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

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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, Universit´eGrenoble-Alpes, 2nd year of Master's degree in Subatomic Physics and Cosmology Supervisor : Vincent ROSSETTO, Laboratoire de Physique et de Mod´elisationdes Milieux Condens´es 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 ^(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 ρ^(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 infin- ity, 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 !1 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 ^(x) verifies the probability distribution: By giving the expression of WH as a inverse Fourier trans- 1 form, it gives: with probability ρ0dV ρ^(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 hρ^(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!1 N jx − xj3 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 1 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 ! 1, we now use the Taylor expansion of the expo- 15 nential function (142) giving: By using the equation (143): 2 X (−1)n Z 1 W (F ) = β−(3n=2+1) x3n=2+1 sin xdx p H πF n! Z 1 du Z 1 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 1 H 1 h 3=2i 8F0 π πF0 WH (F) = 2 3 dx(x sin x) exp −(x/β) (17) 2π F 0 IV.
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