Assortative Exchange Processes
P. L. Krapivsky




Department of Physics, Boston University, Boston, MA 02215, USA


In exchange processes clusters composed of elementary building blocks, monomers, undergo binary exchange in which a monomer is transferred from one cluster to another. In assortative exchange only clusters with comparable masses participate in exchange events. We study maximally assortative exchange processes in which only clusters of equal masses can exchange monomers. A mean-field framework based on rate equations is appropriate for spatially homogeneous systems in sufficiently high spatial dimension. For diffusion-controlled exchange processes, the mean-field approach is erroneous when the spatial dimension is smaller than critical; we analyze such systems using scaling and heuristic arguments. Apart from infinite-cluster systems we explore the fate of finite systems and study maximally assortative exchange processes driven by a localized input. 

Assortative Exchange Processes
    P. L. Krapivsky
    
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Assortative Exchange Processes
    P. L. Krapivsky
    
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§ INTRODUCTION


Exchange processes arise in numerous natural phenomena such as droplet growth via evaporation and re-condensation <cit.>, island growth <cit.> and phase ordering <cit.>.  Exchange processes have been applied to social sciences, e.g., to modeling segregation of heterogeneous populations <cit.>, studying the distribution of wealth through asset exchange <cit.>, mimicking growth of urban populations <cit.> and aggregation behaviors in job markets <cit.>. Exchange processes are also used as toy microscopic models which are simple enough to allow the derivation of the macroscopic `hydrodynamic' equations and explore other fundamental aspects of non-equilibrium statistical mechanics, see e.g. <cit.> and references therein.

In mass exchange processes, clusters interact by transferring mass from one to another. Cluster are usually assumed to be composed of an integer number of elemental building blocks (`monomers'). We shall denote by A_j a cluster of `mass' j, that is, a cluster which is made of j monomers. Clusters are thus labelled solely by their masses; other characteristics (e.g., their shape) are ignored. We assume that in each exchange event, a monomer is transferred from one cluster to another.  Symbolically, the mass exchange process is represented by the reaction scheme 

    A_i ⊕ A_j⟶^K_i,j(A_i± 1, A_j∓ 1)

A cluster disappears when its mass vanishes. Thus in an exchange involving a monomer the number of clusters may decrease; it certainly decreases in an exchange between monomers, one monomer disappears and another becomes a dimer. 

Exchange processes characterized by symmetric migration rates K_i,j have been mostly investigated, e.g., models with homogeneous rates K_i,j=i^aj^b+i^bj^a have been studied through asymptotic and scaling analyses (see e.g. <cit.>). Even in the simplest situation when the system is spatially uniform and remains well-mixed throughout the evolution, the governing (mean-field) rate equations form an infinite set of coupled non-linear differential equations which could not be solved. The exchange processes characterized by the generalized product kernel K_i,j=(ij)^a are special since the governing equations can be linearized, and the models with a=0,1,2 have been solved exactly, see <cit.>. 

In assortative exchange processes, interactions between clusters with disparate masses is suppressed. Here we consider the maximally assortative processes in which exchange can occur only between clusters of the same mass. The matrix of migration rates becomes diagonal, K_i,j=K(i) δ_i,j, and the set of reaction channels (<ref>) narrows to

    A_m ⊕ A_m⟶^K(m)(A_m-1, A_m+1)


We now outline the mathematical framework and announced a few chief results. The rate equations governing the evolution of the general maximally assortative exchange process are 

    dc_m/d t=K(m+1) c_m+1^2-2K(m) c_m^2+K(m-1) c_m-1^2

Here c_m(t) is the density  of clusters containing m monomers at time t. The density of monomers obeys ċ_1 = K(2)c_2^2-2K(1)c_1^2 which is consistent with the first equation  (<ref>) after setting c_0≡ 0, or introducing an extra rate K(0)=0. One can verify that Eqs. (<ref>) agree with mass conservation:

    ∑_m≥ 1 mc_m(t) = 1

Hereinafter we set the conserved mass density M to unity; this can always be done by rescaling: c_m→ Mc_m.

Equations (<ref>) have not been solved; the only exception are somewhat pathological models in which for a certain mass j the corresponding rate vanishes, K(j)=0, so only clusters up to mass j are present. In the following we ignore such models and assume that K(m)>0 for all m≥ 1, so that the number of interacting cluster species is infinite. The most interesting long time behavior of such models can be probed through asymptotic and scaling analyses. For instance, when rates are mass-independent, the density of monomers decays according to

    c_1 ∼
    t^-5/8    d=3
    
    t^-5/8 (ln t)^5/8    d=2
    
    t^-7/18    d=1


To appreciate these results we first recall that Eqs. (<ref>) are applicable only if clusters remain perfectly mixed throughout evolution. The analysis of Eqs. (<ref>) with kernel K(m)=1 leads to c_1∼ t^-5/8 as we show in Sect. <ref>. Now one can ask about actual physical process which could be mathematically described by Eqs. (<ref>) with kernel K(m)=1. In the case of diffusive transport, the natural candidate is the point cluster process in which


  * Each clusters occupies a single lattice site of a d-dimensional lattice. 

  * Clusters hop to neighboring sites and  hopping rates are mass-independent. 

  * When a cluster hops to a site containing a cluster with the same mass, an exchange (<ref>) instantaneously occurs. 


A critical dimension for this diffusion-controlled exchange process is d_c =2, that is the rate equations Eqs. (<ref>) with mass-independent kernel describe the evolution when d>d_c=2, particularly in three dimensions. At the critical dimension there is a logarithmic correction to the mean-field behavior; below the critical dimension the decay is slower than the mean-field prediction (similarly to other diffusion-controlled processes, see <cit.>). It is often useful to treat d as a continuous parameter. The decay exponents are universal when d>d_c=2 and become dimension-dependent when d<d_c where, as we argue in Sect. <ref>, the density of monomers decays as

    c_1/n_0∼[n_0 (Dt)^d/2]^-3d+4/(d+2)^2

Here we write the answer in the dimensionally-correct form which demonstrates the dependence on the diffusion coefficient D and the initial density n_0; the latter is defined via c_m(0)=n_0δ_m,1 if the system is initially composed of monomers. 

Further, the cluster density 

    N(t) = ∑_m≥ 1 c_m(t)

decays according to

    N ∼
    t^-1/4    d=3
    
    t^-1/4 (ln t)^1/4    d=2
    
    t^-1/6    d=1

More generally, below the critical dimension, d<2, the cluster density decays as 

    N/n_0∼[n_0 (Dt)^d/2]^-1/d+2


In Sect. <ref> we study the asymptotic behavior of the solutions to Eqs. (<ref>) with mass-independent migration rates, K(m)=1. In Sect. <ref> we extend results of Sect. <ref> to a one-parameter family of rates varying algebraically with mass, K(m)=m^a. Such rates are particularly suitable for scaling techniques which we employ. The analysis of Sects. <ref> and <ref> relies on mean-field equations (<ref>) which are valid if the initial state is spatially homogeneous and if the system remains well-mixed throughout the evolution. In Sect. <ref> we discuss the behavior of the simplest diffusion-controlled maximally assortative exchange process in which each cluster occupies a single lattice site (the point cluster process) and hops with mass-independent rate. The critical dimension is d_c=2 for such exchange processes, and we analyze asymptotic behaviors of these processes in one and two dimensions. Maximally assortative exchange processes with a finite mass generically do not condense in a single cluster, but reach a non-trivial final state with numerous clusters with different masses. In Sect. <ref> we describe these final states. Diffusion-controlled maximally assortative exchange processes driven by a localized input of monomers are investigated in Sect. <ref>. In Sect. <ref> we discuss approaches which may lead to the progress in understanding of maximally assortative exchange processes with quickly growing rates where scaling is violated.  




§ MASS-INDEPENDENT RATES



For the maximally assortative exchange process with mass-independent rates, K(m)=1, Eqs. (<ref>) reduce to

    dc_m/d t = c_m+1^2 - 2 c_m^2 + c_m-1^2

This neat infinite system of non-linear coupled ordinary differential equations (ODEs) appears intractable. The most interesting large time behavior can be established, however, since the typical mass exhibits an unlimited growth when t→∞ thereby allowing us to employ asymptotic and scaling approaches. The chief idea is to treat m as a continuous variable. If c(m,t)≡ c_m(t) slowly varies with m, the right-hand side in (<ref>) can be replaced by the second derivative to yield

    ∂ c/∂ t = ∂^2 /∂ m^2 c^2

One then seeks a scaling solution to (<ref>):

    c(m,t) = t^-2β F(x),      x=m/t^β

The scaling form (<ref>) agrees with mass conservation, the choice (<ref>) of the initial mass density implies

    ∫_0^∞ dx xF(x) = 1

By inserting the scaling form (<ref>) into (<ref>) we find that the scaling form is consistent only when β=1/4, and in that case the governing PDE turns into an ODE

    4(F^2)” + xF' + 2F = 0

where (·)' = d(·)/dx. Multiplying (<ref>) by x and integrating we get

    4 d/dx(F^2/x) + F = 0

where the integration constant was chosen to be zero to assure that F vanishes as x→∞. Re-writing Eq. (<ref>) as a product of two factors, F[8F'/x+1-4F/x^2]=0, we immediately extract the special solution F(x)=0, and then from 8F'/x+1-4F/x^2=0 we find a one-parameter family of solutions

    F = 1/12 √(x) (x_0^3/2-x^3/2)


The solution is the combination of (<ref>) and F=0. Since the density is non-negative, the scaled mass distribution is given by (<ref>) when 0≤ x≤ x_0 and F(x)=0 for x>x_0.  The parameter x_0 is found from the normalization requirement, 1=∫_0^x_0 dx xF(x), yielding x_0=(80)^1/4. Hereinafter it proves convenient to use a renormalized scaled mass variable y = x/x_0 which in the present case equals to y=m/(80 t)^1/4 in terms of the original variables. Collecting previous results we write the scaling solution in the form

    c_m(t) = t^-1/2 G(y)

with 

    G(y) = 1/3×√(5y) (1-y^3/2)      0≤ y≤ 1
    
    0                                             y>1

Note also simple asymptotic formulas 


    c_1(t)    = B_1  t^-5/8
    
    N(t)    = B   t^-1/4


for the density of monomers and the total cluster density N(t)=∑_m≥ 1c_m(t). The amplitudes in (<ref>)–(<ref>) are 

    B_1 = 5^3/8/3√(2) ,      B = 2· 5^3/4/9


At first sight, the compact shape of the mass distribution seems paradoxical given that the governing equations are parabolic PDEs. Our intuition is based on linear parabolic PDEs for which perturbation propagates with infinite speed preventing the formation of compact solutions. For non-linear parabolic PDEs like Eqs. (<ref>), however, compact solutions may and do arise as was discovered many years ago <cit.>, see <cit.> for review and <cit.> for recent examples of such non-linear parabolic PDEs appearing in the context of lattice gases. 

The compactness is a drawback of the continuum approximation. In the realm of the original discrete system (<ref>), the mass distribution is positive for all m. The front is extremely steep, however, so the discrepancy between the actual solution of the discrete system and the prediction of the continuum approach, viz. c(m,t)=0 in the region m> m_*(t)=(80 t)^1/4, is tiny. To appreciate this we take into account a very sharp decay and simplify (<ref>) in the m> m_*(t) region to
dc_m/d t≃ c_m-1^2 from which we deduce a double exponential decay

    ln(1/c_m)∝ 2^m-m_*(t)





§ ARBITRARY HOMOGENEOUS RATES



In this section we investigate maximally assortative exchange processes with algebraically varying migration rates K(m)=m^a. The governing rate equations read 

    dc_m/d t=(m+1)^a c_m+1^2-2m^a c_m^2+(m-1)^a c_m-1^2

Note that a rate equation for the cluster density

    dN/d t= - c_1^2

is independent on the exponent a. 



 §.§ Scaling approach



Algebraically varying migration rates are physically natural and convenient for analysis since they are compatible with the scaling approach. Thus we immediately focus on the large time behavior, treat again m as a continuous variable and turn an infinite set of ODEs, Eqs. (<ref>), into a single PDE

    ∂ c/∂ t = ∂^2 /∂ m^2 m^a c^2

The scaling solution to this PDE has the form (<ref>) with β=(4-a)^-1; the scaled mass distribution obeys

    (4-a) (x^a F^2)” + xF' + 2F = 0

Multiplying (<ref>) by x and integrating once we obtain

    (4-a) d/dx(F^2/x^1-a) + F = 0

This equation admits the special solution F(x)=0 and a one-parameter family of solutions

    G = x_0^2-a/(3-a)(4-a) y^1-a/2(1-y^3-a/2)

where we have used again the renormalized scaled mass variable y=x/x_0. The scaled mass distribution is given by (<ref>) when 0≤ y≤ 1, while G(y)=0 for y>1. The parameter x_0 is fixed by normalization:

    x_0 = (4-a)^2/(4-a) (5-a)^1/(4-a)


Gathering previous results we arrive at 

    c_m(t) = t^-2/4-a G_a(y) 
        y = x/x_0 = [(4-a)^2(5-a)t]^-1/4-a m

The scaled mass distribution reads 

    G_a(y) = C(a)×
    y^1-a/2(1-y^3-a/2)      0≤ y≤ 1
    
    0                                                                         y>1

The amplitude is

    C(a) = x_0^2-a/(3-a)(4-a)

with x_0(a) given by (<ref>). 

The density of monomers and the total cluster density exhibit algebraic long time behaviors

    c_1 = B_1(a)  t^-(5-a)/(8-2a),    N = B(a)   t^-1/(4-a)

with amplitudes 

    B_1(a) = x_0^(3-a)/2/(3-a)(4-a) ,    B(a) = x_0^3-a/(3-a)^2 (4-a)







On Fig. <ref> we plot the positive part of the scaled mass distribution for a=0,1,2,3:

    G_0    = √(5y) (1-y^3/2)/3
    
    G_1    = 6^-1/3(1-y)
    
    G_2    = (y^-1/2-1)/2
    
    G_3    = (4y)^-1ln(1/y)




 §.§ Marginal case of a=3


The results of Sect. <ref> are applicable only when the homogeneity index satisfies a<3. Taking the a↑ 3 limit in (<ref>) we obtain consistent results, namely the scaled mass distribution becomes 

    c_m(t) = t^-2 G_3(y),      y=m/2t

with

    G_3(y) = 
    
    (4y)^-1ln(1/y)      0<y<1
    
    0                             y≥ 1

Specializing (<ref>)–(<ref>) to m=1 we obtain 


    c_1≃ln t/2t

The decay law for the total cluster density also acquires a logarithmic correction

    N ≃(ln t)^2/4t


To establish this decay law we use (<ref>)–(<ref>) and find

    N = ∑_m≥ 1 mc_m(t) ≃2t/t^2∫_1/(2t)^1dy ln(1/y)/4y

Computing the integral yields (<ref>) in the leading order. Note that the integral in (<ref>) diverges in the y→ 0 forcing us to keep the lower limit finite. This makes the replacement of the summation by integration somewhat doubtful, but it actually does not cause the problem since the divergence is logarithmic and hence the prediction should be correct. As an independent check we can use the exact rate equation (<ref>) and verify that it is consistent with (<ref>)–(<ref>).




 §.§ Non-scaling regime: a>3


The scaling solution (<ref>)–(<ref>) does not make sense when a>3. The behavior in this region is non-scaling, so it is harder to probe analytically. Analogously to ordinary exchange processes, one may expect gelation. For instance, in ordinary exchange processes with generalized product kernels K_i,j=(ij)^λ, it has been shown <cit.> that (i) scaling holds when λ≤3/2; (ii) an infinite cluster (`gel') is formed at a finite time if 3/2<λ≤ 2; (iii) a gel is formed at time  t=+0 when λ>2 and gelation is complete, i.e. c_m(t)=0 for all m≥ 1 at t>0. 

Below in Sect.<ref> we consider the maximally assortative exchange process with a=∞. In this extremal model K(m)=∞ for all m≥ 2 and the the emerging mass distribution has certain features resembling instantaneous gelation, yet there is no gel. More precisely, in the infinite-system limit the rate equations are mathematically ill-defined for the extremal model. We thus consider the extremal model with finite total mass ℳ and establish [see (<ref>)–(<ref>)] the following mass distribution

    c_m =
     
    (1+t)^-1    m=1
    ℳ^-1    2≤ m≤√(2ℳt/1+t)

Thus in the ℳ→∞ limit all cluster densities apart from the monomer density vanish: c_m(t)=0 for all m≥ 2 at t>0. This is similar to instantaneous gelation. There is no gel, however. 

The knowledge of the behavior in the a≤ 3 range and the behavior at a=∞ allows one to make rather plausible guesses about the behavior in the  a>3 range. For instance, the cluster density decays as 

    N ≃
    B(a) t^-1/(4-a)    a<3
    
    (4t)^-1(ln t)^2          a=3
    
    t^-1    a=∞

with B(a) appearing in (<ref>). Thus the upper and lower bounds for N(t) are

    (4t)^-1(ln t)^2 < N(t) < t^-1

when a>3. Logarithmic corrections usually appear in the marginal cases, like a=3 in our situation, so for all a>3 we anticipate a simple decay

    N ≃B^+(a)/t

The unknown amplitude B^+(a) should decrease from lim_a→ 3+0B^+(a)=∞ to lim_a→∞B^+(a)=1. 

Similarly for the density of monomers we have established the following decay laws:

    c_1 ≃
    B_1(a) t^-1/(4-a)    a<3
    
    (2t)^-1ln t                   a=3
    
    t^-1    a=∞

with B_1(a) appearing in (<ref>). Thus the bounds are 

    (2t)^-1ln t < c_1(t) < t^-1

for a>3, and we actually expect a simple decay

    c_1 ≃B_1^+(a)/t

Similarly to B^+(a), the amplitude B^+_1(a) should decrease from lim_a→ 3+0B^+_1(a)=∞ to lim_a→∞B^+_1(a)=1, and the inequality B^+_1(a)≤ B^+(a) should be valid.








§ EXCHANGE PROCESSES IN LOW SPATIAL DIMENSIONS



Here we probe the behavior of diffusion-controlled exchange processes. We assume that clusters hop on a lattice and each cluster occupies a single lattice. The number of clusters at a site is unlimited, but all such clusters are such that they cannot mutually participate in an exchange. When a cluster hops to a site that contains another cluster which can participate in an exchange process, the exchange instantaneously occurs; if then another exchange becomes possible it also occurs instantaneously. Overall, an avalanche of exchanges may happen. 

If exchange between clusters of arbitrary masses is allowed, the diffusion-controlled exchange process is actually identical to the diffusion-controlled aggregation process: When a cluster A_i hops to a site with cluster A_j, an avalanche of exchanges occur till eventually a single cluster A_i+j is formed, this happens instantaneously so the process is indeed a merging event A_i ⊕ A_j→ A_i+j. We shall always assume that the hopping rates are mass-independent. The diffusion-controlled aggregation process in which clusters occupy single sites and hop with mass-independent rates is well-understood, the critical dimension is known to be d_c=2; the computations of the critical dimension in various diffusion-controlled processes are described e.g. <cit.> and <cit.>. The mean-field framework reproduces the asymptotic behavior above the critical dimension, d>d_c=2, e.g. it correctly predicts the N∼ t^-1 decay of the cluster density. In two dimensions, the mean-field framework is almost correct as it only misses logarithmic factors, e.g. N∼ t^-1ln t. In one dimension, the deviations from the mean-field behavior are most pronounced, e.g. N∼ t^-1/2.  The one-dimensional diffusion-controlled aggregation process is actually solvable, see <cit.>. 

Let us now consider the maximal assortative exchange in which a monomer can be transferred only between clusters with equal masses: A_m ⊕ A_m →(A_m-1, A_m+1). In principle, an avalanche of exchanges can occur. As an example, consider what happens when A_2 hops to a site containing clusters (A_1, A_2, A_3). After the hop there are four clusters (A_1, A_2, A_2, A_3) at a site and an avalanche of exchanges (participating pairs  are shown) leads to 

    A_1A_2 A_2 A_3 →A_1 A_1A_3 A_3→A_2 A_2 A_4 → A_1 A_3 A_4

Thus effectively A_2⊕ (A_1, A_2, A_3)→ (A_1, A_3, A_4). 

In the long time limit the density of clusters approaches to zero and avalanches become exceedingly rare. Indeed, the exchange A_m ⊕ A_m →(A_m-1, A_m+1) results in two clusters at a site, but these clusters quickly separate so that when t≫ 1 an occupied lattice site is almost surely occupied by a single clusters.

This problem appears analytically intractable even in one dimension, so we focus on the simplest characteristics, the decay exponents, and rely on heuristic arguments. As a check of such arguments let us first recover the exponents describing the decay of the monomer and cluster densities which we know from the asymptotically exact analysis, see (<ref>)–(<ref>). 

In the long time limit the left-hand side in (<ref>) decays faster than the terms in the right-hand side, so the right-hand side must asymptotically vanish. We thus get c_m+1^2 - 2 c_m^2 + c_m-1^2=0. The general solution is c_m^2=A m + B; recalling the convention c_0=0 we conclude that B=0. Using c_m∝√(m) in conjunction with the scaling form (<ref>) we obtain F(x)∼√(x) and then c_m∼√(m/t^5β) when m≪ t^β.   Combining c_1∼ t^-5β/2 and N∼ t^-β with the exact rate equation

    dN/dt = - c_1^2

we find β=1/4 and therefore

    c_1∼ t^-5/8 ,      N∼ t^-1/4

recovering the exponents in (<ref>)–(<ref>).  

Consider now the maximal assortative exchange on the one-dimensional lattice. (In one dimension the lattice version is not necessary, we can treat clusters as point particles performing independent Brownian motions with the same diffusion coefficient D.) To estimate the decay of the total density, consider two adjacent monomers. They are separated by distance ℓ∼ c_1^-1, and it takes time T∼ℓ^2/D∼ D^-1c_1^-2 for these monomers to meet. Thus the cluster density decays according to

    dN/dt∼ - c_1/T∼ -Dc_1^3

We now use rate equations similar to (<ref>), but with c_m^3 instead of c_m^2 in the right-hand side. (This step involves an uncontrolled approximation, but such approximations have been used in various diffusion-controlled processes and they always lead to correct asymptotic behaviors.) In the long time limit we thus obtain c_m+1^3 - 2 c_m^3 + c_m-1^3=0, from which c_m∝√(m). Combining with the scaling form (<ref>) we obtain  c_m∼√(m/t^7β) when m≪ t^β. Plugging c_1∼ t^-7β/3 and N∼ t^-β into (<ref>) we deduce β=1/6. Thus

    c_1∼ t^-7/18 ,      N∼ t^-1/6

In the dimensionally correct form the decay laws read 

    c_1∼ n_0^2/9(Dt)^-7/18 ,    N∼ n_0^2/3(Dt)^-1/6

where n_0 is the initial density of monomers. 

In two spatial dimensions, we similarly get

    dN/dt∼Dc_1^2/ln[c_1a^2]

where a is the lattice spacing. The same argument as before leads to c_m+1^2 - 2 c_m^2 + c_m-1^2=0; more precisely one should write c_m^2/ln[c_m a^2], but since c_m decay in time according to the same law, the logarithmic factor is independent on m in the leading order. Thus c_m∝√(m) and more precisely c_m∼√(m/μ^5), where we use

    c(m,t) = μ^-2 F(x),      x=m/μ

We denote the typical size by μ rather than t^β since in addition to the algebraic factor μ has a logarithmic factor. Plugging c_1∼μ^-5/2 and N∼μ^-1 into (<ref>) we find μ∼ (t/ln t)^1/4 leading to

    c_1∼(ln t/t)^5/8 ,    N∼(ln t/t)^1/4

In the dimensionally correct form the decay laws read 

    c_1∼ n_0^3/8(ℓ/Dt)^5/8 ,    N∼ n_0^3/4(ℓ/Dt)^1/4

with logarithmic factor

    ℓ = ln(Dt a^8/n_0^3)


Generally for d<2 the proper generalization of (<ref>) reads (see <cit.> for such arguments)

    dN/dt∼ - c_1/T∼ -Dc_1^1+2/d

The same arguments as before give c_m∝ m^d/(d+2) and after the same steps as above one gets the announced asymptotic behaviors (<ref>) and (<ref>).




§ FINAL STATES AND EVOLUTION IN FINITE SYSTEMS



Here we explore the ultimate fate of finite systems undergoing a maximally assortative exchange process. In this setting, the difference between maximally assortative and ordinary exchange processes is even more pronounced than for infinite systems. Indeed, in ordinary exchange processes all mass eventually accumulates in a single cluster. In a maximally assortative exchange process in a finite system, the final outcome is a jammed state containing clusters of different masses, so the exchange is no longer possible. 



 §.§ Final states


A state (m_1,…,m_p) with cluster masses satisfying

    1≤ m_1<…<m_p,      m_1+…+m_p = ℳ

is a jammed state of the system with total mass ℳ. The number of jammed states J_ℳ increases with ℳ.  For small ℳ one easily computes these numbers by hand; Table <ref> shows these numbers in the range ℳ≤ 20.  


 

Contemplating about J_ℳ, i.e. the number of solutions of (<ref>), one realizes that J_ℳ is the number of partitions of ℳ into distinct parts. Such partitions appear in combinatorics <cit.>, often under the name of strict partitions; recently they have been also called Fermi partitions <cit.>. Strict partitions were first studied by Euler (see <cit.>) who expressed the generating function for such partitions through an infinite product

    ∑_ℳ≥ 0 J_ℳ Q^ℳ = ∏_k≥ 1(1+Q^k)

(Here we have used the convention J_0=1.) Using (<ref>) and analyzing the Q→ 1 behavior one can extract the asymptotic behavior:  ln J_ℳ≃π√(ℳ/3) as ℳ→∞.  A more comprehensive analysis <cit.> gives the Ramanujan asymptotic formula

    J_ℳ≃1/4· 3^1/4 ℳ^3/4 exp[π√(ℳ/3)]



Despite of this growth of the total number of jammed states, the fate of the system is surprisingly deterministic, e.g., for the most natural initial condition when all clusters are initially monomers the final state is unique. This outcome is universal—the details of the exchange process are irrelevant, only the requirement that it is maximally assortative matters. Furthermore, the final state remains the same for many other initial conditions, e.g. if the initial number  N_m(0) of clusters of mass m satisfies N_m(0)>0 for all m=1,…,m_max and N_m(0)=0 for m>m_max; only if the initial mass distribution has big `holes' more complicated jammed states may arise. 

The final state is particularly simple when the initial mass is a triangular number, ℳ=T_n=n(n+1)/2 with arbitrary integer n. In this case, (1,2,…,n) is the final state. If the initial mass is not a triangular number, T_n-1 < ℳ < T_n, the final state differs from (1,2,…,n) by a single hole: If we parametrize ℳ = T_n - ℓ with some 1≤ℓ<n,  then the final state is 

    (1,…,ℓ-1,ℓ, ℓ+1,…,n)

where ℓ implies that the cluster with mass ℓ is absent. As an example, take ℳ = 18 = T_6 - 3. Equation (<ref>) tells us that the final state is (1,2,4,5,6). 




 §.§ Completion time


The evolution towards the final state depends on the details of the dynamics, and even for the fixed dynamics the duration varies from realization to realization, that is, the time t_final to reach the final state is a random variable. First, we estimate the completion time for the simplest maximally assortative exchange process with mass-independent migration rates. When ℳ≫ 1, the behavior is initially the same as the behavior of the infinite system (Sect. <ref>). Therefore the total number N_m of clusters of mass m is 

    N_m(t) = ℳ/√(t) G(y)

with y=m/(80t)^1/4 and G(y) given by (<ref>). These formulas formally apply when N_m≫ 1, but we can employ them up to N_m = O(1) in estimates. Thus we use the criterion ℳ∼√(t_final) to estimate 

    t_final∼ℳ^2


A similar argument for maximally assortative exchange processes with algebraic migration rates K(m)=m^a gives ℳ∼ t_final^2/(4-a). Therefore the completion time scales with total mass according to  

    t_final∼ℳ^2-a/2

This is valid when a≤ 1. The population of monomers exceeds the population of clusters of any other mass when a>1, so we cannot use continuum predictions when N_1 = O(1) since other cluster densities are negligible at such times and the continuum approach cannot be trusted in this domain. 



 §.§ The extremal model (a=∞)



Let us look at the maximally assortative exchange process with a=∞, equivalently a process with infinitely fast migration rates K(m)=∞ for all m≥ 2. In this extremal model the composition of the system is remarkably simple: When t<t_final, we still have a lot of monomers, N_1≫ 1, while the rest of the population is composed like (<ref>), namely N_m=1 for 2≤ m≤ m_0(t) with at most a single hole inside. 

To describe the evolution we notice that the merging of two monomers takes a positive time and it may trigger an avalanche of other exchanges which proceed instantaneously. Symbolically 1⊕ 1→ 2 and there will be no other instantaneous exchanges if in the preceding configuration the dimer was absent; otherwise 2⊕ 2→ (1,3) will occur, perhaps followed by a longer avalanche of instantaneous exchanges. Focusing on the population of monomers we have N_1→ N_1-2 in the first case and N_1→ N_1-1 in the second. In the long time limit a hole (if it exists) is usually far away, so that N_1→ N_1-1 dominates. As long as N_1≫ 1, we can use the rate equation 

    dc_1/dt = - c_1^2

for the monomer density c_1=N_1/ℳ. This is very simple, but conceptually remarkable result. Recall that (<ref>) predicts that the density of monomers satisfies 

    dc_1/dt = 2^a c_2^2 - 2c_1^2

It is not immediately obvious how to interpret the first term on the right-hand side of (<ref>) when a=∞. The above analysis shows that we must drop this term and divide by two the pre-factor of the second term. This reminds taking the zero-viscosity limit in turbulence, e.g. in Burgers turbulence one keeps the dissipation rate finite and justifies it by appearance of shocks (see e.g. <cit.>). 

Solving (<ref>) we get c_1=(1+t)^-1 and hence


    N_1 = ℳ/1+t

The rest of the mass distribution is

    N_m = 1,    2≤ m≤ m_0(t)≃√(2ℳ t/1+t)


The largest mass m_0(t) is established from the requirement of mass conservation. In the interesting 1≪ t ≪ℳ time range where we can employ deterministic rate equations the fraction of mass carried by monomers decreases as t^-1.  We also notice that the total number of clusters 

    𝒩 = N_1 + m_0(t) = ℳ/1+t + √(2ℳ t/1+t)

has an interesting behavior: The monomers provide the dominant contribution when t ≪√(ℳ), while for t≫√(ℳ) the total number of clusters saturates to 

    𝒩_final = √(2ℳ)



Combining (<ref>) and the criterion N_1 = O(1) we estimate the completion time

    t_final∼ℳ

It is worth mentioning that for the extremal maximally assortative exchange process with a=∞ one can derive much more precise results about the completion time. Indeed, in the most interesting case when ℳ≫ 1, we established that the dominant channel describing the decrease of monomers is N_1→ N_1-1, i.e. monomers effectively undergo the coalescence process: A_1+A_1→ A_1. This stochastic process is well-understood and the probability distribution for the completion time is known (see <cit.>). For instance, the leading behaviors of the two basic moments of t_final are

    ⟨ t_final⟩ = ℳ,     ⟨ t_final^2⟩/⟨ t_final⟩^2 = π^2/3-2

Thus fluctuations do not die even in the thermodynamic limit ℳ→∞. To appreciate it suffices to note that the process N_1→ N_1-1 occurs with rate N_1(N_1-1)/ℳ, so its average duration is ℳ/N_1(N_1-1). Therefore last steps when N_1=O(1) take time O(ℳ) and this explains the non-self-averaging nature. Up until the very end, however, the evolution is essentially deterministically. 

Overall, the extremal maximally assortative exchange process exhibits a very peculiar behavior. There is no gel (which by definition is a giant cluster containing a finite fraction of mass of the entire system). On the other side, in non-gelling systems or non-gelling phases, the largest cluster usually has a mass of the order of lnℳ, while in  the extremal maximally assortative exchange process there are numerous clusters with masses of the order of √(ℳ), and these clusters actually contain most of the mass is actually. 






§ EXCHANGE PROCESSES DRIVEN BY A LOCALIZED INPUT OF MONOMERS



Reaction-diffusion processes driven by localized input often occur in Nature and they are also used in various industrial applications. Some of these processes involve a few species of atoms; as examples we mention electropolishing <cit.>, dissolution <cit.>, corrosion <cit.>, and erosion <cit.>. These processes are rather tractable  <cit.> and well understood. Other processes involve numerous interacting sub-species, e.g. clusters in aggregation <cit.> and ordinary mass exchange <cit.>; the analysis of these systems is much more challenging and usually relies on non-rigorous tools. 

Here we study maximally assortative exchange processes driven by a localized input. The densities c_m(𝐫,t) obey an infinite system of non-linear coupled PDEs

    ∂ c_m/∂ t   =    
    K(m+1) c_m+1^2-2K(m) c_m^2+K(m-1) c_m-1^2 
       +   D_m∇^2 c_m  + J δ_m,1δ(𝐫)θ(t)

The terms on the top line on the right-hand side of (<ref>) account for exchange. The first term on the bottom line describes mixing due to diffusion and the following term represents the input of monomers source  (J is the strength of the monomer flux) at the origin. We are interested at the behavior on distances greatly exceeding the size of the region where monomers are injected and hence we model the flux using the delta function δ(𝐫). The source is turned at t=0, as specified by θ(t) on the right-hand side of  (<ref>); before that moment the system is assumed to be empty. In the following t>0 and we do not explicitly write θ(t)=1. 



 §.§ Mass-independent hopping rates


Here we study the model with mass-independent migration rates and diffusion coefficients. For the diffusion-controlled point cluster exchange process on the lattice, the migration rates are proportional to the corresponding hopping rates, K(m)∼ D_m, so if diffusion coefficients are mass-independent the  migration rates are also mass-independent. Equations (<ref>) for this model become

    ∂ c_m/∂ t = ∇^2 c_m + c_m+1^2-2c_m^2+c_m-1^2 + J δ_m,1δ(𝐫)

where we have set K(m)=1 and D_m=1. 

The mass density M(𝐫,t)=∑_m≥ 1 m c_m(𝐫,t) is now spatially dependent and it also depends on time. The mass is not affected by the exchange, so it satisfies the diffusion equation with a localized source

    ∂ M/∂ t = ∇^2 M  + Jδ(𝐫)

which can be solved in arbitrary dimension.  



  §.§.§ Three dimensions



In the most physically relevant three-dimensional case the rate equation approach is applicable. An extra simplification is that in three dimensions (and generally when d>2), the mass density is stationary; more precisely, the mass density coincides with Coulomb potential generated by `charge' J, viz.

    M = J/4π r

Since the source is turned on at t=0 and clusters propagate diffusively, the stationarity ceases to hold when r∼√(t), and M(r,t) quickly approaches to zero as r t^-1/2 increases. 

Other natural quantities do not even satisfy closed equations. For instance, the total cluster density N(𝐫,t)=∑_m≥ 1 c_m(𝐫,t) evolves according to 

    ∂ N/∂ t = ∇^2 N  - c_1^2 + Jδ(𝐫)


It is reasonable to assume that in the long time limit the densities become stationary. More precisely, they are stationary as long as the distance is not too far from the source, namely r≪√(t).  In the stationary regime in three dimensions (<ref>) becomes 

    1/r^2 d/dr(r^2 dN/dr) - c_1^2 + Jδ(𝐫)=0

Further, in the stationary regime in three dimensions  Eqs. (<ref>)  read

    1/r^2 ∂/∂ r(r^2 ∂  c/∂ r) + ∂^2 /∂ m^2 c^2 = 0

where we have replaced c_m+1^2-2c_m^2+c_m-1^2 by the second derivative which should be asymptotically exact when m≫ 1. We seek a solution to (<ref>) in a scaling form

    c(m,r) = c_m(r) = r^-2β-1Φ(x),      x=m/r^β

The pre-factor r^-2β-1 is chosen to be consistent with (<ref>). Indeed 

    M(r) = ∑_m≥ 1 m c_m(r)≃ r^-1∫_0^∞ dx xΦ(x)

has correct spatial dependence, and the complete match is obtained if 

    ∫_0^∞ dx xΦ(x)=J/4π


By inserting (<ref>) into (<ref>) we deduce β=1/4 and

    16 (Φ^2)” + x^2 Φ”+9xΦ'+12Φ = 0

This non-linear second-order ODE with non-constant coefficients is soluble. First we notice that (<ref>) admits an integrating factor: Multiplying (<ref>) by x we obtain

    [16x(Φ^2)' - 16Φ^2 + x^3 Φ'+6x^2Φ]' = 0

which we integrate and write the outcome as

    16 d/dx(Φ^2/x) + x d Φ/dx+6Φ = 0

(The integration constant was to zero to assure that Φ(x) vanishes as x→∞.) We simplify the first-order ODE (<ref>) by making the transformation

    Φ(x) = √(x) Ψ(x)

Using Y=x^3/2 instead of x we find that ψ(Y)=Ψ(x) satisfies 

    3dψ/dY + 13ψ/32Ψ + Y =0

This equation simplifies if instead of ψ(Y) we consider the inverse function Y(ψ): 

    13/3 dY/dψ = -32 - Y/ψ

Solving this equation we arrive at

    Y = 6(C ψ^-3/13 - ψ)

where C is an integration constant. Returning to the original variables we obtain an implicit solution

    x^2/6 = C[x^8/Φ^3]^1/13 - Φ




The limiting behaviors of the scaled mass distribution are (see also Fig. <ref>)

    Φ≃
    C^13/16 √(x)    x→ 0
    
    (6C)^13/3 x^-6    x→∞


The constant C is fixed by (<ref>). To compute the integral in (<ref>) we first re-write it as

    ∫_0^∞ dx xΦ(x) = 2/3∫_0^∞ dY Y^2/3ψ(Y)

Equation (<ref>) shows that 0≤ψ≤ψ_0=C^13/16. We can now compute the integral in (<ref>) using integration by parts, Eq. (<ref>) and straightforward transformations: 

    2/3∫_0^∞ dY Y^2/3ψ(Y)     =    2/5∫_0^ψ_0 dψ  Y^5/3
        =    2/5 6^5/3∫_0^ψ_0 dψ (C ψ^-3/13 - ψ)^5/3
        =    2/5 6^5/3 ψ_0^8/3∫_0^1 du (u^-3/13 - u)^5/3
        =     6^5/3 C^13/6 13√(π) Γ(8/3)/40 Γ(19/6)

Combining this with (<ref>) we express the amplitude C through the flux:

    C=(AJ)^6/13,    A = 10 Γ(19/6)/13π^3/2 6^5/3 Γ(8/3)


Let us also compute the cluster density. We have

    N(r) = ∑_m≥ 1 c_m(r)≃ r^-5/4∫_0^∞ dx Φ(x)

The last integral is computed using the same tricks as in the computation in Eq. (<ref>). We get

    ∫_0^∞ dx Φ(x)    =    2/3∫_0^∞ dY ψ(Y) 
        =    2/3∫_0^ψ_0 dψ  Y 
        =      4∫_0^ψ_0 dψ (C ψ^-3/13 - ψ) 
        =    16/5 ψ_0^2 = 16/5 C^13/8

leading to


    N(r) =  165 (AJ)^3/4  r^-5/4

The monomer density is found from (<ref>) and the x→ 0 asymptotic of Φ(x), see (<ref>), to give

    c_1(r) =  (AJ)^3/8  r^-13/8


As a useful consistency check we note that (<ref>) and (<ref>) agree with (<ref>). 



  §.§.§ High dimensions


The three-dimensional case is most relevant, but it is amusing to explore the behavior in dimension d=4 and higher. It turns out that the exchange is barely relevant at the `upper' critical dimension d=d^c=4 and asymptotically irrelevant in higher dimensions. To see this let us treat d as a continuous parameter. The rate equation approach is generally applicable when d>2. The mass density is stationary and given by  

    M = J/(d-2)Ω_d  r^d-2

where Ω_d = 2π^d/2/Γ(d/2) is the `area' of unit sphere 𝕊^d-1. This expression suggests that the relevant generalization of the three-dimensional scaling form (<ref>) is

    c(m,r) = c_m(r) = r^-2β-d+2Φ(x),      x=m/r^β

Plugging this form into 

    1/r^d-1 ∂/∂ r(r^d-1 ∂  c/∂ r) + ∂^2 /∂ m^2 c^2 = 0

we deduce β=1-d/4 and determine the scaled mass density (see Appendix <ref>). 

The small mass behavior is again Φ∼√(x), and the monomer density decays according to

    c_1(r) =  (A_d J)^3/8  r^-(3d+4)/8

The cluster density is given by  

    N(r) =  16(3d-4)(4-d) (A_d J)^3/4  r^-(3d-4)/4

The decay law (<ref>) can be extracted from (<ref>) and 

    1/r^d-1 d/dr(r^d-1 dN/dr) - c_1^2 = 0
 

The scaling form (<ref>) is applicable when 2<d<4. Indeed, when d≤ d_c=2 we cannot use mean-field rate equations. The upper bound d<d^c=4 is obvious from the above formulas, e.g. the exponent β=1-d/4 must be positive, yet it vanishes at d=4 and becomes negative when d>4. In sufficiently high dimensions, d>4, clusters essentially do not `see' each other. More precisely, some exchange processes occur near the source, but then clusters hardly meet. Therefore both the monomer density and the total cluster density decay in the similarly to the mass density:

    c_1 ∼ r^-(d-2),      N ∼ r^-(d-2)


The exponent d-2 approaches to two as d→ 4. Since β=0  at the upper critical dimension d=d^c=4, we 
anticipate that m scales logarithmically. Thus we seek the mass distribution in the form 

    c_m(r) = r^-2C_m(ρ),   ρ=ln r

Plugging this ansatz into the governing equations 

    1/r^3 d/dr(r^3 dc_m/dr) +c_m-1^2 - 2c_m^2 + c_m+1^2 = 0

we obtain

    2d C_m/d ρ + d^2 C_m/d ρ^2  = C_m-1^2 - 2C_m^2 + C_m+1^2

The interesting behavior occurs far from the source where the second term on the right-hand size of (<ref>) is negligible in comparison with the first term. (This is asymptotically true; however, the ratio of these two terms vanishes as ρ^-1, and since ρ=ln r the ratio decays very slowly.) Dropping the second term on the right-hand size of (<ref>) we arrive at

    2d C_m/d ρ   = C_m-1^2 - 2C_m^2 + C_m+1^2

This set of equations can be identified with (<ref>) after the transformation

    C_m(ρ)=J/4π^2 c_m(Jρ/8π^2)

which also matches (<ref>) with ∑_m≥ 1mC_m = J/4π^2 following from (<ref>) at d=4. Using previous results we deduce that when y<1 the mass density distribution is given by 

    c_m(r) = √(5J/18π^2) √(y)-y^2/r^2 ρ^1/2,    
    y = m(10Jρ/π^2)^-1/4

In particular


    N(r)     = B J^3/4/r^2 ρ^1/4
    
    c_1(r)    = B_1 J^3/8/r^2 ρ^5/8


with

    B_1= 5^3/8/3· 2^5/8·π^3/4,      B = 5^3/4/9· 2^1/4·π^3/2




  §.§.§ Low dimensions


When d≤ 2 the rate equation approach becomes erroneous. There are no closed form exact equations for cluster densities, but modified rate equations provide qualitatively correct results and lead to exact scaling. We now outline the results for d=1 and d=2. 

In one dimension, we seek the scaling solution in the form c(m,r)=m^-αF(m/r^β). Plugging this ansatz into the analog of (<ref>), namely ∂^2 c/∂ r^2+∂^2 c^3/∂ m^2=0, we deduce the relation β=(1+α)^-1 between the scaling exponents. Estimating ∑_m≥ 1mc_m∼ r^(2-α)β and noting that it should scale as r we deduce the second relation β(2-α)=1. From these relations α=1/2 and β=2/3. One can also establish proper powers of the source strength (omitted above). The scaling form reads  

    c_m(r) =  √(J/m) Φ(x),      x=m/J^1/3r^2/3

Using this expression we estimate the cluster density 


    N(r)∼ J^2/3r^1/3

Equations (<ref>) and (<ref>) are consistent  with d^2 N/dr^2∼ c_1^3 if Φ(x)∼ x^5/6 as x→ 0. Thus

    c_1(r) ∼ J^2/9r^-5/9



In two dimensions, we obtain


    N(r)   ∼ J^3/4ρ^1/4 r^-1/2
    
    c_1(r)    ∼ J^3/8ρ^5/8 r^-5/4


where we again shortly write ρ=ln r. 



  §.§.§ Total numbers of monomers and clusters


The total number of monomers 𝒞_1(t) is estimated by integrating the stationary density till r=√(t). Thus

    𝒞_1(t) ∼∫_0^√(t) dr r^d-1 c_1(r)

Using (<ref>), (<ref>), (<ref>) and (<ref>) we obtain

    𝒞_1 ∼
    J^2/9 t^2/9    d=1
    
    J^3/8 t^3/8  (ln t)^5/8    d=2
    
    J^3/8 t^11/16    d=3
    
    J^3/8 t  (ln t)^-5/8    d=4
    
    J t                                             d>4

Similarly the total number of clusters is estimated from 

    𝒩(t) ∼∫_0^√(t) dr r^d-1 N(r)

Using (<ref>), (<ref>), (<ref>) and (<ref>) we obtain

    𝒩∼
    J^2/3 t^2/3    d=1
    
    J^3/4 t^3/4  (ln t)^1/4    d=2
    
    J^3/4 t^7/8    d=3
    
    J^3/4 t  (ln t)^-1/4    d=4
    
    J t                                             d>4




 §.§ Mass-dependent rates


Diffusion coefficients generally decrease with mass. An algebraic decay, D_m∼ m^-ν, often occurs, e.g., the mobility exponents ν=1 and ν=3/2 arise in problems involving two-dimensional clusters <cit.>.  For the diffusion-controlled point cluster exchange processes on the lattice K(m)∼ D_m suggesting to study the models with D_m∼ K(m)∼ m^-ν. The behavior of such models driven by a local source can be treated using the same scheme as before, namely assuming the emergence of a stationary mass distribution and the validity of scaling. 

As a concrete example, let us consider the model with D_m= K(m)=m^-1. The rate equations read

    ∂ c_m/∂ t   =     (m+1)^-1c_m+1^2-2m^-1c_m^2+ (m-1)^-1c_m-1^2
        +     m^-1∇^2 c_m + J δ_m,1δ(𝐫)

The mass density now varies according to 

    ∂ M/∂ t = ∇^2 N  + Jδ(𝐫)

In the most physically relevant three-dimensional case, Eq. (<ref>) gives a simple expression 

    N = J/4π r

for the cluster density in the long time limit. 

Let us explore the stationary regime in three dimensions in more detail. 
We simplify Eqs. (<ref>) to

    1/r^2 ∂/∂ r(r^2 ∂  c/∂ r) + m ∂^2 /∂ m^2 c^2/m = 0

and seek a solution to (<ref>) in a scaling form

    c(m,r) = c_m(r) = r^-β-1Φ(x),      x=m/r^β

The pre-factor r^-β-1 is consistent with (<ref>). Indeed

    N(r) = ∑_m≥ 1 c_m(r)≃ r^-1∫_0^∞ dx Φ(x)

assures the correct spatial decay of the cluster density, and the constraint 

    ∫_0^∞ dx Φ(x)=J/4π

provides the complete match with (<ref>). By inserting (<ref>) into (<ref>) we deduce β=1/3 and

    9x (x^-1Φ^2)” + x^2 Φ”+6xΦ'+4Φ = 0

which is integrated to yield 9 (x^-2Φ^2)' + Φ'+4x^-1Φ = 0. The implicit solution to this equation reads

    13x^2 = C (x^6/Φ)^1/5-Φ

with C being an integration constant. The limiting behaviors of the scaled mass distribution are 

    Φ≃
    C^5/6 x                x→ 0
    
    (3C)^5 x^-4    x→∞

Using (<ref>) we compute the integral in (<ref>) and extract the amplitude C=(J/16π)^2/5. In particular,

    c_1(r) = (J/16π)^2/5 r^-5/3




To determine the spatial size R of the region where the densities have become stationary, we first compute the mass density:

    M(r) = ∑_m≥ 1 mc_m(r)≃ r^-2/3∫_0^∞ dx xΦ(x)

The integral is calculated using (<ref>) to yield 

    M(r) =   729/56(J/16π)^4/3 r^-2/3

Ignoring numerical factors we estimate the total mass 

    ℳ∼∫_0^R dr r^2 M(r) ∼ J^4/3  R^7/3

Since ℳ = Jt we have R∼ J^-1/7 t^3/7. Using this result and (<ref>) we estimate the total number of clusters 
    𝒩(t) ∼∫_0^R dr r^2 N(r) ∼ J R^2 ∼ J^5/7  t^6/7

Note also the asymptotic growth law for the total number of monomers

    𝒞_1(t) ∼ J^22/105  t^4/7




§ DISCUSSION



Maximally assortative exchange processes are mathematically challenging and not a single one has been solved so far. We have shown that for a class of models with algebraic migration rates, K(m)=m^a, we relied on scaling to establish the most interesting asymptotic behaviors in the a≤ 3 range. It would be interesting to understand the behavior when a>3 where scaling is violate. 

When scaling holds, a single typical mass characterizes the mass distribution. The mass distribution in the extremal model (a=∞) has two scales: m=1 corresponding to the monomers and the scale m_0, see (<ref>), characterizing the rest of the system. This suggests that when 3<a<∞ there may be two scales, an inner region m∼ t^β_- and much larger outer region with t^β_-≪ m ∼ t^β_+. Mass distributions with two scales, and even three, scales have appeared in a few models of aggregation with uniform input, see <cit.>. In the present situation, however, we haven't succeeded in establishing a consistent a boundary layer structure of the mass distribution. 

A strange feature of the mass distribution in the extremal model is that the outer scale m_0 is asymptotically independent on time, but depends on the total mass of the system: m_0≃√(2ℳ). Thus for infinite systems, ℳ=∞, the extremal model provides little insight for guessing the behavior when 3<a<∞, or perhaps the message is well hidden. Overall, the extremal model resembles taking the zero-viscosity limit in turbulence—the terms containing a=∞ formally disappear, yet they affect the evolution.   

The behavior of maximally assortative exchange processes substantially differs from the behavior of ordinary exchange processes. To study the interpolation between these two extremes one can introduce parameter r∈ [0,1] measuring the degree of assortatitivity by postulating that the reaction channel (<ref>) to operate only when r≤i/j≤ r^-1. With this definition, r=1 corresponds to maximally assortative exchange processes and r=0 corresponds to ordinary exchange processes. The extreme behaviors are known for simple rates rates such as K_i,j=(ij)^a/2; ordinary exchange processes with these rates were studied in <cit.>, while for maximally assortative exchange processes we recover the rates K(m)=m^a.  We know that e.g. the cluster density decays as

    N∼
    t^-1/(3-a)   when  r=0
    
    t^-1/(4-a)   when  r=1
 
These asymptotic results are valid when a<3. One would like to understand how r affects the decay law for the cluster density and behaviors of other quantities. 




§ MASS DENSITY IN HIGH DIMENSIONS (D>2)




To determine the scaled mass density in d dimensions, we insert the scaling ansatz  (<ref>) into the governing equation (<ref>) and deduce the scaling exponent β=1-d/4 together with the ODE for the scaled mass density

    (1-d4)^-1(Φ^2)” + (1-d4) x^2 Φ”+ (3-d4)xΦ'+dΦ = 0

Multiplying by x and integrating we obtain 

    (1-d/4)^-2d/dx(Φ^2/x) + x d Φ/dx+2d/4-d Φ = 0

Making the same transformation (<ref>) and using again Y=x^3/2 we obtain

    Y = 6/4-d(C ψ^-δ - ψ),   δ = 3 4-d/4+3d

indicating that the results are applicable when d<4.

The same computation as in Eq. (<ref>) allows one to fix the amplitude:

    J/(d-2)Ω_d    =    2/5∫_0^ψ_0 dψ  Y^5/3
        =    2/5(6/4-d)^5/3∫_0^ψ_0 dψ (C ψ^-δ - ψ)^5/3
        =    2/5(6/4-d)^5/3ψ_0^8/3∫_0^1 du (u^-δ - u)^5/3
        =    2/5(6/4-d)^5/3C^4+3d/6 Γ(Δ) Γ(8/3)/(1+δ) Γ(Δ+8/3)

where Δ = (1+δ)^-1 - 5δ/3. Thus 

    C = (A_dJ)^6/(4+3d)

with a cumbersome expression for the numerical factor A_d following from above formulas. 

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