Self-gravitating Brownian particles in two dimensions: the case of particles
Systems with long-range interactions have recently been the object of considerable interest . One usually considers isolated systems in which the particles evolve according to deterministic Hamiltonian equations. These systems are described by the microcanonical ensemble. Examples of such systems include self-gravitating systems, two-dimensional point vortices, the Hamiltonian mean field (HMF) model etc. However, one may also consider dissipative systems in which the particles, in contact with a thermal bath, evolve according to stochastic Langevin equations. These systems are described by the canonical ensemble. The statistical mechanics of Hamiltonian and Brownian systems with long-range interactions is discussed in  at a general level. In this paper, we consider the case of Brownian particles in gravitational interaction. It is known that this system bears deep analogies with simple models of bacterial populations experiencing chemotaxis in biology
The paper is organized as follows. In Section 2 we recall the -body coupled stochastic equations describing the evolution of self-gravitating Brownian particles and specifically consider the case . We introduce the center of mass and the reduced particle. We show that the center of mass undergoes a pure Brownian motion and that the reduced particle undergoes a Brownian motion in a central potential . We also recall the “naive” virial theorem obtained in  and discuss, with a new light, the distinction between the critical temperatures and . In Section 3, we study the motion of a Brownian particle (reduced particle) in an attractive central potential in . We show that the corresponding Fokker-Planck equation is equivalent to a Schrödinger equation (with imaginary time) with a potential . This equation can be solved analytically in terms of Bessel functions. Then, we can obtain various analytical results such as the probability to find the reduced particle at position at time , the probability that the particle reaches the origin for the first time between and , the probability that the particle has reached the origin at time and the variance of the distribution. We find that the reduced particle has a normal diffusion behavior for small times with a gravity-modified diffusion coefficient and an anomalous diffusion for large times . In particular, the variance increases with time when and tends to zero for when . In Section 4, we consider the case of two self-gravitating Brownian particles in a bounded domain and discuss the differences with the case of an infinite domain. Finally, in Section 5 we show that our study also describes the large time asymptotics of the Smoluchowski-Poisson system (or Keller-Segel model) for . Indeed, in the post-collapse regime, the system is made of a growing central Dirac peak (condensate) surrounded by a dilute halo whose dynamical evolution is eventually described by a Fokker-Planck equation similar to the one studied in the case of particles. We find that the saturation of the mass of the condensate to the total mass is algebraic in an infinite domain and exponential in a bounded domain and we characterize it precisely. In Section 6, we briefly generalize our results to the logarithmic Fokker-Planck equation in dimensions. The Appendices provide complements such as the deterministic limit (Sec. Appendix C), the van Kampen classification (Sec. Appendix F), the correlation functions (Sec. Appendix G) and the general form of the virial theorem for Brownian particles with power law interaction (Sec. Appendix I).
We may note that our study bears some similarities with the stochastic motion (induced by viscosity) of two point vortices studied by Agullo & Verga . However, there also exists crucial differences between the two problems since in our case the interaction is radial leading to the formation of Dirac peaks while in the case of point vortices the interactions is rotational leading to the formation of a spiral structure.
We may also note that the statistical mechanics of particles in gravitational interaction has been considered by Padmanabhan  in (and generalized by Chavanis  for the dimensions and ) in the microcanonical and canonical ensembles. However, these authors consider the equilibrium statistical mechanics of self-gravitating particles in a box, and with a small-scale cut-off, while we consider here the dynamical evolution of self-gravitating Brownian particles in a finite or infinite domain without small-scale cut-off. Therefore, we address the time dependent problem and investigate the formation of Dirac peaks.
Finally, the particular character of the dimension in gravity is well-known. We refer for example to  for more details and further references.
2The position of the problem
2.1The -body problem
We consider a system of overdamped Brownian particles with mass in gravitational interaction in a space of dimension . Their motion is described by the coupled stochastic equations :
for . Here, is the friction coefficient and is a white noise satisfying and where refers to the particles and to the coordinates of space. The diffusion coefficient is given by the Einstein relation
where is the temperature. We assume that the friction is the same for all the particles.
From these stochastic equations, it is possible to derive the Fokker-Planck equation for the -body distribution and then write the BBGKY-hierarchy for the reduced distributions . Let us consider for brevity the single-species system. The proper thermodynamic limit corresponds to in such a way that the normalized temperature is of order unity. In that limit, it can be shown that the mean field approximation becomes exact so that the -body distribution factorizes in a product of one-body distributions: . Furthermore, the one-body distribution, or equivalently the smooth density field , is solution of the Smoluchowski-Poisson system :
Up to a change of notations, these equations are isomorphic to a simplified version of the Keller-Segel model of chemotaxis that is valid in the limit of large diffusivity of the chemical and in the absence of degradation .
2.2The case : the reduced particle
with and . Like for the standard two-body problem, we introduce the center of mass
and the reduced particle
Concerning the motion of the center of mass, we have
where the noise satisfies
Therefore, the center of mass undergoes a pure Brownian motion of the form
with a diffusion coefficient
Concerning the motion of the reduced particle, we have
where the noise satisfies
Therefore, the reduced particle undergoes a Brownian motion in a central potential of the form
with a diffusion coefficient
2.3The naive virial theorem
Let us introduce the total moment of inertia
with the critical temperature
It is instructive to recover this result in a different manner. The positions of the particles and can be expressed in terms of (reduced particle) and (center of mass) as
Substituting these relations in Eq. (Equation 19), we obtain after straightforward algebra
a relation which was of course expected. Now, the Fokker-Planck equation associated with the stochastic motion (Equation 17) of the reduced particle is
Taking the time derivative of
and using simple integrations by parts, we naively
This relation exhibits a critical temperature
Introducing the moment of inertia of the reduced particle , we can rewrite Eq. (Equation 26) as
The mean square displacement of the reduced particle satisfies
This is a normal diffusion with a gravity modified diffusion coefficient
The variance increases for and tends to zero in a finite time for . On the other hand, the Fokker-Planck equation associated to the stochastic motion (Equation 13) of the center of mass is simply
and we classically obtain the relation
Finally, summing Eqs. (Equation 26) and (Equation 32) and using Eq. (Equation 23), we recover Eq. (Equation 20). We now clearly see the origin of the two temperatures and that were reported in . In the case , the critical temperature is associated to the dynamics of the reduced particle while the critical temperature enters in the expression of virial theorem for the total moment of inertia (reduced particle and center of mass). This distinction is further discussed in Appendix A in the general case of particles.
In fact, there is a flaw in the above derivation of the virial theorem because we have naively assumed that the normalization is conserved in time. However, as we shall see, this is not correct. The normalization is not conserved in time because the reduced particle can reach the origin and be “lost” by the system (if it reaches the origin, it remains there for ever). This corresponds to the coalescence of the two particles, resulting in the formation of a Dirac peak, i.e. a new particle of mass . As a result of these “trapping” events
and we must reconsider the problem in more detail.
3Brownian particle in a Newtonian potential in two dimensions
3.1The Fokker-Planck equation
Let denote the probability density of finding the reduced particle in at time . The evolution of is governed by the Fokker-Planck equation
The initial distribution is normalized such that . Introducing
the Fokker-Planck equation can be rewritten
In the absence of small and large scale cut-offs, this equation has no steady state since the distribution is not normalizable. We assume that the initial distribution is radially symmetric, so that is radially symmetric for all times. Therefore, we can write the Fokker-Planck equation in the form
As discussed previously, the probability is not conserved because the reduced particle may reach the origin and form a Dirac peak (the two particles coalesce). The probability that the particle has not reached at time is
Taking the time derivative of this quantity and using the Fokker-Planck equation (Equation 37) we obtain
which is non zero since . Therefore, the probability for the particle to form a Dirac peak between and (i.e. to reach for the first time between and ) is
and the probability for the particle to have formed a Dirac peak at time (i.e. to have reached at time ) is
We obviously have . We can now obtain the proper form of the virial theorem associated to the Fokker-Planck equation (Equation 34). Introducing the moment of inertia of the reduced particle
we easily obtain the virial theorem
3.2The associated Schrödinger equation
Let us consider a general Fokker-Planck equation of the form
For a spherically symmetric distribution in dimensions, it can be rewritten
As is well-known , we can transform this Fokker-Planck equation into a Schrödinger equation (with imaginary time) by setting
with the potential
For a spherically symmetric distribution, the Schrödinger equation (Equation 47) can be rewritten
Making the separation of variables
we obtain the eigenvalue equation
Let us note the eigenvalues and the corresponding eigenfunctions. The eigenfunctions are orthogonal with respect to the scalar product
We also normalize them so that . Then, any function can be expanded in the form
If the spectrum is continuous, the sum over must be replaced by an integral over .
3.3The general solution
We consider the Green function which corresponds to the initial condition
The solution on the Fokker-Planck equation (Equation 45) can be expanded on the eigenfunctions in the form
and using the initial condition (Equation 55), we finally obtain
3.4The case of a logarithmic potential in
Therefore, if we assume that initially
the solution of the Fokker-Planck equation (Equation 37) can be written
where is solution of the differential equation
Equation (Equation 62) is a Bessel differential equation that can be solved analytically. The solutions that are finite at the origin are of the form
Using the closure relation
Using the identity
valid for , we find that it can finally be written
The distribution is plotted in Figure 1 at different times and for (corresponding to ).
Using the identity
we get for :
which tends to Eq. (Equation 60) as expected. On the other hand, for , the probability tends to zero meaning that the particle has been absorbed in after a sufficiently long time so that it is ultimately lost by the system. For , using the identity (Equation 72), we have
For , using the identity
Finally, for , the probability density (Equation 71) becomes
which is the solution of the diffusion equation in . Indeed, in the limit of infinite temperature, the gravity is negligible with respect to diffusion.
3.5The probability to form a Dirac peak
is the incomplete Gamma function. The probability decays because, as time goes on, the particle has more and more chance to reach and form a Dirac. The probability that the particle reaches for the first time between and is given by Eq. (Equation 40). Combining this relation with Eq. (Equation 76), we obtain
Integrating Eq. (Equation 80), we obtain the probability that the particle has formed a Dirac peak at time t:
For , using the expansion
We see that the probability for due to the exponential factor. This tendency is reinforced by the algebraic factor for () while it is reduced for ().
For , using the expansion
Therefore, the probability that the particle has not formed a Dirac at time decreases algebraically as . Equation (Equation 85) can be written in the form
where the time
gives an idea of the rapidity at which the Dirac forms as a function of the temperature . The function is represented in Figure 4 and its asymptotic behaviors are given in Appendix Appendix E. We find that for and for . We note that tends to a finite value for while we know that the system does not form a Dirac peak for (indeed for ). Therefore, the physical interpretation of should be considered with care. Another measure of the effect of the temperature on the formation of the Dirac is provided by the quantity
Finally, the normalized probability density can be written
3.6The moment of inertia
The moment of inertia of the reduced particle is defined by
and the variance of the distribution (mean square displacement) is
For the density distribution given by Eq. (Equation 71), the integral can be calculated explicitly yielding
In Appendix B, we check that this relation is consistent with the virial theorem (Equation 43). The evolution of the moment of inertia is represented in Figure 6 for different values of the temperature. For , the moment of inertia increases, for the moment of inertia is constant and for the moment of inertia decreases. This will become clear from the asymptotic behaviors.
For , we get
which can be rewritten
In that case, we have a normal diffusion with a gravity-modified diffusion coefficient
For the variance increases with time while for it decreases. This expression agrees with the naive virial theorem (Equation 29). Indeed, for small times, the probability for the particle to reach is exponentially small so that the probability is conserved and the naive virial theorem holds since there is no Dirac peak.
For , using the expansion (Equation 84), we get
This corresponds to
For , i.e. , the variance increases and goes to for large times. In that case, we have an anomalous diffusion with an exponent . The evolution is always sub-diffusive. The origin of the anomalous diffusion is related to the fact that the particle can be trapped at (and form a Dirac peak). For , the variance decreases and goes to for .
3.7The most probable position
The most probable value of the distribution is obtained by maximizing , or equivalently , with respect to . This gives
Using the recurrence relation
This equation can be rewritten in the parametric form
which gives . Using the asymptotic expansions of , we find that
Therefore, the radius is decreasing for any temperature and it goes to zero in a finite time depending on the temperature. Some curves are represented in Figure 7.