Kinetic Theory of Cluster Dynamics
Robert I. A. Patterson
Weierstrass Institute Berlin
Mohrenstr. 39, 10117 Berlin, Germany
Zentrum Mathematik, TU München
Boltzmannstr. 3, 85748 Garching, Germany
Weierstrass Institute Berlin
Mohrenstr. 39, 10117 Berlin, Germany
In a Newtonian system with localized interactions the whole set of particles is naturally decomposed into dynamical clusters, defined as finite groups of particles having an influence on each other’s trajectory during a given interval of time. For an ideal gas with short–range intermolecular force, we provide a description of the cluster size distribution in terms of the reduced Boltzmann density. In the simplified context of Maxwell molecules, we show that a macroscopic fraction of the gas forms a giant component in finite kinetic time. The critical index of this phase transition is in agreement with previous numerical results on the elastic billiard.
Keywords: low–density gas; Boltzmann equation; cluster dynamics; Maxwell molecules.
As a proposal to gain insight on the statistical properties of large systems in a gaseous phase, N. Bogolyubov suggested to investigate a simple notion of cluster decomposition characterizing the collisional dynamics Bo46 (). When the evolution is determined by a sequence of single, distinct, two–body interactions, a natural partition of the system can be defined in terms of groups of particles connected by a chain of collisions, so that a “cluster” consists of elements having affected each other’s trajectory.
This notion has been developed later on, in connection with the problem of the Hamiltonian dynamics of an infinite system. In a Newtonian system, particles with rapidly increasing energies at infinity may generate instantaneous collapses for special initial configurations La68 (). Mathematically, one needs to prove that such initial data form a set of measure zero in the phase space of infinitely many particles. In fact, one possible strategy to construct the dynamics is to show that, at properly fixed time, the system splits up into an infinite number of clusters which are moving independently as finite-dimensional dynamical systems. After some random interval of time, the partition into independent clusters changes, and one iterates the procedure. This dynamics is known as cluster dynamics and its existence has been proved first in Si72 () for some one–dimensional models (see Si74 () for generalizations).
In more recent years, the statistical properties of cluster dynamics of a system obeying Newton’s law have been studied numerically GKBSZ08 (). In this reference, the authors focus on the frictionless elastic billiard in a square two–dimensional box with reflecting walls, and show that the dynamics undergoes a phase transition. This occurs in a way reminiscent of problems in percolation theory. Namely, the maximal (largest) cluster starts to increase dramatically at some critical time. At the critical time, the fraction of mass in the maximal cluster is rather small ( for disks at small volume density). After the critical time, it approaches the total mass of the system with exponential rate. Moreover, the transition is distinguished by a power–law behaviour for the cluster size distribution with exponent . Such critical index is believed to be universal, since it has been observed for several different models (see also McFB10 ()).
The cluster dynamics concept, together with the above described statistical behaviour, appear as well in a number of applied papers, e.g. geophysics, economics, plasma physics: see GKBSZ08 () and references therein.
Kinetic theory often provides successful methods for the computation of microscopic quantities related to properties of the dynamical system, for instance Lyapunov exponents or Kolmogorov–Sinai entropies vZvBD00 (); vBDDP97 (); Co97 (); Sp91 (). In the present work, we are concerned with the cluster dynamics of an ideal gas where the kinetic description of Boltzmann based on molecular chaos applies.
Our setting is given by a density function describing the amount of molecules having position and velocity at time , and evolution ruled by
where , is a pair of velocities in incoming collision configuration and is the corresponding pair of outgoing velocities when the scattering vector is :
The time–zero density is fixed. For simplicity, the gas moves in the square dimensional box of volume , with reflecting boundary conditions. The microscopic potential is assumed to be short–ranged and the cross–section satisfies (“Grad’s cut–off assumption”).
The precise connection with a dynamical system of particles interacting at mutual distance , such as the one studied in GKBSZ08 (), can be established locally in the low–density limit
(“Boltzmann–Grad regime”) as the convergence of correlation functions to the solution of the Boltzmann equation La75 () (see also Uc88 (); Sp91 (); CIP94 (); GSRT12 (); PSS13 ()). In the regime (1.3), the gas is so dilute that only two–body collisions are relevant. Furthermore, the collisions are completely localized in space and time. The limit transition (1.3) explains the microscopic origin of irreversible behaviour Gr58 ().
Our purpose here is to describe how the cluster size distribution is constructed from the solution to the Boltzmann equation. This is done in Section 2 by means of a suitable tree graph expansion, which is inspired by previously known formulas representing the Boltzmann density as a sum over collision sequences Wi51 (). In Section 3, we indicate how to derive formally the introduced expressions as the limiting cluster distributions of a system of hard spheres in the Boltzmann–Grad scaling. Finally, in Section 4, we restrict to the simplest nontrivial (and paradigmatic) case in kinetic theory, i.e. the model of Maxwellian molecules. We show that the cluster distribution exhibits a phase transition characterized by a breakdown of the normalization condition at finite time. This implies that the “percolation” survives in the Boltzmann–Grad limit, with same qualitative behaviour and same critical index of the elastic billiard analyzed in GKBSZ08 ().
2 cluster distributions
2.1 Bogolyubov clusters
We start with a formal notion of cluster. Let be a given positive time.
(i) Two particles are neighbours if they collided during the time interval .
(ii) A Bogolyubov cluster is any connected component of the neighbour relation (i).
The definition can be generalized to generic time intervals . However, in what follows we will study the notion of cluster only, which is no restriction, and drop often the -dependence in the nomenclature.
Notice that each particle has collided with at least one other particle of its Bogolyubov cluster, while it has never collided with particles outside the cluster, within the time interval . In particular, if , any particle of the gas forms a singleton (cluster of size ). At , the mass of singletons starts to decrease and clusters of size start to appear. We therefore expect to see (and do observe in the experiments) some “smooth” exponential distribution in the cluster size.
2.2 Backward clusters
where , and . The sequences are in one–to–one correspondence with binary tree graphs, e.g.
for respectively. In (2.1), is interpreted as the contribution to the probability density due to the event: the backward cluster of has structure . By “backward cluster” we mean here the group of particles involved directly or indirectly in the backwards–in–time dynamics of particle . Operatively, in a numerical experiment, we select particle at time , run the system backwards in time, and collect all the particles which collide with and with “descendants” of in the backwards dynamics, following (2.2). In other words, (2.1) is an expansion on sequences of real collisions111Observe that recollisions (e.g. the pair colliding twice in the backward history), certainly possible in an experiment, do not affect the notion of backward cluster, which is based on sequences of collisions involving at least one “new” particle; see APST14 () for details on the numerical procedure..
Formulas of this kind have been previously studied in the context of Maxwellian molecules with cut–off under the name of Wild sums Wi51 (); Mc66 (); CCG00 (), and are written in APST14 () for a gas of hard spheres in a homogeneous state. It is not difficult to generalize such a representation to inhomogeneous states and general interactions. The formula for reads
– , (position, velocity);
– is the initial density (and );
– the “free–flight rate” of particles is given by
where the function depends on the solution of the Boltzmann equation itself:
– the “trajectory of the backward cluster” is constructed as follows:
(a) fix , with
(b) construct first the sequence of velocities , , defined iteratively by:
where, at step , the pair are the pre–collisional velocities (in the collision with impact vector ) of the pair (which are post–collisional, as ensured by the fact that only for ), according to the transformation (1.2);
(c) construct the trajectory of the backward cluster iteratively by
and, for and ,
with the convention .
The term in (2.2) is
and density of free particles in . Similarly, density of particles having a backward cluster of size in .
We can see as an integral over trajectories of a Markov process with collisions in specified order. Each trajectory has probability density given by the integrand function, that is:
We remind that, in a backward cluster, the trajectory of particle is specified only in the time interval , where .
The density of Bogolyubov clusters can be obtained by “adding” the missing information, namely the future history of the particles in the time intervals respectively, together with the complete history of the particles with whom they collide. This amounts, for instance, to extend in the future, by free motion, the trajectories of
where the dotted lines correspond to free–flight. However, this example is not enough, since we need to take into account additional trajectories, e.g.
In other words, we extend the history of the backward cluster to provide full knowledge of the trajectory of the particles in the time interval .
We make this more precise in the rest of this Section. Our goal is to write a formula for the density of Bogolyubov clusters (Definition 2 below) starting from (2.2) & (LABEL:eq:pdtr). Before that, we need to introduce notions of ‘collision graph’ and of the associated ‘trajectory of clusters’. The density of clusters will be indeed expressed as an integral over such trajectories.
Let be a labelled tree with vertices, i.e. a connected graph with vertices and edges. For instance for
The vertices are labelled and (non ordered set of pairs). Each vertex represents a particle and each link represents a collision. We refer to as collision graph of the Bogolyubov cluster.
A trajectory of the Bogolyubov cluster , where (position, velocity), is constructed as follows.
(a) Fix , with ;
(b) let be the permutation of such that ; construct the sequence of velocities , , defined iteratively by:
where, at step , the pair are the pre–collisional velocities (in the collision with impact vector ) of the pair (assumed post–collisional), according to the transformation (1.2);
(c) define the trajectory of particle of the Bogolyubov cluster iteratively by
and, for ,
with the convention . Notice that the positions at time are uniquely determined as soon as we fix .
With respect to the definition of trajectory used in (2.2), the essential differences are summarized in the following table.
|sequence of collisions||times of collisions||history of particle|
|Backward cluster, size||tree graph||specified in|
|Bogolyubov cluster, size||tree graph||specified in|
In both cases, there are exactly collisions, since recollisions are forbidden in the assumed kinetic regime (see page 3, item (2) for a precise statement). Such collisions are specified, respectively, by the binary tree graph and by the ordinary tree graph .
The Bogolyubov cluster can be described as the time–symmetrized version of the backward cluster. Alternatively, this can be seen as a symmetrization in the labelling of the particles, since no particle plays a special role anymore.
2.4 Size distribution of kinetic clusters
Motivated by the previous discussion, we introduce here an explicit written in terms of the Boltzmann density solving (1.1), which should be interpreted physically as the fraction of particles of the gas belonging to a dynamical cluster of size . A formal argument identifying as the kinetic limit of the corresponding quantity in a system of finitely many particles will be presented below (see Claim 1); a rigorous derivation remains an open problem.
Let and , then
with the conventions .
Compared to , see (2.2) and (LABEL:eq:pdtr), has been replaced by the symmetric integral , and the integrand function is now the probability density of a trajectory of a Bogolyubov cluster with collision graph .
By looking at (2.10) in a simple example, we will show in Section 4 that this distribution is not normalized for all times. This is due to the development of giant clusters () at some critical time . Therefore, with respect to backward clusters (often associated to the description of correlations APST14 ()), the Bogolyubov clusters exhibit a more interesting statistics.
The following rescaled version is also relevant.
The kinetic fraction of Bogolyubov clusters with size is
is the normalization constant.
The functions just introduced are the kinetic counterparts of the functions and examined in GKBSZ08 (). This connection with the finite dynamical system is clarified in the next Section.
3 Derivation of (2.10)
We consider here, for simplicity, a system of identical hard spheres. These particles have unit mass and diameter and move inside the box with reflecting boundary conditions. The dynamics is given by free flow plus collisions at distance , which are governed by the laws of elastic reflection. We label the particles with an index . The complete configuration of the system is then given by a vector , where collects position and velocity of particle . Let us assign a probability density on the particle phase space, assuming it symmetric in the exchange of the particles. For , we call the particle marginal of .
Let be the dynamical flows, considered in isolation, of the groups of particles and respectively. Set ( only up to the time of the first collision between the two groups). We denote , the time of the first collision of particle with the set of particles in the dynamics . Furthermore, we set , that is the time of the first collision of the group with the group .
Let us focus on Definition 2 and let
By the symmetry in the particle labels, the average of with respect to is
where is the characteristic function of the set , and the last line in (3.2) defines the conditional probability.
More generally, we indicate by a conditional probability given and . Similarly, is the conditional probability density of , given and . Using again the symmetry,
so that by iteration we obtain
The term with is defined to be .
Next we evaluate . Let be the velocity of particle at time , the index of the particle that particle will hit, and the normalized relative displacement of particle with respect to particle . Let be position and velocity of particle along the flow . By definition of , we must find particle in a cylinder of volume for some . We remind that, for hard spheres, the cross–section is . It follows that, assuming the Boltzmann approximation and 222As follows from the statistical independence of particle from any finite collection of given particles, in the limit ., the conditional probability will be close to
Inserting this into (3.2), we find
We further assume:
(1) the marginals of the initial distribution factorize in the Boltzmann–Grad limit: ;
(2) we may neglect recollisions; namely, the lower order term in (3.2) is given by trajectories of the flow showing exactly collisions.
If this is true, then where the tree graph specifies which pairs of particles collide in the cluster. It follows that
Applying collision transforms, the measure can be rewritten as
where are position of particle and velocities of the particles of the cluster at time , while are the collision times, is the relative distance of and at the collision time , is the outgoing relative velocity at collision, and the corresponding cross–section (see Si13 () for a similar computation).
In the notations of the previous Section, this implies
By (1.3), . Therefore, we conclude that
This heuristic discussion leads to the following claim.
Assume as , where is a solution of the Boltzmann equation with initial datum . The average density of clusters with size at time , satisifies
as , where is given by (2.10).
This is a law of large numbers type of result.
Similarly, the total number of clusters has fractional average
given by (2.12), while the average fraction of clusters
given by (2.11).
This relation, however, breaks in the Boltzmann–Grad limit. In fact, we will show in the following section that is not normalized.
It would be interesting to write a rigorous derivation of (2.10) along the above or similar arguments, using the known results of convergence to the Boltzmann equation. This will be matter of further investigation. An exact perturbative expression for the path measure of one tagged sphere in a gas at thermal equilibrium can be found in LS82 ().
4 Phase transition for Maxwell molecules
It has been observed in Reference GKBSZ08 () that the cluster size distribution of an elastic billiard consisting of disks with volume density is well described by a power law with exponential damping
The strength of damping turns out to be monotone decreasing for small times and such that as . Close to the critical time , the distribution transforms sharply from exponential to pure power law. Moreover, after the critical time, a distinct gap appears between the largest cluster and the rest of the distribution. The maximal cluster starts to dominate as in a percolation model.
Consider now the Boltzmann equation model