Gammalike mass distributions and mass fluctuations in conserved-mass transport processes
aa r X i v : . [ c ond - m a t . s t a t - m ec h ] M a r Gammalike mass distributions and mass fluctuations in conserved-mass transportprocesses
Sayani Chatterjee , Punyabrata Pradhan and P. K. Mohanty Department of Theoretical Sciences, S. N. Bose National Centre for Basic Sciences,Block - JD, Sector - III, Salt Lake, Kolkata 700098, India CMP Division, Saha Institute of Nuclear Physics, 1/AF Bidhan Nagar, Kolkata 700064, India
We show that, in conserved-mass transport processes, the steady-state distribution of mass in asubsystem is uniquely determined from the functional dependence of variance of the subsystem masson its mean, provided that joint mass distribution of subsystems is factorized in the thermodynamiclimit. The factorization condition is not too restrictive as it would hold in systems with short-ranged spatial correlations. To demonstrate the result, we revisit a broad class of mass transportmodels and its generic variants, and show that the variance of subsystem mass in these models isproportional to square of its mean. This particular functional form of the variance constrains thesubsystem mass distribution to be a gamma distribution irrespective of the dynamical rules.
PACS numbers: 05.70.Ln, 05.20.-y
Introduction. – Understanding fluctuations is funda-mental to the formulation of statistical mechanics. Un-like in equilibrium, where fluctuations are obtained fromthe Boltzmann distribution, there is no unified principleto characterize fluctuations in nonequilibrium. In thisLetter, we provide a statistical mechanics framework tocharacterize steady-state mass fluctuations in conserved-mass transport processes.Nonequilibrium processes of mass transport whichhappen through fragmentation, diffusion and coales-cence are ubiquitous in nature, e.g., in clouds [1], fluidscondensing on cold surfaces [2], suspensions of colloid-particles [3], polymer gels [4], etc. To study these pro-cesses, various models with discrete as well as continuoustime dynamics have been proposed on a lattice where to-tal mass is conserved [5–14]. These models, a paradigmin nonequilibrium statistical mechanics, are relevant notonly for transport of mass, but can also describe seem-ingly different nonequilibrium phenomena, as diverse asdynamics of driven interacting particles on a ring [7],force fluctuations in granular beads [15, 16], distributionof wealth [17, 18], energy transport in solids [19], trafficflow [20, 21], and river network [22], etc.A striking common feature in many of these processesis that the probability distributions of mass at a sin-gle site are described by gamma distributions [7, 8, 12–14, 16]. In several other cases, e.g., in cases of wealth dis-tribution in a population [17, 18, 23] or force distributionin granular beads [15, 16], the distribution functions arenot always exactly known, but remarkably they can oftenbe well approximated by gamma distributions. Althoughthese models have been studied intensively in the pastdecades, an intriguing question [24] - why the gamma-like distributions arise in different contexts irrespectiveof different dynamical rules - still remains unanswered.In this Letter, we address these issues in general andexplain in particular why mass-transport processes of-ten exhibit gamma-like distributions. Our main result is that, in the thermodynamic limit, the functional depen-dence of variance of subsystem mass on its mean uniquelydetermines the probability distribution of the subsystemmass, provided that (i) total mass is conserved and (ii)the joint probability distribution of masses in subsystemshas a factorized form as given in Eq. 2. In other words,if the conditions (i) and (ii) are satisfied, the probabil-ity distribution P v ( m ) of mass m in a subsystem of size v can be determined from the functional form of thevariance σ v ≡ ψ ( h m i ) where h m i the mean. In fact, ψ ( h m i ) in systems with short-ranged spatial correlationscan be calculated by integrating two-point spatial corre-lation function. An important consequence of the mainresult is the following. When the variance of subsys-tem mass is proportional to the square of its mean, i.e., ψ ( h m i ) = h m i /vη with a parameter η that depends onthe dynamical rules of a particular model, the subsystemmass distribution is a gamma distribution, P v ( m ) = 1Γ( vη ) (cid:18) vη h m i (cid:19) vη m vη − e − vηm/ h m i , (1)where Γ( η ) = R ∞ m η − exp( − m ) dm the gamma func-tion. Indeed, we find that ψ ( h m i ) is proportional to h m i in a broad class of mass-transport models, which explainswhy these models exhibit gamma distributions.It might be surprising how the variance alone could de-termine the probability distribution P v ( m ) as an analyticprobability distribution function is uniquely determinedonly if all its moments are provided. However, the re-sult can be understood from the fact that, for a systemsatisfying the above conditions (i) and (ii), there existsan equilibrium-like chemical potential and consequentlya fluctuation-response relation that relates mass fluctua-tion to the response due to a change in chemical poten-tial. This relation, analogous to equilibrium fluctuation-dissipation theorem, provides a unique functional depen-dence of the chemical potential on mean mass and con-strains P v ( m ) to take a specific form. Proof. – Let us consider a mass-transport process on alattice of V sites with continuous mass variables m i ≥ i = 1 , . . . , V . With some specified rates, massesget fragmented and then the neighboring fragments ofmass coalesce with each other. At this stage, we need notspecify details of the dynamical rules, only assume thatthe total mass M = P Vi =1 m i is conserved. We partitionthe system into ν subsystems of equal sizes v = V /ν andconsider fluctuation of mass M k in k th subsystem. Weassume that the joint probability P ( { M k } ) of subsystemshaving masses { M , M , . . . M ν } ≡ { M k } has a factorizedform in steady state, P ( { M k } ) = Q νk =1 w ( M k ) Z ( M, V ) δ ν X k =1 M k − M ! (2)where weight factor w ( M k ) depends only on mass M k of k th subsystem and Z ( M, V ) = Z ( M, vν ) = Q νk =1 [ R dM k w ( M k )] δ ( P νk =1 M k − M ) the partition sum.Probability distribution P v ( m ) of mass M k = m in the k th subsystem of size v is obtained by summing over allother subsystems k ′ = k , i.e., P v ( m ) = w ( m ) Z ( M, V ) Y k ′ = k (cid:20)Z dM k ′ w ( M k ′ ) (cid:21) δ X k M k − M ! . After expanding Z ( M − m, V − v ) in leading order of m and taking thermodynamic limit M, V ≫ ρ = M/V fixed, we get P v ( m ) = w ( m ) Z ( M − m, V − v ) Z ( M, V ) = w ( m ) e µ ( ρ ) m Z ( µ ) , (3)where Z ( µ ) = R ∞ w ( m ) exp( µm ) dm and chemical po-tential µ ( ρ ) = df ( ρ ) dρ (4)with Z ( M, V ) = exp[ − V f ( ρ )] [9, 12, 28, 29]. Using twoequalities for mean of the subsystem mass h m i = vρ = ∂ ln Z /∂µ and its variance σ v ( h m i ) = ( h m i − h m i ) = ∂ ln Z /∂µ , a fluctuation-response relation is obtained d h m i dµ = σ v ( h m i ) . (5)For a homogeneous system, the mean and the varianceshould be independent of i . Moreover, when mass is con-served, the variance is a function of mean mass h m i orequivalently density ρ . The analogy between Eq. 5 andthe fluctuation-dissipation theorem in equilibrium is nowevident. Now Eqs. 4 and 5 can be integrated to obtain Z ( M, V ) = exp( − V f ( ρ )) and then its Laplace transform˜ Z ( s, V ) = R ∞ Z ( M, V ) e − sM dM. Since [ ˜ Z ( s, V )] /ν =˜ w ( s ) , the Laplace transform of w ( m ) , one can calculate w ( m ) straightforwardly and use it in Eq. 3 to get P v ( m ). We demonstrate this procedure explicitly in a specificcase where the variance of mass in a subsystem of size v is proportional to the square of its mean, i.e., σ v ( h m i ) ≡ ψ ( h m i ) = h m i vη , (6)with η a constant depending on parameters of a partic-ular model. By integrating Eq. 5 w.r.t. h m i = vρ andusing Eq. 4 we get µ ( ρ ) = − ηρ − α ; f ( ρ ) = − η ln ρ − αρ − β. (7)The integration constants α and β do not ap-pear in the final expression of mass distribution.Finally, we get the partition sum Z ( M, V ) =exp[ − V f ( ρ )] = ( M/V ) ηV exp ( αM + βV ) . Its Laplacetransform ˜ Z ( s, V ) = e βV Γ( ηV + 1) / [ V ηV ( s − α ) ( ηV +1) ]can be written as˜ Z ( s, V ) ≃ e βV √ πηV ( ηV ) ηV e − ηV V ηV ( s − α ) ( ηV +1) = const . ( s − α ) ( ηV +1) (8)using asymptotic form of the gamma function Γ( z + 1) ≃√ πzz z e − z for large z . The constant term in the numer-ator is independent of s and thus [ ˜ Z ( s, vν )] /ν gives˜ w ( s ) = const . ( s − α ) vη (9)in the thermodynamic limit ν → ∞ . Consequently itsinverse Laplace transform is w ( m ) ∝ m vη − e αm . Theweight factor w ( m ) , along with Eqs. 3 and 7, leadsto P v ( m ) which is a gamma distribution as in Eq. 1.This completes the proof for the functional form ψ ( x ) ∝ x . The proof follows straightforwardly for discrete-massmodels. Note that different classes of mass distributions P v ( m ) can be generated for other functional forms of ψ ( x ) (see Supplemental Material, section I). In all thesecases, P v ( m ) serves as the large deviation function formass in a large subsystem.Though the above proof relies on the strict factoriza-tion condition Eq. 2, the results are not that restrictiveand are applicable to systems when the joint subsystemmass distribution is nearly factorized. In fact, the near-factorization of the joint mass distribution can be realizedin a wide class of systems as long as correlation length ξ is finite, i.e., spatial correlations are not long-ranged.In that case, subsystems of size much larger than ξ canbe considered statistically independent and thus well de-scribed by Eq. 2 [25–27]. Models and Discussions. – We now illustratethe results in the context of a broad class of mass-transport models where exact or near factorizationcondition holds. First we consider driven lattice gases(DLG) on a one dimensional (1 D ) periodic latticeof L sites with discrete masses or number of parti-cles m i ∈ (0 , , , . . . ) at site i where the total mass M is conserved. A particle hops only to its rightnearest neighbor with rate u ( m i − , m i , m i +1 ) whichdepends on the masses at departure site i and its near-est neighbors. For a specific rate u ( m i − , m i , m i +1 ) = g ( m i − , m i − g ( m i − , m i +1 ) / [ g ( m i − , m i ) g ( m i , m i +1 )] , the steady-state mass distribution of themodel is pair-factorized [11], i.e., P ( { m i } ) ∼ [ Q Li =1 g ( m i , m i +1 )] δ ( P i m i − M ). Unlike a site-wise fac-torized state, i.e., Eq. 2 with ν = V , the pair-factorizedsteady state does generate finite spatial correlations. Fora homogeneous function g ( x, y ) = Λ − δ g (Λ x, Λ y ) , thetwo-point correlation for the rescaled mass m ′ i = m i /ρ can be written as h m ′ i m ′ i + r i ≃ A ( r ) where A ( r ) = Q k (cid:2)R ∞ dm ′ k g ( m ′ k , m ′ k +1 ) (cid:3) m ′ i m ′ i + r δ ( P k m ′ k − L ) Q k (cid:2)R ∞ dm ′ k g ( m ′ k , m ′ k +1 ) (cid:3) δ ( P k m ′ k − L )is independent of ρ . The variance of mass m = P i ∈ v m i in a subsystem of size v ≫ σ v ≃ v P ∞ r = −∞ ( h m i m i + r i − ρ ) = h m i /ηv where η − = P ∞ r = −∞ [ A ( r ) − . Thus, in DLG with homogeneous g ( x, y ), ψ ( h m i ) is proportinal to h m i ; in fact this pro-portionality is generic in models where steady state isclusterwise factorized with g a homogeneous function ofmasses at several sites (see Supplemental Material, sec-tion II.B). In all these cases, P v ( m ) should be a gammadistribution.We now simulate DLG for two specific cases with g ( x, y ) = ( x δ + y δ + cx α y δ − α ) : Case I. δ = 1, c = 0and Case II. δ = 2, c = 1 and α = 1 . . We then cal-culate the variance σ v ≡ ψ ( h m i ) as a function of meanmass h m i . As shown in Fig. 1(a), in both the cases, ψ ( h m i ) ∝ h m i as in Eq. 6 with η ≃ . η ≃ . η , corresponding P v ( m )obtained from simulations are also in excellent agreementwith Eq. 1 as seen in Fig. 1(b). Interestingly, the valueof η can be calculated analytically for case I where δ and α are integers (see Supplemental Material, section II.A).Next we consider a generic variant of paradigmaticmass-transport processes, called mass chipping models (MCM) [7, 8, 12–14]. These models are based on massconserving dynamics with linear mixing of masses atneighboring sites which ensures that σ v ≃ h m i /vη whenthe two-point correlations are negligible. Note that, fac-torizability of steady state necessarily implies vanishingof two-point correlations, but not vice versa . However,when higher order correlations are also small, which isusually the case in these models, the steady state is nearlyfactorized and the resulting P v ( m ) can thus be well ap-proximated by gamma distribution for any v (including v = 1). We demonstrate these results considering mainlythe asymmetric mass transfer in MCM; the symmetriccase is then discussed briefly.In 1 D, asymmetric MCM is defined as follows. On aperiodic lattice of L sites with a mass variable m i ≥ σ υ
Driven lattice gases : (a) Variance σ v of subsystem mass vs. its mean h m i (lines - fit to the formin Eq. 6) and (b) corresponding mass distribution P v ( m ) forCase I. δ = 1, c = 0 and v = 10 (red circles) and Case II. δ = 2, c = 1, α = 1 . v = 15 (magenta squares). In bothcases ρ = 10 and L = 2000 . Mass chipping models : Massdistribution P v ( m ) vs. mass m with (c) v = 1 and (d) v = 10for the model with λ = 1 / p = 0 (red squares), 0 . Wealth distributionmodels : Mass distribution P v ( m ) vs. mass m with (c) v = 1and (d) v = 5 for the model with λ = 0 . . . ρ = 1 and L = 1000. Simulations - points, gammadistributions (Eq. 1) - dotted lines. at site i, first (1 − λ ) fraction of mass m i is chipped off,leaving the rest of the mass at i . Then a random fraction r i of the chipped-off mass (1 − λ ) m i is transferred to theright nearest neighbor and the rest comes back to site i .At each site, the chipping process occurs with probabil-ity p ; thus the extreme limits p = 0 and 1 correspond re-spectively to random sequential (i.e., continuous-time dy-namics) and parallel updates. Effectively, at time t , mass m i ( t ) at site i evolves following a linear mixing-dynamics m i ( t + 1) = m i ( t ) − (1 − λ )[ γ i m i ( t ) − γ i − m i − ( t )] , where γ i = δ i r i with δ i and r i are independent random vari-ables drawn at each site i : δ i = 1 or 0 with probabilities p and 1 − p respectively and r i is distributed accord-ing to a probability distribution φ ( r i ) in [0 , h m i m i − i ≈h m i ih m i − i = ρ , a very good approximation in thiscase, the variance of mass σ = h m i i − ρ at a singlesite ( v = 1) can be calculated using the stationarity con-dition h m i ( t + 1) i = h m i ( t ) i . Then the variance takes asimple form σ = ρ /η with η = η ( λ, p, µ , µ ) = µ − (1 − λ ) µ (1 − λ )( µ − pµ ) (10)where µ k = R r k φ ( r ) dr moments of φ ( r ) . Moreover,in these models, as the two-point correlation function h m i m i + r i − ρ ≃ | r | >
0, the variance ofsubsystem mass is given by σ v ≃ vσ = h m i /vη. A special case of asymmetric MCM with λ = 0 and p = 1 is the ‘ q ’ model of force fluctuations [15, 16]which has a factorized steady state for a class of dis-tribution φ ( r ) [12]. In this case, P ( m ) can be immedi-ately obtained by using η = ( µ − µ ) / ( µ − µ ) (fromEq. 10) and v = 1 in Eq. 1. The mass distribution is inperfect agreement with that obtained earlier [12] usinggenerating function method. As a specific example, weconsider φ ( r ) = r a − (1 − r ) b − /B ( a, b ) with B ( a, b ) =Γ( a )Γ( b ) / Γ( a + b ) for which the first two moments are µ = a/ ( a + b ) and µ = ab/ ( a + b ) ( a + b +1) − a / ( a + b ) , and thus η = a + b. Corresponding mass distributions isin agreement with that obtained in [13]. For λ = 0 and p <
1, the generalized asymmetric MCM becomes theasymmetric random average process [7, 12, 13]. We con-sider a specific case, when r is uniformly distributed in[0 , P ( m ) is not known [8]. However, since thetwo-point correlations vanish [8], we assume the steadystate to be nearly factorized and obtain P ( m ) , a gammadistribution with η = 2 / (4 − p ) . We verified numeri-cally that this simple form agrees with the actual P ( m )remarkably well, except for small m ≪ ρ. For generic λ and p and for a uniform φ ( r ) = 1 with r ∈ [0 , λ = 1 / p = 0 [14]. However, the spatial correlationsare small and gamma distribution provides in general agood approximation of P v ( m ). In Fig. 1(c), P ( m ) versus m is plotted for λ = 1 / ρ = 1 and for various p = 0 , . P ( m ) agrees quite well withEq. 1 with respective values of η = 2, 5, and 8. Thedeviation for m ≪ ρ is an indication of the absence ofstrict factorization on the single-site level. In Fig. 1(d),distribution P v ( m ) of mass m in a subsystem of volume v = 10 is plotted as a function of m and it is in excellentagreement with Eq. 1 almost over five orders of magni-tude. Note that, although Eq. 2 does not strictly holdon the single-site level, it holds extremely well for subsys-tems - a feature observed in MCM or wealth distributionmodels (discussed later) for generic values of parameters.In symmetric MCM’s, with parallel update rules, afraction λ of mass m i at site i is retained at the siteand fraction (1 − λ ) of the mass is randomly and sym-metrically distributed to the two nearest neighbor sites [14]: m i ( t +1) = λm i ( t )+(1 − λ ) r i − m i − ( t )+(1 − λ )(1 − r i +1 ) m i +1 ( t ) where r i uniformly distributed in [0 , λ = 0, the steady state is factorized [14] and P ( m ) isexactly given by Eq. 1 with η = 2. Clearly, when λ = 0,both symmetric and asymmetric MCM’s with parallelupdates result in η = 2, which explains why P ( m ) inthese two cases are the same [14]. Due to the presence offinite spatial correlations, P ( m ) with other update rulesare not described by Eq. 1.Our results are also applicable to models of energytransport [19] and wealth distributions [17, 18, 23, 30, 31]defined on a 1 D periodic lattice of size L . Here, (1 − λ )fraction of the sum m s ( t ) = m i ( t ) + m i +1 ( t ) of individ-ual masses (equivalent to ‘energy’ or ‘wealth’) at nearest-neighbor sites i and i + 1 is redistributed : m i ( t + dt ) = λm i ( t ) + r (1 − λ ) m s ( t ) and m i +1 ( t + dt ) = λm i +1 ( t ) +(1 − r )(1 − λ ) m s ( t ) where r is uniformly distributed in[0 , h m i m i − i ≈ ρ , the variance is written as σ ( ρ ) ≈ ρ /η ( λ ) with η ( λ ) = (1 + 2 λ ) / (1 − λ ), in agree-ment with that found earlier numerically [23]. For λ = 0,i.e., Kipnis-Marchioro-Presutti model in equilibrium [19],the steady state is factorized and P ( m ) = exp( − m/ρ ) /ρ (with η = v = 1) is exact. For non-zero λ , as the spatialcorrelations are small, the mass distributions, to a goodapproximation, are gamma distributions. In Fig. 1(e), P ( m ) versus m is plotted for λ = 0 .
3, 0 . . ρ = 1 and L = 1000. Except for m ≪ ρ , P ( m ) agreeswell with Eq. 1. For a subsystem of size v = 5, thedistributions P v ( m ), plotted in Fig. 1(f) for the sameparameter values as in the single-site case, are in excel-lent agreement with Eq. 1 for almost over five orders ofmagnitude. Summary. – In this Letter, we argue that subsystemmass fluctuation in driven systems, with mass conserv-ing dynamics and short-ranged spatial correlations, canbe characterized from the functional dependence of vari-ance of subsystem mass on its mean. As described inEq. 2, such systems could effectively be considered as acollection of statistically independent subsystems of sizesmuch larger than correlation length, ensuring existence ofan equilibrium-like chemical potential and consequentlya fluctuation-response relation. This relation along withthe functional form of the variance, which can be calcu-lated from the knowledge of only two-point spatial cor-relations, uniquely determines the subsystem mass dis-tribution. We demonstrate the result in a broad class ofmass-transport models where the variance of the subsys-tem mass is shown to be proportional to the square of itsmean - consequently the mass distributions are gammadistributions which have been observed in the past indifferent contexts. From a general perspective, this workcould provide valuable insights in formulating a nonequi-librium thermodynamics for driven systems.
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