γ F0 ≈ 2.674F0 (127) 2 2π 3 the total force F. Giving ui and βi independent random variables, the average hβi = 0. The statistical distribution For F → ∞, reporting the first two terms of the Taylor for the radial projection of the total gravitational force ex- expansion of W1(F ): erted by a set of N nearest neighbors in dimension d = 3 is then given by the density W (β). We note that the total 1/2 15 2 −5/2 75 −4 force modulus is given by the absolute value |β|, where the W1(F ) ∼ β − β (128) 8F0 π 16πF0 distribution W (|β|) can be assimilated to the Holtsmark distribution when N → ∞. We can first calculate the av- We get exactly the first term of the Holtsmark distribution erage hβ2i, which corresponds exactly to the moment of Taylor expansion for β 1 in equation (27). That implies order 2 of the distribution W (|β|). We get: that for regime given by F → ∞, the largest force on the test particle is due to the nearest neighbour. For low F , N N X 2 X the exponential term cannot admit any Taylor expansion. hβ2i = hβ2u2i = hβ2i (135) i i 6 i More generally, for any n, we get for β 1 the Taylor i=1 i=1 expansion for W (F ): " N 2 # n 2 X σW,i = + hβ i2 (136) √ 6 F i n k k i=1 0 [5/2 2π] 3 X (−1) 5 −(1+3(n+k)/2) Wn(β) = √ β F0(n − 1)! 2 k! 2 k=0 2 2π We note that hβ i diverges to infinity because the disper- (129) sion σW is infinite, regardless the total number of parti- √ 1 [5/2 2π]n 3 5 cles N. Then, the non-finite dispersion of the distribution ≈ β−(1+3n/2) − √ β−(5/2+3n/2) W (|β|) is due to The first nearest neighbor contribution. F0(n − 1)! 2 2 2π For n > 1, the pair correlation for radial coordinate ρ(n) (130) gets close to 1, the n-th nearest neighbors accumulate at At first order, we retrieve for a given index k the exponent equivalent distance and to lower the contribution of the of β in the series development of the Holtsmark distribu- force modulus, whereas the first nearest neighbor is much tion for large β 1, given by the general term β−bn with nearer and exerts in average larger force on the test particle. The fluctuations of the force that follows density bn = 1 + 3n/2 in equation (26). W1(F ) are also larger than any others and show statistical weights much larger than densities W (β) when β 1. VII.b. Joint density for the total force modulus n>1

We can write the joint probability to find the set {βi}1≤i≤N VIII. Conclusions knowing the position r of each particle: The Holtsmark distribution provides group effect informa- 1 dr1...drN tion on the statistics of the gravitational force in a random W (β1, ..., βN ) = N w(r1, ..., rN ) (131) F0 dβ1...dβN homogeneous gas with Poisson-like spatial distribution of N −3/2 N 5β point particles. We discussed the statistical properties of 15 Y −5/2 − √N = √ β e 2 2π (132) location in a d-dimension space of constant density ρ of 4 2πF i 0 0 i=1 a set of N particles considering the neighborhood of a ar- where the Heaviside term in the density is: bitrary center particle. We generalized the joint probability density function for finding the N nearest neighbors at N respective location. Side-by-side particle joint probability Y Θ(βj − βi) (133) density function of location validates that the radial coor- i>j=1 dinates of 2 nearest neighbor with successive index n are strongly correlated when n is high. In dimension d = 3, we Considering the new random dimensionless variable β as analyzed the n-th contribution to the total gravitational the radial component of the total gravitational force ex- force exerted on a test particle. We noticed that the non- erted of the center particle following an arbitrary direction: finite dispersion of the Holtsmark distribution is due to

N the non-finite dispersion of the distribution of force ex- X erted by the first nearest neighbor. We may pursue the β = βiui (134) i=1 study of the gravitational field by using the joint probabil-

11 ity distribution for any number of neighbor to investigate The Taylor expansion of the exp(x) function is: the instabilities in a gas of gravitational-bounded particles X xn with homogeneous initial spatial distribution, leading to exp[x] = (142) n! the emergence of coherent structures (filament). The influ- n=0 ence of a continuous mass distribution within the gas may be implemented to the study of statistics of gravitation. Using the complex integral, we have: Z ∞ απ IX. Mathematical appendix xα sin xdx = − cos Γ(1 + α) (143) 0 2 The Euler-gamma function given by: We give the following Beta integral: Z ∞ u−1 Z 1 Γ(u) = x exp(−x)dx (137) dn−1 d N−n Γ(N + 1 − n)Γ(n) 0 d x (1 − x ) dx = (144) 0 Γ(N + 1) More generally for the exponents α and β: Z ∞ 1 + α β α β Γ = 1+α x exp −(x/K) dx (138) References β K 0 The incomplete upper Euler-gamma function is defined as: [1] A. Gabrielli, F.Sylos Labini, M. Joyce, L. Petronero, Statistical Physics for Cosmic Structures, (Springer Z ∞ Verlag, Berlin, 2004) Phys.Rev E (69) 031110. Γ(u, p) = xu−1 exp[−x]dx (139) p [2] S. Chandrasekhar, Rev. Mod. Phys. 15, 1(1943). The lower Euler-gamma function is defined as: [3] S. Mazur, J. Chem.Phys., 97, pp. 9276-9282 (1992). Z p γ(u, p) = Γ(u) − Γ(u, p) = xu−1 exp(−x)dx (140) 0 [4] Appel W. Probabilitéspour les non-probabilistes, M & K éditions,Paris. The Taylor expansion of the sin(x) function is: [5] P. Chavanis, European Physical Journal B: Condensed X (−1)n sin(x) = x2n+1 (141) Matter and Complex Systems, Springer-Verlag, 2009, (2n + 1)! n=0 70 (3), pp.413-433.