We study a class of mass transport models where mass is transported in a
preferred direction around a one-dimensional periodic lattice and is globally
conserved. The model encompasses both discrete and continuous masses and
parallel and random sequential dynamics and includes models such as the
Zero-range process and Asymmetric random average process as special cases. We
derive a necessary and sufficient condition for the steady state to factorise,
which takes a rather simple form.Comment: 6 page

Angel A G

Coppersmith S N

Evans M R

Godrèche C

Großkinsky S

Levine E Mukamel D Ziv G

M R Evans

O'Loan O J

R K P Zia

Rajesh R

Satya N Majumdar

Zielen F

English

arXiv

We study a class of mass transport models where mass is transported in a preferred direction around a one-dimensional periodic lattice and is globally conserved. The model encompasses both discrete and continuous masses and parallel and random sequential dynamics and includes models such as the Zero-range process and Asymmetric random average process as special cases. We derive a necessary and sufficient condition for the steady state to factorise, which takes a rather simple form

evans, M.

majumdar, Satya

zia, R.

Edinburgh Research Explorer

     Edinburgh Research Explorer                                      Factorised Steady States in Mass Transport ModelsCitation for published version:evans, M, majumdar, S & zia, R 2004, 'Factorised Steady States in Mass Transport Models', Journal ofPhysics A: Mathematical and General, vol. 37, no. 25, pp. L275-L280. https://doi.org/10.1088/0305-4470/37/25/L02Digital Object Identifier (DOI):10.1088/0305-4470/37/25/L02Link:Link to publication record in Edinburgh Research ExplorerPublished In:Journal of Physics A: Mathematical and GeneralGeneral rightsCopyright for the publications made accessible via the Edinburgh Research Explorer is retained by the author(s)and / or other copyright owners and it is a condition of accessing these publications that users recognise andabide by the legal requirements associated with these rights.Take down policyThe University of Edinburgh has made every reasonable effort to ensure that Edinburgh Research Explorercontent complies with UK legislation. If you believe that the public display of this file breaches copyright pleasecontact openaccess@ed.ac.uk providing details, and we will remove access to the work immediately andinvestigate your claim.Download date: 04. Jan. 2021arXiv:cond-mat/0406524v1  [cond-mat.stat-mech]  22 Jun 2004Factorised Steady States in Mass Transport ModelsM. R. Evans1, Satya N. Majumdar2,3, R. K. P. Zia4,51School of Physics, University of Edinburgh,Mayfield Road, Edinburgh EH9 3JZ, UK2 Laboratoire de Physique Theorique et Modeles Statistiques,Universite Paris-Sud, Bat 100, 91405, Orsay-Cedex, France3Laboratoire de Physique Theorique (UMR C5152 du CNRS),Universite Paul Sabatier, 31062 Toulouse Cedex, France4Department of Physics andCenter for Stochastic Processes in Science and Engineering,Virginia Tech, Blacksburg, VA 24061-0435, USA;5FB-Physik, Universität Duisburg-Essen, 45117 Essen, Germany.Abstract. We study a class of mass transport models where mass is transportedin a preferred direction around a one-dimensional periodic lattice and is globallyconserved. The model encompasses both discrete and continuous masses and paralleland random sequential dynamics and includes models such as the Zero-range processand Asymmetric random average process as special cases. We derive a necessary andsufficient condition for the steady state to factorise, which takes a rather simple form.PACS numbers: 05.70.Fh, 02.50.Ey, 64.60.-iFactorised Steady States in Mass Transport Models 2Mass transport models form a general class of lattice models defined by dynamicsin which mass is transferred (without loss) stochastically from site to site. Theyhave attracted much recent attention, especially in connection with “condensationtransitions” [1, 2, 3, 4, 5, 6]. Examples include the Zero-Range Process (ZRP) [1] andAsymmetric Random Average Process (ARAP) [7, 8], which have been used to modelsuch diverse situations as traffic flow, clustering of buses [4], phase separation dynamics[9] and force propagation through granular media [10]. In general, it is difficult todetermine the steady state distribution of such models. Thus, it is remarkable that, notonly are the steady states of many models found, they often share a very convenientproperty, namely, a factorised steady state (also referred to as a product measure). Ofcourse, such a property greatly facilitates the analysis of interesting behaviour, e.g.,condensation.In this letter we determine the requirement for a factorised steady state in avery broad class of mass transport models. The form of this necessary and sufficientcondition, stated in (15), turns out to be appealingly simple. Encompassing bothrandom sequential and parallel dynamics, this class includes both the ZRP and ARAP.We discuss the salient features of the approach leading to (15) and recover somepreviously known cases.We consider a one-dimensional lattice of L sites with periodic boundary conditions(site L+ 1= site 1): associated with each site is a mass mi, i = 1 . . . L. The total massis given by M =∑Li=1mi. We shall most generally consider mi as continuous variables.The dynamics is as follows: from time t to t + 1, at each site i, mass µi drawn from adistribution ϕ(µi|mi) ‘chips off’ the mass mi, and moves to site i+ 1. Thus the masterequation for the weights (unnormalised probabilities) Ft(m) isFt+1(m) =L∏i=1∫∞0dm′i∫ m′i0dµi ϕ(µi|m′i)L∏j=1δ(mj−m′j+µj−µj−1)Ft(m′) , (1)where m ≡ {m1, m2, . . . , mL}. Note that this dynamics conserves the total mass, M , sothat Ft(m) may be considered as a function of only L− 1 variables. The integral of theweights, subject to the constraint of globally conserved mass,Z (M,L) ≡L∏i=1∫∞0dmi δ(M −L∑i=1mi)Ft(m) , (2)should be finite and serves as a “partition function,” so that F/Z is a probability density(or “canonical distribution”).In the t → ∞ limit, Ft(m) approaches a t-independent function, which we denotesimply by F (m) and refer to as the steady state. Defining the Laplace transformG(s) =[L∏i=1∫∞0dmie−simi]F (m) , (3)and transforming (1), we findG(s) =[L∏i=1∫∞0dm′i∫ m′i0dµi ϕ(µi|m′i)e−si(m′i−µi+µi−1)]F (m′) . (4)Factorised Steady States in Mass Transport Models 3We now assume that the steady state weight factorisesF (m) =∏if(mi) (5)which impliesG(s) =∏ig(si) where g(s) =∫∞0dmf(m)e−sm . (6)Then (4) becomes∏ig(si) =∏i[∫∞0dm′if(m′i)∫ m′i0dµi ϕ(µi|m′i)e−si(m′i−µi+µi−1)]. (7)Changing variables to σ ≡ m− µ (the mass remaining after the move), we writef(m)ϕ(µ|m) = P(µ, σ) . (8)Note that no assumption on the form of f(m) or ϕ(µ|m) is implied at this point. Withthis notation (7) becomes∏ig(si) =∏i[∫∞0dµi∫∞0dσi P(µi, σi)e−siσi−si+1µi]. (9)A necessary and sufficient condition for the solution of (9), is∫∞0dµi∫∞0dσi P(µi, σi)e−siσi−si+1µi = ℓ(si)k(si+1) (10)where the two functions, k and ℓ, must satisfyk (s) ℓ (s) = g (s) . (11)That (10) is necessary and sufficient may be seen by taking the logarithm of (9) thentaking derivatives with respect to si and si+1.Condition (11) implies via the convolution theorem thatf (m) = [v ∗ w] (m) ≡∫ m0dµ v (µ)w (m− µ) (12)wherek (s) =∫∞0dµ e−sµv (µ) ; ℓ (s) =∫∞0dσ e−sσw (σ) . (13)Then, equations (10) and (13) implyP (µ, σ) = v (µ)w (σ) . (14)Finally we obtain from (8) and (12)ϕ(µ|m) =v (µ)w (m− µ)[v ∗ w] (m). (15)Let us emphasize that the condition for a factorised stationary distribution for thewhole lattice precisely reduces to the condition that ϕ(µ|m) has the form (15). Thus,equation (15) is the central result of this paper: for chipping rules of the form (15),one has a factorised steady state (5) with weights given by (12). Let us comment onFactorised Steady States in Mass Transport Models 4several important points. Equations (11-15) allow us to define “equivalence classes”of chipping distributions—those leading to the same stationary state—by dividing k (s)and multiplying ℓ (s) by any (well behaved) function of s. In particular, their roles can be“reversed” to form a “dual” ϕ, i.e., w (µ) v (m− µ) / [w ∗ v] (m). Now, we can obviouslyinterpret the factors in (14) as a function for µ, the mass which moves, and a function ofσ, the mass which stays. In this sense, “duality” reverses these two portions of the mass,without changing F (m). If we further perform a Galilean transformation (shifting theentire lattice by one site in a time step) and a parity transformation (i⇔ L+1−i), thenwe recover the original system. Finally, note that both ϕ and the steady state (F/Z)are invariant under shifts of ln v and lnw by a linear function (i.e., there are arbitraryamplitudes or exponential factors in v and w: aµ, aσ).In addition to treating models with parallel dynamics, manifest in (1), we canextend the approach outlined above to models with random sequential dynamics. Letthe probability of a chipping event in a time step ∝ dt so that, to leading order in dt,at most one chipping event over the whole lattice occurs per update. Furthermore, wecan let the duration of a time step be dt and take dt → 0 to obtain a continuous timelimit where chipping events occur with rates per unit time. Thus, we writev(µ) = δ(µ) + x(µ)dt, (16)where δ(µ) is the Dirac delta function. Then (12) and (15) yieldf(m) = w(m) + dt[x ∗ w](m) andϕ(µ|m) =1w(m) + dt[x ∗ w](m){δ(µ)w(m) + dt x(µ)w(m− µ)}= δ(µ)[1 −dtw(m)[x ∗ w] (m)]+ dtx(µ)w(m− µ)w(m)+O(dt2) .Taking dt → 0 we obtain the continuous time limit where mass µ moves from a sitewith mass m with rate x(µ)w(m− µ)/w(m) and f(m) = w(m).Let us illustrate how this approach unifies two seemingly unrelated models – ARAPand ZRP. First we consider the ARAP [7, 8, 11, 12], a model in which each site containsa continuous amount of mass and at each time step a random fraction of the massmoves to the next site to the right. Its precise definition lies in ψ(r|m) = ϕ(µ|m)m, thedistribution for r, the fraction of mass that moves to the neighbouring site. A knownfamily of distributions where one has a factorised steady state is ψ(r|m) = (n− 1)rn−2[10, 11] which becomesϕ(µ|m) = (n− 1)µn−2mn−1. (17)In our approach, the results are particularly simple:v(µ) = µn−2, w(σ) = 1, (18)f(m) = mn−1/ (n− 1) . (19)Note that, to relate this f (m) to relevant quantities in the literature (e.g., [11]), thesingle site mass distribution, defined as the full distribution integrated over the restFactorised Steady States in Mass Transport Models 5of the mass variables, is p (m) = f (m)Z (M −m,L− 1) /Z (M,L). In this case,Z (M,L) = MnL−1 [Γ(n− 1)]L /Γ(nL), so that our p (m) reduces, e.g., to equation (37)of [11] in the thermodynamic limit.Another well-known case is the Zero-Range Process, reviewed in [1]. A focus ofmajor interest (for recent developments see for example [13, 14, 15, 16]), it is a masstransport model where mi takes integervalues and a unit mass moves from site i to sitei+1 with probability u(mi). Within our approach, this model appears as a very specialcase, with Dirac delta distributions for both v and w. Since the moved mass can takeonly two values while the one remaining can be of any integer, the most general formsarev(µ) = δ(µ) + aδ(µ− 1), w(σ) =∞∑k=0wkδ(σ − k) , (20)where a and wk are arbitrary weights. As overall amplitudes are irrelevant, we havechosen the coefficient of δ (µ) to be unity and will set w0 = 1. From f = v ∗ w, we seethatf(m) = w(m) + aw(m− 1) (21)= δ(m) +∞∑k=1[awm−1 + wm] δ(m− k) . (22)With a little care, we obtainϕ(µ|m) =wmδ(µ) + awm−1δ(µ− 1)wm + awm−1. (23)The coefficient of δ(µ−1) is precisely the chipping probability, denoted by u (m) above.From here, we easily find the wm in terms of the u:wm = amm∏n=11 − u(n)u(n). (24)Substituting this expression into (22) yields for the weights,f(m) =∞∑k=0fkδ(m− k) (25)wherefk =ak1 − u(k)k∏n=11 − u(n)u(n)for k ≥ 1 , (26)= 1 for k = 0 . (27)This result was previously obtained by a more complicated approach [17]. Note thatthe factors ak will drop out when we consider the probability density itself: F (m)/Z.We close this paragraph by noting the case with random sequential dynamics, which isobtained by letting a = ãdt and u(m) = x(m)dt where dt→ 0 yieldingfk = ãkk∏n=11x(k)k ≥ 1 . (28)Factorised Steady States in Mass Transport Models 6Finally, the results presented here may be generalised to the case of heterogeneousmass transfer where ϕi(µ|m) depends on the site i. A necessary and sufficient conditionfor a factorised steady state is thatϕi(µ|m) =v (µ)wi (m− µ)[v ∗ wi] (m)(29)where v and wi are arbitrary functions but v must be the same for each site. The weightfunctions are given byfi(m) = [v ∗ wi] (m) . (30)To conclude, we have determined the condition for the steady state in a generalclass of mass transport models to factorise. This class encompasses both continuousand discrete mass, as well as parallel and random sequential dynamics. Not only doesthis approach provide a unified perspective of all previously known models, it opensavenues to construct new models with this property (e.g., binomial chipping processand generalized Zero-Range Processes). In addition, we believe this approach wouldfacilitate a deeper understanding of the existence and nature of condensates and possiblyreveal novel forms of phase transitions. Implications of the gauge-like transformationsshould also be explored. Further work is in progress and will be published elsewhere.AcknowledgmentsWe thank Max Planck Institute, Dresden where this work was initiated for hospitality.One of us (RKPZ) thanks H.W. Diehl for his hospitality at the University of Duisburg-Essen, where some of this research was carried out, and acknowledges the supportfrom the Alexander von Humboldt Foundation and the US National Science Foundationthrough DMR-0088451.[1] M.R. Evans, Braz. J. Phys. 30, 42 (2000)[2] M. R. Evans, Europhys. Lett. 36, 13 (1996)[3] P. Bialas, Z. Burda, and D. Johnston, Nucl. Phys. B 493, 505 (1997).[4] O.J. O’Loan, M.R. Evans and M.E. Cates, Phys. Rev. E 58, 1404 (1998)[5] S.N. Majumdar, S. Krishnamurthy and M. Barma, Phys. Rev. Lett. 81, 3691 (1998); J. Stat. Phys.99 (2000) 1.[6] E. Levine, D. Mukamel, G. Ziv (2004) Condensation transition in zero-range processes withdiffusion cond-mat/0402520[7] J. Krug and J. Garcia, J. Stat. Phys., 99 31 (2000)[8] R. Rajesh and S. N. Majumdar, J. Stat. Phys., 99 (2000) 943; Phys. Rev. E 64 (2001) 036103.[9] Y. Kafri, E. Levine, D. Mukamel, G.M. Schütz and J. Török, Phys. Rev. Lett. 89, 035702 (2002)[10] S.N. Coppersmith, C.-h. Liu, S. Majumdar, O. Narayan, T.A. Witten Phys. Rev. E., 53, 4673(1996)[11] F. Zielen and A. Schadschneider, J. Stat. Phys 106, 173 (2002)[12] F. Zielen and A. Schadschneider, J. Phys. A 36, 3709 (2003)[13] M. R. Evans and T. Hanney, J. Phys. A 36, L441 (2003)[14] S. Großkinsky, G.M Schütz and H. Spohn, J. Stat. Phys 113, 389 (2003)[15] C. Godrèche, J. Phys. A: Math. Gen. 36, 6313 (2003)[16] A. G. Angel, M. R. Evans and D. Mukamel, J. Stat. Mech.:Theor. Exp. P04001, (2004)[17] M. R. Evans, J. Phys. A: Math. Gen. 30 5669 (1997)

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