# Ballistic aggregation for one-sided Brownian initial velocity

## Abstract

We study the one-dimensional ballistic aggregation process in the continuum limit for one-sided Brownian initial velocity (i.e. particles merge when they collide and move freely between collisions, and in the continuum limit the initial velocity on the right side is a Brownian motion that starts from the origin ). We consider the cases where the left side is either at rest or empty at . We derive explicit expressions for the velocity distribution and the mean density and current profiles built by this out-of-equilibrium system. We find that on the right side the mean density remains constant whereas the mean current is uniform and grows linearly with time. All quantities show an exponential decay on the far left. We also obtain the properties of the leftmost cluster that travels towards the left. We find that in both cases relevant lengths and masses scale as and the evolution is self-similar.

###### keywords:

Adhesive dynamics, Ballistic aggregation, Inviscid Burgers equation, non-equilibrium statistical mechanics^{1}

## 1 Introduction

We consider in this article the continuum limit of a one-dimensional ballistic aggregation process, for the case of Brownian initial velocities (i.e. the initial velocity field is a Brownian motion). In such a model, point particles of identical mass move on a line and perform completely inelastic collisions, that is, in binary collisions particles (or clumps) merge to form a single larger aggregate under conservation of mass and momentum (but dissipation of energy). Between collisions clumps move at constant velocity (free motion). Thus, without external forcing, the stochasticity is only due to the randomness of the initial velocities. This model was introduced in Carnevale et al. (1990), for the case of uncorrelated initial velocities (i.e. white-noise case in the continuum limit), as a simple test-case for scaling arguments used in more general hydrodynamical or statistical systems. Indeed, this ballistic aggregation process can be seen as a simple model for the merger of coherent structures, such as vortices, thermal plumes, or cosmic dust into planetesimals within proto-planetary disks.

In this context it is natural to investigate the late-time asymptotic scaling regime obtained for the case of uncorrelated initial velocities. Thus, one finds that the average cluster mass grows with time as with a large-mass tail for the universal mass distribution of the form Carnevale et al. (1990); Frachebourg (1999); Frachebourg et al. (2000). When the number of particles is finite, at long times the system reaches a stationary “fan” state, where the velocities of the final clusters increase from left to right. This final state also shows many universal properties, such as the number and size distributions of final clusters and the size of the leftmost and rightmost clusters Suidan (2000); Majumdar et al. (2008). On the other hand, when the initial particle velocities are given by a Brownian motion this ballistic aggregation process can be related to a simple additive coalescent model (which does not take into account positions nor velocities), where each pair of clusters merges with a rate proportional to its total mass, independently of other pairs Bertoin (2000). This also provides results for the statistics of dislocation of clusters in the time-reversed fragmentation process.

In the continuum limit, at fixed uniform initial density , it is well known that this system can be mapped onto the Burgers equation in the inviscid limit Burgers (1974); Gurbatov et al. (1991); Kida (1979); Frachebourg et al. (2000),

(1) |

Then, shock locations in the Eulerian velocity field describe particle aggregates of finite mass whereas regular points correspond to infinitesimal particles. It is clear that away from shocks Eq.(1) corresponds to free motion (in the limit where the right-hand side vanishes) and it can be shown that shocks conserve momentum Burgers (1974), which explains the relation with the ballistic aggregation process.

The Burgers equation (1) itself is a nonlinear evolution equation that appears in many physical problems, such as turbulence studies Burgers (1974); Kida (1979), the propagation of nonlinear acoustic waves Gurbatov et al. (1991), or the formation of large-scale structures in cosmology Gurbatov et al. (1989); Vergassola et al. (1994), see the recent review Bec & Khanin (2007) for a detailed discussion. In particular, the study of the statistical properties of the dynamics, starting with random Gaussian initial conditions, is also referred to as “decaying Burgers turbulence” in the hydrodynamical context Gurbatov et al. (1997), or as the “adhesion model” in the cosmological context Gurbatov et al. (1989). In these frameworks, where the random initial velocity applies over all space, and may be homogeneous or only have homogeneous increments, one is interested in Eulerian quantities such as the velocity structure functions, the -point velocity distributions, the matter density distribution, the mass function of shocks, or Lagrangian quantities such as the distribution of the displacement field. Then, it is customary to consider power-law initial energy spectra, , where the initial velocity fluctuations scale as over size (the white-noise case is and the Brownian case is ). Then, for , a self-similar evolution develops Gurbatov et al. (1991); Molchanov et al. (1995). The integral scale of turbulence, which measures the typical distance between shocks and the correlation length, grows as whereas the tails of the cumulative shock distribution and velocity distribution satisfy , , for , see She et al. (1992); Molchan (1997). In such a context, the white-noise case, , corresponds to initial velocity fluctuations that are dominated by high wavenumbers, whereas they are governed by low wavenumbers in the Brownian case. Then, this latter case is of particular interest since in many hydrodynamical systems the power is generated by the larger scales. For instance, the Kolmogorov spectrum of turbulence, , shows such an infrared divergence, whereas in cosmology the velocity fluctuations are also governed by scales that are larger than the nonlinear scales where structures have already formed in the density field.

The connection with the Burgers equation (1) allows us to derive many results for the continuum limit of the ballistic aggregation process, taking advantage of its well-known Hopf-Cole solution Hopf (1950); Cole (1951). (This also corresponds to the late-time evolution of the system if we keep a finite particle mass.) In particular, using the geometrical description of this solution in terms of first-contact points between the initial velocity potential and parabolas Burgers (1974), as recalled below in (8), or the equivalent description in terms of the convex hull of the Lagrangian potential Bec & Khanin (2007), it is possible to derive closed analytical results for the specific cases of white-noise and Brownian initial velocities. Indeed, in these two cases the velocity or potential fields are Markovian which allows us to greatly simplify the analysis Bertoin (1998); Frachebourg & Martin (2000); Valageas (2008). Moreover, a specific property obtained for Brownian initial conditions is that the inverse Lagrangian map, , where is the initial Lagrangian position of the particle located at at time , keeps homogeneous increments at all times (on the right side for one-sided initial conditions and far from the origin for two-sided initial conditions) Bertoin (1998); Valageas (2008). This can actually be extended to some Lévy processes with no positive jumps Bertoin (1998).

Then, for both white-noise and Brownian one-sided cases, Bertoin et al. (2001) found that the flux through the origin is a pure jump time-inhomogeneous Markov process and obtained its statistical distribution. While for the white-noise case clusters (shocks) are in finite number per unit length, which implies that the mass that has flowed to the the left side increases through a finite number of jumps per unit time, for the Brownian case clusters are dense (on the right side), which implies that on any time interval some small-mass clusters have crossed the origin She et al. (1992); Bertoin et al. (2001). Some properties of the limit clusters travelling towards the left were obtained in Isozaki (2006) for the white-noise case, while for the Brownian case it was found that shocks are dense to the right of the leftmost cluster, as on the right side, but no results for the statistics of the latter were obtained. Statistical properties of limit clusters (i.e. at the infinite time limit) were also obtained in Winkel (2002) for initial velocities that are given by a Lévy process with no positive jumps.

For the white-noise case, Frachebourg et al. (2000, 2001) studied the late-time dynamics reached when the “excited” particles are restricted to the semi-infinite right side, or to a finite interval, and expand into empty space or a medium at rest. Many explicit analytical results can be derived in the continuum limit Frachebourg et al. (2001). If the initial “excited” interval is finite and surrounded by empty space, the late-time evolution is ballistic and the characteristic length scales as . Indeed, since the total mass is finite, the system eventually reaches a “fan” state with a finite number of clusters that move freely without anymore collisions. If the “excited” particles expand into a medium of uniform density at rest, the latter slows down the motion and the characteristic length scales as . For one-sided initial conditions, where the initial white-noise velocities apply to the semi-infinite right axis, one recovers the scaling law associated with the homogeneous turbulent case recalled above, and . Thus, nontrivial mass density and current profiles develop over this scale, with a power-law tail on the far left. In addition, other nontrivial asymptotic mass profiles are obtained over scales , with , that interpolate between the natural scaling and the ballistic regime . This corresponds to forerunners that carry a mass , interpolating between masses of order and of order unity. Finally, if the “excited” particles expand into a filled medium at rest, other profiles develop on the natural scale , but there is no more propagation on larger scales, , because of the slowing down by the left-side particles, and the density decays exponentially fast on the left side.

In this article, we study how these results are modified when the initial velocities on the right side are given by a Brownian motion, instead of a white-noise spectrum, and we also consider the statistical properties of the leftmost (“leader”) cluster that has formed on the left side. As noticed above, the Brownian case is a template for large-scale forcing as opposed to small-scale forcing in the initial velocity field. Thus, we consider the case where the initial velocity field, at time , is a Brownian motion on the semi-infinite axis , while the initial density is constant over , and we write

(2) |

Here we introduced the velocity potential , with , and a Gaussian white-noise , which we normalize by

(3) |

where is the average over all realizations of . In Eq.(2) we normalized the initial velocity and potential by and at the origin. We consider two cases, “” and “”, where the left semi-infinite axis, , is either filled with particles at the same density but with zero initial velocity (medium at rest), or empty (zero density, ). Therefore, we complete the definition (2) by

(4) | |||||

(5) |

Here, as in Isozaki (2006), we used the fact that the empty case, “”, of (5), can be obtained by keeping the same uniform initial density over , while giving to these particles a velocity that goes to . Then, these particles immediately escape to the infinite left at and the Brownian particles with spread into empty space. These initial conditions can be summarized by stating that the initial velocity potential is either continuous and constant out of the Brownian region (filled case “”) or goes to (empty case “”).

At any point on the left part, the system remains unchanged (at rest or empty) until some particles that originate from the right-side Brownian region have managed to travel down to position . Note that once some particles have entered the left part they will keep travelling with a negative velocity forever. However, their velocity can change as they may overtake slower particles or may be overtaken by faster particles that escaped at a later time from the Brownian region. Since particles do not cross, the leftmost cluster is associated with the particle, , that was initially at the left boundary of the Brownian domain. Once the latter has entered the left side, it keeps moving to the left and in case it draws along all the matter that was initially at rest.

In the context of the ballistic aggregation process studied in this article, and contrary to the hydrodynamical context where the Burgers equation (1) is used to investigate statistically homogeneous turbulence, the systems defined by Eqs.(2)-(5) are clearly statistically inhomogeneous and a current develops towards the left side as particles escape into the left part of the system and then keep travelling to the left forever. Therefore, we mainly focus on quantities that express this out-of-equilibrium propagation of matter towards the left. Indeed, the conditional probabilities to the right of the leftmost cluster are identical to the ones obtained for the two-sided Brownian-motion initial velocity Valageas (2008). In particular, the distribution of velocity increments and of the matter density are the same on the right side , see also Bertoin (1998).

We recall in section 2 the geometrical construction in terms of first-contact parabolas of the solution of Eq.(1) and the associated Brownian propagators. Next, we first consider the case “” of a filled left-side at rest, and we study the velocity distribution as well as the probability that matter from the right side has already reached the position on the left side by time . We consider the mean density profile and current in section 3.2. Then, we derive the Lagrangian displacement field in section 3.3 and we obtain in section 3.4 the properties of the leftmost cluster. Finally, we consider the case “” of the empty left side in section 4.

## 2 Geometrical construction and Brownian propagators with parabolic absorbing barrier

As recalled above, in the continuum limit, at fixed density , the ballistic aggregation dynamics is fully described by the Burgers equation (1) in the limit of zero viscosity. As is well known Hopf (1950); Cole (1951), substituting for the velocity potential , with , and making the change of variable , transforms the nonlinear Burgers equation into the linear heat equation. This provides the explicit solution of Eq.(1) for any initial condition, and in the limit a saddle-point method gives

(6) |

where we introduced the Lagrangian coordinate defined by

(7) |

The Eulerian locations where there are two solutions to the minimization problem (7) correspond to shocks (and all the matter initially between and is gathered at ), that is to clumps of particles of finite mass. The application is usually called the Lagrangian map, and the inverse Lagrangian map (which is discontinuous at shock locations). For the case of Brownian initial velocity that we consider in this paper, it is known that the set of regular Lagrangian points has a Hausdorff dimension of Sinai (1992), whereas shock locations are dense in Eulerian space Sinai (1992); She et al. (1992), in the Brownian region.

As is well known Burgers (1974), the minimization problem (7)
has a nice geometrical solution. Indeed, let us consider the
downward^{2}

(8) |

Then, starting from below with a large negative value of , such that the parabola is everywhere well below (this is possible thanks to the scaling for the integral of the Brownian motion, which shows that only grows as at large ), we increase until the two curves touch one another. Then, the abscissa of the point of contact is the Lagrangian coordinate and the potential is given by .

This geometrical construction clearly shows that a key quantity is the conditional
probability density,

,
for the Markov process
, starting from at ,
to end at at , while staying above the
parabolic barrier, , for . Following
Frachebourg & Martin (2000) (who studied the case of two-sided white-noise initial
velocity) and Valageas (2008) (who studied the case of two-sided Brownian initial
velocity), we shall obtain the properties of the system from this propagator.
It was derived in Valageas (2008) who obtained

(9) |

with

(10) |

Here we introduced the dimensionless coordinates (which we shall note by capital letters in this article)

(11) |

For completeness, we give in Appendix A the expression of the reduced propagator and of two associated kernels.

We can note that, thanks to the scale invariance of the Brownian motion, the scaled initial potential has the same probability distribution as , for any . Then, using the explicit solution (6) we obtain the scaling laws

(12) |

where means that both sides have the same probability distribution. Indeed, the initial conditions on the left side, or , do not break these scalings. This implies that all our results can be written in terms of the dimensionless variables (11), as we shall check below. This would no longer hold if the Brownian domain were restricted to a finite interval , since the size would add a new length scale into the problem which can give rise to other scalings. In particular, at late times one would find a simple ballistic propagation to the left or right side in the empty case, when all high-velocity particles have already escaped from the Brownian domain.

Finally, we may note that we defined the continuum limit as at fixed density . The same limit also describes the large time behavior of the system (i.e. ) at fixed reduced lengths and masses and , as in (11). In the first point of view, we consider the properties of the system at any finite time, arising from a distribution of infinitesimal particles, whereas in the second point of view we keep the discrete nature of the initial system but we look at the asymptotic late-time distribution over large scales and masses that grow as (so that discrete effects become subdominant); see Frachebourg et al. (2000) for more detailed analysis in the case of white-noise initial velocity.

## 3 Case “”: expansion into an uniform medium at rest

We first investigate the case “”, defined in (4), where the left side, , is initially filled with particles at the same uniform density with zero velocity.

### 3.1 Eulerian velocity distribution and probability of having been shocked

We first consider the one-point velocity distribution, , at the Eulerian location , as well as the distribution, , of the Lagrangian coordinate associated with the particle that is located at position at time . From Eq.(6) they are related by

(13) |

since is well defined for any except over a set of zero measure in Eulerian space associated with shocks She et al. (1992). As in Valageas (2008), we first consider the cumulative probability, , that the Lagrangian coordinate is within the range . From the geometrical construction (8), this is the integral over the parabola height of the bivariate probability distribution, , that the first-contact point of the potential with the family of downward parabolas , with increasing from , occurs at an abscissa in the range , with a parabola of height between and . In terms of the propagator introduced in Eq.(9) this probability density reads as

(14) |

Here we used the Markovian character of the process , which allows us to factorize the probability into two terms, which correspond to the probabilities that i) stays above , but does not everywhere remain above , over the range , while reaching an arbitrary value at , over which we will integrate, and ii) stays above for . We show in Fig. 1 the geometrical interpretation of Eq.(14) for a case with (we did not try to draw on the right side an actual Brownian curve which has no finite second derivative).

The constraint associated with the left part of the potential at merely translates into an upper bound for the parabola height . Thus, if , we must integrate over up to the value where the parabola runs through the origin . Indeed, it is clear that all points on the negative axis are still located above hence they cannot be the minimum associated with (6). In fact, we can note that for any first contact always occurs before reaching because the initial potential has a zero derivative at (). This also means that no rarefaction interval opens at (nor at any other location, see Bertoin (1998)). On the other hand, for we must clearly integrate up to . Moreover, if the first contact is only reached at for , it means that no particles from the right part have reached the position yet. (For the case of two-sided Brownian initial velocity one would need to add a third factor of the form in Eq.(14) to take into account the left part of , see Valageas (2008).)

Using the expressions given in Appendix A, as well as the results of appendices A and B of Valageas (2008), we obtain from Eq.(14) for , after integration over , the probability density

(15) |

which we expressed in terms of the dimensionless variables (11). Of course, the distribution vanishes over , since particles from the left side cannot travel to the right side .

In particular, at the origin this yields

(16) |

where we used for . The probability vanishes for and , as particles cannot come from the left side. Thus we recover the results of Bertoin (1998), who obtained Eqs.(15)-(16) from probabilistic tools. The distribution of the time increments of , i.e. of , was obtained in Bertoin et al. (2001). We can note that Eq.(16) is identical to the large-velocity tail of the distribution obtained at for the case of two-sided Brownian initial conditions Valageas (2008). Hence, for rare events the tail of the distribution does not strongly depend on the initial conditions on the opposite side of the origin.

On the other hand, at large we obtain for fixed velocity, ,

(17) |

Here we used the relationship between the probability distributions of the velocity and of the Lagrangian coordinate , and the explicit expression (15). Thus, as for the two-sided case, we recover at leading order the initial Gaussian distribution on large scales, here at . This is related to the “principle of permanence of large eddies” encountered in the hydrodynamical context Gurbatov et al. (1997), that holds for more general energy spectra, , with . This states that regions of size , where is the integral scale of turbulence (here , see the scalings (11)), have not been strongly distorted by smaller scale motions yet (since the relative distance between particles has changed by an amount of order at time ). Thus, as checked in numerical simulations Aurell et al. (1993); Gurbatov & Pasmanik (1999), the stability of large-scale structures is not only a statistical property but actually holds on an individual basis, that is for each random realization of the velocity field. The properties of the velocity and Lagrangian increments on the right part of the system, , were already obtained in Bertoin (1998) and are also identical to those obtained far from the origin in Valageas (2008) for two-sided initial conditions. In particular, it can be seen that the -point distributions factorize as , where we note and the relative distances, for and . Thus, the increments of the inverse Lagrangian map, , are independent and have a simple distribution, which is given by the expression (15) without the factor . Then, over the properties of the density field are identical to those described in detail in Valageas (2008) far from the origin, for the case of two-sided Brownian initial conditions.

On the left side, , we must integrate Eq.(14) over up to , as explained above. This yields

(18) |

We can note that from Eqs.(15), (18), the tails at large and read as

(19) |

which hold for both and . Thus, at any finite the tail of the velocity distribution simply follows the exponential decay obtained at in (16), multiplied by a prefactor .

The distribution (18) corresponds to realizations where some particles coming from the right side have already passed by position . Thus, integrating Eq.(18) over gives the probability, , that matter coming from the right side has already passed by the position . This yields

(20) |

This could also be obtained directly by computing as in (14) the probability that the curve goes below the parabola at some point . Then, to the contribution (18) we must add the contribution , with , that corresponds to realizations where no particles from the right side have already passed by position , so that the medium has remained at rest at until time . Finally, Eq.(20) gives the asymptotic behaviors

(21) |

We can check that at , while it shows an exponential tail at large negative . Since all quantities can be expressed in terms of the scaled variables (11), as we noticed from (12) and can be checked in the results above, this exponential decay can be obtained from simple scaling arguments. Thus, for particle to reach the Eulerian position at time , we can expect its initial velocity to be of order . Since the initial velocity is Gaussian, of variance given by Eq.(3), this corresponds to a probability of order . This is maximum at , which gives a weight . Hence we recover an exponential over . Of course, such a reasoning cannot give the numerical factor within the exponential. We show in Fig. 2 our results for the probability , given by Eq.(20), for the present case “” where the left side is initially filled by particles at rest (solid line) (we also display for comparison the result associated with the empty case “”, that we shall obtain below in Eq.(43) (dashed line)). We clearly see the fast decline with larger obtained in Eq.(21).

### 3.2 Mean density profile and mean current

From Eq.(15) we also obtain the mean Lagrangian coordinate , and velocity , at Eulerian location on the right side, as

(22) |

As expected, since particles gradually leak into the left side the mean velocity is negative, and particles that occupy the Eulerian position come from increasingly far regions on the right as time increases. We also obtain the mass, , of excited particles (i.e. with initial Brownian velocity and which were initially located at ) that are located to the left of the Eulerian position by noting that it is given by since particles do not cross each other. This yields from Eq.(22)

(23) |

where we introduced the density and the current at position of excited particles. Therefore, we obtain a uniform mean flow from the right, with the mean current , while the mean density remains equal to . Note that , which implies that the fluctuations of the density and velocity are correlated. In fact, the velocity field is associated with Eulerian regular points, since shocks have a zero measure, but the flow of matter is associated with shocks since all the mass of excited particles is contained within shocks She et al. (1992); Sinai (1992); Bertoin et al. (2001); Valageas (2008). Therefore, it is not surprising to find that , since these quantities probe different aspects of the dynamics.

Thus, even though particles keep escaping into the left part , particles coming from the right semi-infinite axis keep replenishing the system and manage to maintain a constant mean density over , through the mean uniform current that grows linearly with time. The linear growth with time of the mean velocity and current is due to the fact that at later times particles coming from more distant regions have been able to reach the boundary . Again, this exponent can be obtained from simple scaling arguments. Thus, at time we can expect to see at the boundary, , particles coming from a distance with an initial velocity of order . Since the initial velocity scales as , see (3), this gives and , whence , assuming that matter flows through the boundary in a well-ordered fashion, so that the velocity of these particles has not been significantly damped by nearer lower-velocity particles, as the latter have already escaped into the left side.

On the left side, , we obtain from Eqs.(18)-(20) the mean Lagrangian coordinate , and velocity , as

(24) |

and . This gives for the mean velocity the asymptotic behaviors

(25) |

We show our results for the mean velocity profile in Fig. 3. We can check that shows a monotonic decrease for larger on the left side. As expected, we recover the same asymptotic exponential decay as for the probability obtained in Eq.(21).

The average (24) takes into account both the contributions (18) and . However, since only the part (18) contributes to the mass of excited particles located to the left of , we no longer have , as was the case for in (23), where we introduced the dimensionless mass defined by

(26) |

In the last two relations in (26) we used the property that only depends on the reduced variable . Integrating over the contribution (18) gives

(27) |

as well as

(28) |

and

(29) | |||||

We show in Figs. 4, 5, our results for the mean density and current. Then, we can check that the mean density and current are continuous at the boundary , and we obtain the asymptotic behaviors

(30) | |||||

From Eq.(30), we can note that the amplitude of the mean density and current first increases to the left of the boundary (contrary to the mean velocity which showed a monotonic decrease into the left side). This is due to the dragging effect of the matter that was initially at rest on the left side, which slows down the leftmost cluster associated with particles coming from the right side. Then, this deterministic friction leads to a transitory growth for the mean density and current to the left of the boundary . Figures 4 and 5 show that although this feature can be clearly seen for the mean density (although it remains modest, of order ), it is almost invisible for the current, in agreement with the higher power instead of in Eq.(30), which leads to a suppression by a factor at small . As we shall check in section 4.2 below, this feature disappears when the left side is initially empty. The same behavior was obtained in Frachebourg et al. (2001) for the case of white-noise initial velocity. At large distance from the boundary we obtain the exponential decays

(31) |

Again, the characteristic length scale is the reduced variable of (11), and the exponential decay is the same as the one obtained in Eq.(21) for the shock probability .

### 3.3 Lagrangian displacement field

We now consider the dynamics from a Lagrangian point of view. Thus, labelling the particles by their initial position at time , we follow their trajectory . Since particles do not cross each other, the probability, , for the particle to be located to the right of the Eulerian position at time , is equal to the probability, , for the Eulerian location to be “occupied” by particles that were initially to the left of particle . In terms of dimensionless variables, this gives for right-side particles, ,

(32) |

where the integration contour runs to the right of the pole, i.e. , and

(33) |

where the integration contour obeys . Note that for we must take into account both the probability, , that no particles from the right side have reached yet, and the probability, , that particles from the right side, with , have already passed by point . Then, probability densities are obtained from Eqs.(32)-(33) by differentiating with respect to .

### 3.4 Leftmost cluster

We can identify the Eulerian position, , of the particle that was initially located at the origin (), as the position of the leftmost cluster (or “leader”) formed by excited particles that have escaped into the left side. In the present case, where these particles spread into a medium of uniform density that was initially at rest, this cluster also contains a mass of particles that were located in the interval . It acts as a snow-plough while the conditional properties of the system to its right are no longer sensitive to the initial conditions on the left side. From Eq.(32) we can see that vanishes for , which means that at any time the particle has almost surely already passed to the left side and the leftmost cluster has formed with a finite mass, in agreement with the results of Isozaki (2006). On the other hand, for the probability density of the leftmost cluster position reads as

(34) |

Here we used the obvious property , which states that the leftmost cluster is located to the right of point if, and only if, no particles from the right side have reached this point yet. We can check that the result (34) is identical to the one that would be obtained from Eq.(33). Using Eq.(21) we obtain the asymptotic behaviors

(35) |

We show in Fig. 6 our result (34) for . From Eq.(34), we obtain after an integration by parts the mean position of the leftmost cluster as