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Abstract
Smoluchowski’s equation is a macroscopic description of a many particle system with coagulation and shattering interactions. We give a microscopic model of the system from which we derive this equation rigorously. Provided the existence of a unique and sufficiently regular solution of Smoluchowski’s equation, we prove the law of large numbers for the empirical processes. In contrast to previous derivations we assume a moderate scaling of the particle interaction, enabling us to estimate the critical fluctuation terms by using martingale inequalities. This approach can be justified in the regime of high temperatures and particle densities, which is of special interest in astrophysical studies and where previous derivations do not apply.
Stoch. An. Appl. \jvol \jissue \jyear2003
www.dekker.com \cprightMarcel Dekker, Inc.
A rigorous derivation of Smoluchowski’s equation]A rigorous derivation of Smoluchowski’s equation in the moderate limit
Großkinsky et al.]S. Großkinsky, C. Klingenberg and K. Oelschläger
1 Introduction
We consider a system of dust particles of different masses , embeded in a dimensional hot gas. Particles of size are drifting according to the velocity field with a superimposed Brownian motion with diffusion constant . Two particles of size and collide with rate . The material coefficients determine the number of particles of size produced by that collision event, deciding for coagulation or shattering events. In this first model we take a macroscopic viewpoint, where two colliding particles occupy the same position in spacetime , where the function is evaluated. A complete description of the above model is given by Smoluchowski’s equation [15], in our case a system of reaction diffusion equations for the particle densities , (cf. [14], sect. 2) with initial conditions :
We suppose that for any size the particles consist of several atoms of size and we set . Moreover, the masses of the particles are ordered as . The conservation of the total mass of the system under the above dynamics, i.e. , is assured by
(1) 
for all . We also assume that is symmetric in and , i.e. .
The above model is commonly used to describe reaction diffusion systems and there have been rigorous approaches to identify equation (MA) as the limit dynamics of a suitable many particle system. These derivations are restricted to onedimensional systems [1], the spatially homogeneous case [10] or a spatially discretized microscopic model [2, 4]. In [8] there is a derivation accounting for the full space dependence of the problem, using the BoltzmannGrad limit which is applicable for very small particle densities. In [4, 10] existence and uniqueness of a solution of Smoluchowski’s equation are also studied.
In this paper we give a microscopic particle model (MI) in section II, from which we rigorously derive (MA) in the spatially inhomogeneous (general) case without space discretization or restrictions on space dimension. Our many particle system properly describes an astrophysical system recently studied in [6, 14], which is explained in section V.A. It corresponds to a situation of high gas temperatures and particle densities, which is not covered by the derivation in the BoltzmannGrad limit [8].
In this regime the dominating particle interactions are shattering collisions, so it is justified to neglect coagulation events. That means that the mass of each of the two interaction partners may not increase by the collision, but they are shattered into fragments of smaller or equal mass. This constitutes a constraint on the material coefficients given in (9), which is important to ensure compatibility with the microscopic particle model. Our main theorem in section III states the convergence of the empirical processes (4) to a solution of (MA) and is proved in section IV. Before giving a short conclusion in the last section we also discuss two apparent generalizations of the microscopic model (MI).
The most important feature of our approach is the moderate scaling of the collision interaction, which is introduced in section II.B (M3) and discussed on a physical level in section V.A. It enables us to use a technique developed by K. Oelschläger [11], which was previously applied to derive the porous medium equation [12], or in the description of aggregation phenomena in biological populations [9], [13]. With this technique we are able to derive Smoluchowski’s equation in the spatially inhomogeneous form (MA), in a regime where the previous approaches cannot be applied.
2 Microscopic particle model
Given the macroscopic model of section I we present a corresponding microscopic many particle system. The most important modeling assumptions are marked by (M1) to (M4) and are discussed in sections V.A and V.B.
2.1 Dynamics without interaction
Let be the number of particles of species and the number of all particles at time . The system size is characterized by the number of atoms of mass at time :
(2) 
Let be the set of all particles and , , the subsets of particles of species at time , where each particle is identified with a unique integer number.

The particles are considered to be point masses with positions , , at time in a system of size . Each particle of species is given the rescaled mass , which keeps the initial total mass independent of the system size according to (2).

Neglecting the hydrodynamic drag interaction between gas and particles, we consider the latter to move according to the given velocity fields and Brownian motion with diffusion constants , , introduced in the macroscopic equation (MA).
Between two subsequent collision events the system at time is then described by uncoupled stochastic differential equations:
(3) 
The , , are independent Wiener processes modelling the Brownian motion of the particles. We always assume the existence of a filtration , with respect to which the stochastic processes under consideration are adapted (cf. [11], sect. 2.B) and which fulfills the usual conditions [7].
The particle interaction is described by suitable changes of the sets and is explained in the next subsection. A microscopic quantity comparable to the particle density in (MA) is given by the measurevalued, empirical processes:
(4) 
where denotes the space of positive, finite measures on and is the Dirac measure concentrated in . describes the timeevolution of the spatial distribution of particles within the subpopulation of species . It is known by the law of large numbers that the empirical distribution of independent, identically distributed random variables converges to their probability distribution in the limit . In this paper we prove the convergence for stochastic processes which are not independent for times , due to the particle interaction.
2.2 Description of the particle interaction
Due to (M1) we have to specify a model for the ‘collision’ interaction of two point particles.

We take a stochastic model determined by a rate depending on the distance of the interaction partners and . The scaling of this rate is given by
(5) with and a moderate scaling parameter . We assume that is symmetric and positive with . It follows that for all and for all in the sense of distributions.
In contrast to the usual hydrodynamic scaling with this leads to a microscopically large interaction volume. This assumption is motivated and justified in a physical context in section V.A.

Instead of considering pair interactions (see sect. V.B) we assume that every particle interacts with an effective field of all other particles of species with rate
(6) where is the macroscopic collision rate given in section I.
In (6) we used the generalized convolution product
(7) 
By substraction of the term including Kronecker’s delta in (6) selfinteraction is excluded. The rate is bounded uniformly in by a suitable constant , which is specified in condition (C5) in section III.B. This cutoff prevents diverging interaction rates due to high particle concentrations in the limit . Each possible interaction event is described by a jump process
(8) 
where are independent standard Poisson processes with a transformed time argument in the brackets (cf. [11]) and is the indicator function of the set .
The process jumps from to at some time if particle exists in , belongs to species and interacts with a particle of species at time . After the interaction the number is removed from the sets and . The mass of particle is distributed on the interaction products according to the microscopic material coefficient . The latter fulfills conservation of mass and is related to its macroscopic counterpart in the following way:
(9) 
We note that this also constitutes a condition on , corresponding to the absence of coagulation mentioned in section I. The particles resulting from the interaction are located at and obtain new numbers starting with , which were previously not assigned to any particle. These numbers are added to and the subsets corresponding to the various species. We note that any process only jumps once, since after that jump the respective particle disappears, i.e. .
2.3 Complete description of the model
Using a generalized scalar product we can formulate the time evolution of the empirical processes in a weak sense. For all and we have
(10)  
Inserting the expression for from equation (3) and using Itô’s formula [7] we get:
(11)  
The first integral term describes the stochastic fluctuations of the particle positions and the second one particle transport and diffusion, resulting from the interaction free description (3). The next two terms consider the change of the sets in (10) due to the loss of particles of species after interactions with others, and the gain of such particles from products of other interactions. We separate the fluctuation terms due to stochasticity in the free particle dynamics and the interaction in stochastic integrals. So we get for all and the complete description of our microscopic model:
and initial conditions . (MI)
This set of equations combines all features mentioned in the preceding two subsections and is used to derive the macroscopic model (MA), shown in the next section.
3 Derivation of Smoluchowski’s equation
We show how to obtain (MA) heuristically from our microscopic particle model (MI), leading us to a proper formulation of the main theorem.
3.1 Heuristic derivation of the macroscopic equation
The empirical processes are defined as solutions of (MI). For this subsection we assume that for every they converge to limit processes on a compact time interval in a yet unspecified sense. The limit processes are assumed to be absolutely continuous with respect to Lebesgue measure on and therefore have densities , which should be in . With the generalized scalar product defined in (10) we therefore have for all . We also assume the validity of conditions (C1) to (C7) given in the next subsection.
In section IV.D we get the following for the stochastic integrals in (MI) for any :
(12) 
so the fluctuation terms asymptotically vanish in any compact time interval and the limit equation is supposed to be deterministic (see (14)). The convergence of the should be sufficiently strong to assure the following:
(13) 
for all , and . The first condition assures the convergence of the drift and diffusion term in (MI) and the second one is needed for the interaction terms. We formally substitute the above limits into (MI) and notice that the self interaction term in (6) vanishes for . Therefore we get the following deterministic integral equation for all test functions , and :
(14)  
After partial integration in the transport and diffusion terms one immediately recognizes this as a weak version of Smoluchowski’s equation. Using (1) and (9) it is easy to get the last line in the form (MA).
Therefore we showed that, assuming the empirical processes converge, their limit densities fulfill a weak form of Smoluchowski’s equation. In the next subsection we explain how to prove this convergence in an appropriate rigorous limit sense, which can be seen from (3.1) to be of type.
3.2 Convergence theorem
To formulate the convergence theorem we use the following distance function between the empirical processes (MI) and the solution of Smoluchowski’s equation (MA) specified in (C3) below:
(15) 
for all and . The convolution kernel smooths out the empirical processes and obeys the following regularity conditions:

is a different scaling of the interaction function and both have to fulfill:
and and
The scaling parameter plays no role in the dynamics of the manyparticle system. However, by the above assumptions some restrictions on the parameter determining the moderate interaction are introduced.

The unscaled function is symmetric, positive and standardized, i.e. . We also need and the Fourier transform has to fulfill:
A Gaussian probability density is an example for which obeys these conditions. To the knowledge of the authors there is no proof of the existence of a sufficiently smooth solution of the macroscopic equations, therefore we have to assume the following:

There exists a positive, unique solution of Smoluchowski’s equation (MA) in the time interval for some positive . The functions and their partial derivatives are bounded uniformly in .

The macroscopic collision rate given in section I should be Lipschitz continuous, bounded and fulfill the conditions (1) for all . The macroscopic material coefficient should obey condition (1) and together with its microscopic counterpart given in section II.B, it should fulfill (9) and be symmetric in and .

The upper bound for the microscopic interaction rates (6) is given so that the limit equation is not affected, .

The velocity fields of the different particle species have to fulfill

The diffusion constants of all particle species have to be positive, i.e. for all .
We note that our proof only applies if all particles are Brownian. Now we are ready to formulate our main convergence result.
Theorem. With conditions (C1) to (C7) and it is
(16)  
Convergence at time is given if the initial conditions of (MA) and (MI) are compatible. One possibility is to take the particle positions , as independent, identically distributed random variables with suitably normalized densities for all . For discussion of this point see [12] (sect. 4B).
To formulate the result without the smoothing convolution kernel we introduce a metric on by
(17) 
This quantifies a distance between the empirical processes defined in (MI) and the processes given by the solution (C3) of the macroscopic equation. As the theorem states convergence in an sense the convergence in the weak sense (3.2) is easy to conclude.
Corollary. With the conditions of the theorem we have
(18) 
4 Proof of the convergence result
4.1 Preliminaries
The following lemma is useful in central estimates of section IV.
Lemma. With , we have
(19) 
An analogous estimate is true, if and are replaced by and .
For any finite, positive measure on and with there is
(20) 
For any finite, signed measure on it is
(21) 
Proof. see [11] sect. 4A,B and [12] sect. 5B, or [3], sect. 4.3
In the proof of the lemma there is essentially made use of the conditions (C1) and (C2) on the interaction function and the kernel . Due to the conservation of mass in the microscopic system (9) and with (2) we get the following bound on the empirical processes,
(22) 
We also use the following property without explicitly noting it for all suitable and , such that the expressions are well defined:
(23) 
because is symmetric according to (C2). Throughout this chapter , etc. denote suitably chosen constants, whose value can vary from line to line.
4.2 Proof of the theorem
To prove statement (16) we first look at the time evolution of the quantity
(24) 
The dynamics of the first two terms is obtained analogous to (11) using (3), (8), (23) and Itô’s formula:
We just have to replace the test function in (11) by resp. . The expansion of the third term in (24) follows from the macroscopic equation (MA):
Combining the parts suitably by using (9) to express in terms of we get:
(25)  