# Pseudodifferential Equation of Fluctuations of Nonstationary Gravitational Fields

Authors: Vladyslav Litovchenko
Journal: Journal of Mathematics
 (2021)

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## Abstract

Developing Holtzmark’s idea, the distribution of nonstationary fluctuations of local interaction of moving objects of the system with gravitational influence, which is characterized by the Riesz potential, is constructed. A pseudodifferential equation with the Riesz fractional differentiation operator is found, which corresponds to this process. The general nature of symmetric stable random Lévy processes is determined.


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## Body

## 1. Introduction

One of the main problems of celestial mechanics is the analysis of the nature of the force of interaction between objects in one or another star system. Forceℱ acting on a specific star of the system has two components: the first one, K, is the influence of the entire system and the second, F, is the local influence of the immediate environment: ℱ=K+F.The influence of the entire system can be described by the gravitational potentialℛr;t [1] the average spatial distribution of stars of different masses m at time t. Such force, K, related to the unit of mass acting on the star Z0 under consideration by the system (as a whole), is determined by the equality:(1)Kr;t=−gradℛr;t.The forceKr;t is a function with a slow change in space and time since the corresponding potential ℛr;t characterizes the “smoothed” distribution of matter in the stellar system, whereas the other force Ft, related to the unit of mass, has relatively fast, sharp changes caused by instantaneous changes in the local distribution of stars from the environment Z0 at time t. The value of Ft is subject to fluctuations, so we can only talk about its probabilistic value.The famous Danish astronomer Holtzmark studied the stochastic properties ofFt [1, 2]. He used Newton’s gravitational law, according to which(2)Ft=∑j=1NtFj=G∑j=1Ntmjrj2rj∘,where G is the gravitational constant, mj is the mass of a typical star Zj, rj is the radius-vector of Zj, rj∘≔rj/rj, rj2=rj,rj, scalar square in the Euclidean space ℝ3, and Nt is the number of stars that at the moment t form the local environment Z0. He assumed the constancy of the average density nr;m;t≡n of the spatial distribution of stars as well as the fulfillment of the equality:(3)N=43πR3n,∀R>0.Using classical means of probability theory in combination with integral calculus, Holtzmark found a stationary distributionWF of the quantity F in the form(4)WF=12π3∫ℝ3e−iξ,F−aξ3/2dξ≡F−1e−aξ3/2F,where a≔4/152πG3/2nm3/2 is the fluctuation coefficient, in which m3/2 is the average value of m3/2, which corresponds to the observed law of stars distribution in the stellar system, and F is the Fourier transform operator.This Holtzmark distribution belongs to the class of distributions(5)ℒλαx=F−1e−λξαx,λ>0,x∈ℝ3,of symmetric stable random processes described by the French researcher Zolotarev et al. [3, 4] at the beginning of the last century.The appearance of the class of distributionsℒλα⋅, in addition to these Holtzmark studies, was preceded by the studies of Poisson and Cauchy. About a hundred years before the publication of [4], first Poisson and then Cauchy paid attention to the distribution with density:(6)Pλx=λπλ2+x2,λ>0,x∈ℝ,which has the property(7)Pλ1⋅∗Pλ2⋅=Pλ⋅,and the parameter λ is uniquely determined by the values λ1 and λ2. Within his statistical studies, being interested in the cases, when the error of an individual observation might be comparable to the mean error in a series of large numbers of independent observations, Cauchy found that such a case would occur if the observation errors were subject to a probability distribution with a density Pλ⋅. It should be noted that(8)Pλ⋅=ℒλ1⋅.Cauchy [5] knew that each of the functions ℒλα⋅ for α>0 possesses property (7), but he could not select the nonnegative functions from them (which were, obviously, the densities of some probability distributions) except for (8). The famous Hungarian mathematician Polya [6] managed to prove that ℒλα≥0 for α∈0;1 only at the beginning of the last century. The fact that the functions ℒλα⋅ are densities of probability distributions only for α∈0;2 was fully determined by Le´vy in 1924 [4].The Gauss–Wiener distribution of(9)ℒt2x=12tπ3e−x2/4t,t>0,x∈ℝ3.Brownian motion in the spaceℝ3, according to which heat and diffusion propagate in 3D environment, is also a good representative of the class ℒλα⋅. In the scientific literature [7–12], there are many examples of real applications of the Holtzmark, Cauchy, and Gauss distributions in astronomy, nuclear physics, economics, in the industrial and military sectors, etc. Each of these applications characterizes the stochastic features of Le´vy distributions with one or another value of α.However, the symmetric stable random processesℒλα⋅, besides their individual characteristics, have a general nature. In this paper, it is determined that each of such Lévy processes at α∈0;2 can be regarded as a process of local influence of moving objects in the gravitational field of M. Rees, i.e., in a system, in which the interaction between masses occurs according to a certain power law ⋅−β. In particular, the classical Holzmark process α=3/2 corresponds to the interaction with β=2 (the case of Newtonian gravity), while the Cauchy process corresponds to the interaction with β=3. Here, we also consider the Holtzmark problem in a general formulation and obtain a pseudodifferential equation (PDE) with the Riesz operator of fractional differentiation, which corresponds to the Holtzmark random nonstationary processes. The presence of this equation allows one to study Holtzmark stochastic processes in domains with boundary by means of the theory of boundary value problems for the PDE.The main content of the work is as follows. In Section2, the stochastic problem of the local influence of moving objects in the evolutionary gravitational field is studies. The connection of the Holtzmark fluctuations’ distributions of nonstationary gravitational fields with one classical PDE is determined in Section 3. A brief historical overview of this equation study and its further application is also presented here. Section 4 is the conclusions.


## 2. The Problem of the Local Influence of Stars

Let us consider a star system, in which the interaction between masses is subordinate to M. Riesz potential [13]. It means that the gravity between two arbitrary stars of the corresponding masses M and m is described by the law:(10)F=GMmrβr∘,β>0,where G is a certain gravity constant and r is a vector of distance between these stars. Let the star Z0 under consideration be at the origin of the coordinate system and Ft be the force acting on the unit of mass of the star Z0 from its closest environment at time t.Developing Holtsmark’s idea, we will find the nonstationary distributionWβFt for the force Ft. For this, we also assume that the distribution of stars in the neighborhood Z0 is subject to fluctuations according to some empirically established law and that, at each moment of time t, the fluctuations of the density of stars are subject to the condition of the constancy of their average density per unit volume:(11)nr;m;t≡nt.If we now assume that the starZ0 under study is at the origin of the system and its spherical circle of radius R at time t contains Nt stars, then, according to the above,(12)Ft=G∑j=1Ntmjrjβ+1rj≡∑j=1NtFj,(13)Nt=43πR3nt.First, let us fixt and consider the distribution of Wβ,NtFt in the center of the spherical vicinity of the radius R, which contains Nt of stars of the system. Let us find the probability Wβ,NtF°tdF°t that the quantity Ft will be cubed F°t;F°t+dF°t⊂ℝ3. Using the well-known method of characteristic functions, we obtain(14)Wβ,NtF°t=12π3∫ℝ3e−iξ,F°tANtξdξ,where(15)ANtξ≔∏j=1Nt∫0+∞∫KR0eiξ,Fjτjrj;mj;tdrjdmj,Here,KR0 is a sphere of radius R with the center at the origin and τjrj;mj;t is the distribution of probability that, at time t, the star Zj has mass mj and is in position rj.Since it is believed that there are only fluctuations of stars compatible with the constancy of their spatial average density, then(16)τjrj;mj;t=3τm;t4πR3,where τm;t is the frequency, with which stars of different masses meet at time t.Taking it into consideration, we will receive the equality:(17)ANtξ=34πR3∫0+∞∫KR0eiξ,ητm;tdrdmNt,in which(18)η≔Gmrrβ+1.DirectingR⟶+∞ and Nt⟶+∞, according to condition (13), we obtain(19)WβFt=12π3∫ℝ3e−iξ,FtAξ;tdξ,where(20)Aξ;t≔limR⟶∞34πR3∫0+∞∫KR0eiξ,ητm;tdrdm4πR3nt/3.For all values oft, the equality(21)34πR3∫0+∞∫KR0τm;tdrdm=1,is true, so(22)Aξ;t=limR⟶∞1−34πR3∫0+∞∫KR01−eiξ,ητm;tdrdm4πR3nt/3.Furthermore, the absolute convergence in (22) of the integral with the integration variable r in the entire space ℝ3 at β>3/2 allows us to write equation (22) in the form(23)Aξ;t=limR⟶∞1−34πR3∫0+∞∫ℝ31−eiξ,ητm;tdrdm4πR3nt/3.From here, we get the image(24)Aξ;t=e−ntBβξ;t,in which(25)Bβξ;t≔∫0+∞∫ℝ31−eiξ,ητm;tdrdm.The last equality can be written in the form(26)Bβξ;t=4πGξ3/β<m3/β>β∫0+∞ρ−sinρρ−2−3/βdρ.For this, it is necessary to replace the integration variabler with the variable η in the inner integral of this equality according to rule (18) and then to turn to a spherical coordinate system, in which the z-axis is directed along the ξ vector.It should be noted that the integral of the last equation coincides only atβ>3/2. Integrating it by parts, we arrive at the equality(27)Bβξ;t=4βπIβ3β+3Gξ3/βm3/β,t≥0,ξ∈ℝ3,β>32,where(28)Iβ≔β3−βΓ2−3βcos2−3/βπ2,32<β<3,π2,β=3,Γ1−3βsin1−3/βπ2,β>3,where Γ⋅ is Euler’s gamma function.Combining equalities (19), (24), and (27), we finally find(29)WβFt=12π3∫ℝ3e−iξ,Fte−aβtξ3/βdξ,where(30)aβt≔4βπIβ3β+3G3/βntm3/β.Thus, we have proved the following theorem.Theorem 1.
With the above assumptions, for everyβ>3/2, the function(31)WβFt=F−1e−aβtξ3/βF;t,is the probabilities distribution of the force Ft of the local influence of moving objects in the system with the interaction that takes place according to the power law (10).Remark 1.
Forβ≤3/2, the integral Iβ diverges, so the coefficient aβ⋅ becomes immeasurable. Therefore, we conclude that there is no corresponding random Holtzmark process in the sense of the problem of local interaction of moving objects.
Let us introduce the notationℋγF;t≔WβFt, where γ≔2/β. The function ℋγ⋅;⋅ is called the Holtzmark distribution of the order γ of fluctuations of nonstationary gravitational fields. The classical situation considered by Holtzmark corresponds to β=2; in this case, the distribution order γ=1. Taking the ratio β>3/2 into account, this is the only case of the integer order γ, and the rest of the possible values of γ have a nonzero fractional part: γ∈0;4/3.
In Section3, we find the corresponding differential equation, the fundamental solution of which is the Holtzmark distribution ℋγ⋅;⋅.


## 3. Connection with the PDE

It would be more desirable to study the fluctuations in the local interaction of moving objects, especially in a limited environment with certain conditions at the boundary, by reducing to solve the corresponding boundary value problems for differential or pseudodifferential equations. This would allow us to use the advanced computing apparatus of the theory of boundary value problems and use its known results. In this regard, there is a need to obtain an appropriate differential equation that adequately reflects the process under study. Under certain conditions, we will try to derive this equation “starting” from the distribution functionℋγ. To do this, let us first find out the properties of this function.It will be assumed here that the coefficientaβ⋅ is a positive, continuous-differential function on the interval 0;T. It follows directly from [14, 15] that, for all γ∈0;4/3, the function ℋγx;t on the set ℝn×0;T is differentiable by t and infinitely differentiable by the variable x; for its derivatives, the following estimates are fulfilled:(32)∂xkℋγx;t≤c1tt2/3γ+x−3+k+3γ/2,∂t∂xkℋγx;t≤c2t2/γ−1t2/3γ+x−3+k+3γ/2,with some positive constants c1 and c2.Estimate (32) ensures that ℋγ⋅;t belongs to the Lebesgue class L1ℝ3 for each fixed t∈0;T. Therefore, this guarantees the existence of the Fourier transform of the function ℋγ⋅;t and satisfies the equality(33)Fℋγx;tξ;t=e−aβtξ3γ/2,t∈0;T,ξ∈ℝ3.We shall arbitrarily fixt∈0;T, and for △t≠0, we consider(34)ℋγx;t+△t−ℋγx;t=12π3∫ℝ3e−ix,ξ−aβtξ3γ/2e−aβt+△t−aβtξ3γ/2−1dξ.According to Lagrange’s theorem on finite increments, we have(35)aβt+△t−aβt=aβ′t+θ△t△t,θ∈0;1.Hence, given the continuity ofaβ′⋅, we obtain that(36)e−aβt+△t−aβtξ3γ/2−1△t⇉ξ∈KR0−aβ′tξ3γ/2,∀R>0.In addition, using Lagrange’s theorem again, we find(37)e−aβ′t+θ△tξ3γ/2△t−1△t=aβ′t+θ△tξ3γ/2e−aβ′t+θ△tξ3γ/2θ^△t≤≤aξ3γ/2ea△tξ3γ/2,θ^∈0;1,a≔supt∈0;Taβ′t.Then, for all0<△t≤aβt/2a and ξ∈ℝ3, the following estimates are true:(38)e−aβtξ3γ/2e−aβt+△t−aβtξ3γ/2−1△t≤aξ3γ/2e−aβt−a△tξ3γ/2≤≤4ae−aβt/4ξ3γ/2supρ>0ρe−ρ.Relations (36) and (38) ensure the correctness of the equality(39)lim△t⟶0∫ℝ3e−ix,ξ−aβtξ3γ/2⋅e−aβt+△t−aβtξ3γ/2−1△tdξ==∫ℝ3e−ix,ξ−aβtξ3γ/2⋅lim△t⟶0e−aβt+△t−aβtξ3γ/2−1△tdξ,according to which, from (34), we obtain(40)∂tℋγx;t=−aβ′t2π3∫ℝ3ξ3γ/2e−ix,ξ−aβtξ3γ/2dξ,t∈0;T,x∈ℝ3.Now, taking (33) into account, we finally arrive at(41)∂tℋγx;t=−aβ′tF−1ξ3γ/2Fℋγξ;tx;t,t∈0;T,x∈ℝ3.Thus, the Holtzmark distributionHγ is a classical solution of the equation(42)∂tux;t+aβ′tAνux;t=0,t∈0;T,x∈ℝ3,with Riesz operator Aν fractional differentiation of the ν≔3γ/2 order [16, 17].Furthermore, we clear up the question of existence of the limit value of distributionℋγ⋅;t at the point t=0.First, let us consider the caseaβ0≠0. According to equation (33) and the well-known Fourier transform formula for convolution of elements of the Lebesgue class L1ℝ3, we obtain that(43)ℋγx;t=Gν∗ℋ^γx;t,t∈0;T,ξ∈ℝ3,where ℋ^γ⋅≔ℋγ⋅;0 is the corresponding Holtzmark stationary distribution, and(44)Gν⋅;t≔F−1e−∫0taβ′τdτξν⋅;t.We now show that, for every continuous functionℝ3 limited by φ⋅, the boundary relation holds:(45)Gν∗φ⋅;t⟶t⟶+0φ⋅.For that, we shall use the equality(46)∫ℝ3Gνx;tdx=1,t∈0;T,according to which(47)Gν∗φx;t−φx≤∫ℝ3Gνξ;tφx−ξ−φxdξ≡ℑx;t.Sinceφ⋅ is a continuous function on ℝ3, then, for every x∈ℝ3 and arbitrary ε>0, there exists t° such that t°1/2ν<ε and φx−ξ−φx<ε if ξ<t°1/2ν. Then,(48)ℑx;t<ε∫ξ<t°1/2νGνξ;tdξ+∫ξ≥t°1/2νGνξ;tφx−ξ−φxdξ≤εℑ1t+ℑ2x;t,where(49)ℑ1t≔∫ℝ3Gνξ;tdξ,ℑ2x;t≔∫ξ≥t°1/2νGνξ;tφx−ξ−φxdξ.Furthermore, having considered the estimates [14, 15](50)∂xkGνx;t≤c1tt1/ν+x−3+k+ν,k∈ℤ+3,t∈0;T,x∈ℝ3,and the limited function φ⋅ in ℝ3, for all t∈0;T and x∈ℝ3, we find(51)ℑ1t≤c1t∫ℝ3dξt1/ν+ξ3+ν=c1∫ℝ3dz1+z3+ν≡c2,ℑ2x;t≤c3t∫ξ≥t°1/2νξ−3+νdξ=c3t∫t°1/2ν+∞ρ−1+νdρ=c4tt°−1/2.It follows from the last inequality that, for allx∈ℝ3 and t≤t°,(52)ℑ2x;t≤c4t°1/2<c4εν.Consequently,(53)ℑx;t<c2ε+c4εν,∀x∈ℝ3∀ε>0∃t°<ε2ν∀t≤t°,i.e., the boundary relation (45) is satisfied.The functionℋ^γ⋅ is infinitely differentiable and is limited to ℝ3, so, according to (45), for ℋγ⋅;t, the relation is fulfilled:(54)ℋγ⋅;t⟶t⟶+0ℋ^γ⋅.Thus, the distributionℋγx;t is a classical solution of the Cauchy problems (42) and (54).Now, letaβ0=0; then, the next equality follows directly from (33):(55)ℋγ⋅;t=Gν⋅;t,t∈0;T.It should be noted that relation (45) characterizes the property “δ-similarity” of the function Gν⋅;t in the space S′ of Schwartz distributions [18]:(56)Gν⋅;t⟶t⟶+0δ⋅,where δ⋅ is the Dirac delta function. Therefore, for aβ0=0, the Holtzmark distribution ℋγ⋅;t is the solution of the Cauchy problems (42) and (56), which in the usual sense satisfies equation (42), and it satisfies the initial condition (56) in the sense of weak convergence in the space S′. This solution Gν is called Green’s function of the Cauchy problem for equation (42).Let us summarize the above in the form of the following statement.Theorem 2.
Letβ>3/2 and aβ⋅ be positive, continuous-differential functions on the interval 0;T; then, for aβ0≠0, the corresponding Holtzmark distribution ℋ2/β⋅;t on the set ℝ3×0;T is a classical solution of the Cauchy problems (42 and 54). If aβ0=0, then ℋ2/β⋅;t is Green’s function of this problem.Remark 2.
Equation (55) reveals the meaning of Green’s function of the Cauchy problem for equation (42): Gν is the primary Holtzmark distribution of the local influence on the object under consideration by its moving environment characterizing this process from the very beginning of its origin, i.e., from the moment when the elements of local influence first appeared in the environment of the object.
The study of Green’s function of the Cauchy problem for the PDE of form (42) was initiated by S.D. Eidelman and Ya.M. Drin in the early 80s of last century [19, 20]. They proposed a method for constructing and studying the function Gν based on the Fourier transform and obtained the following estimates:(57)∂xkGνx;t≤c1tt1/ν+x−n+k+ν,k∈ℤ+n,t∈0;T,x∈ℝn,where ⋅ is the integer part of a number. However, this method imposes restrictions on the order of ν PDE: ν>1.
The accurate asymptotic behavior of Green’s functionGν⋅;t in the vicinity of infinitely distant points was determined by Fedoryuk in [21]:(58)Gν⋅;t∼⋅−n−ν,t>0.
Subsequently, Schneider [22], effectively using the Mellin transform, expresses the function Gν⋅;t through special Fox’s H functions and, as a consequence, obtains asymptotics (58). It should be noted that long before the publication of [21], the asymptotics (58) for the PDE (42) at aβt=t and ν∈0;1 was described by R.M. Blumenthal and R.K. Getoor in [23].
A new approach to the study of the properties of the functionGν⋅;t, which is based on the use of the elements of the theory of generalized functions and harmonic analysis, was applied by Kochubei in [24]. He was the first to receive estimates (57), in which ν is replaced by ν, in the case when the dimension of the spatial variable is greater than unity and ν≥1.
In the works of [14, 15], developing the idea of [24], Kochubei’s estimates are extended in the case of ν>0.Remark 3.
In a partial caseaβt=c0t+c1,c0>0,c1≥0, equation (42) is known as the equation of random walk with arbitrarily long jumps. It has important applications in modern ecology, biology, economics, and physics and is also closely related to Le´vy’s random flights (see, for example, [25]).
Pseudodifferential equations of type (42) and their generalizations as well as the application of these equations in the theory of stable random processes have been considered by many researchers [25–36].


## 4. Conclusions

According to Theorem1, each Holzmark distribution ℋγ is a Le´vy distribution ℒλα of the order α=3γ/2. Since γ∈0;4/3, then the γ-spectrum of Holtzmark distributions coincides with the α-spectrum of Le´vy distributions with the accuracy up to one nearest boundary value α=2, which corresponds to the diffusion process. This value α=2 corresponds to γ=4/3, which is also the boundary value of the γ-Holzmark spectrum. However, for such γ, there is no random Holzmark process. This means that the process of classical diffusion occurs according to the laws that differ in nature from the laws of random vortices of local gravity of moving objects.Taking into account the results of the research in Section3, equation (42) can be called the pseudodifferential equation of local fluctuation of nonstationary Riesz gravitational fields.



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