Phase Transitions in Parallel Replication Process
aa r X i v : . [ c ond - m a t . s t a t - m ec h ] A p r Phase Transitions in Parallel Replication Process
P. N. Timonin ∗ Physics Research Institute at Southern Federal University, 344090 Rostov-on-Don, Russia
G. Y. Chitov † Department of Physics, Laurentian University, Sudbury, Ontario P3E 2C6, Canada (Dated: November 17, 2018)The one-dimensional kinetic contact process with parallel update is introduced and studied byMonte Carlo simulations. This process is proposed to describe the plant population replication andepidemic disease spreading among them. The phase diagram of the model features the line of thesecond order transitions between absorbing and active phases. The numerical results for the criticalindex β demonstrate its continuous variation along the transition line accompanied by the variationsof the structural characteristics of limiting steady states. We conjecture the non-universality of thecritical behavior of the model. PACS numbers: 05.20.Dd, 64.60.De, 64.60.F-
There has been a steadily growing interest during thelast two decades or so in the kinetics and phase tran-sitions in non-equilibrium systems. The applications ofthose systems range from physics, like, e.g., statisticalphysics, critical phenomena, condensed matter to lessconventional fields, like biology, ecology or quantitativefinance. [1, 2, 3] The essentially non-equilibrium fea-ture of these systems is due to their possibility to irre-versibly enter an absorbing (dead, empty) state. Themere existence of such a state violates the detailed bal-ance. The central problem in studies of non-equilibriumsystems is the transitions they undergo between variousactive phases and the inactive (absorbing) state.The kinetic contact processes which model, e.g., theepidemic disease spreading or population replication,are the simplest kinetic models exhibiting such non-equilibrium phase transitions under variation of their pa-rameters. [1, 2, 3] These variations consist in the changeof limiting probability distribution for the site occupa-tion numbers in replication process, or the infected sitenumbers for epidemic models. Mostly these models pos-sess two phases: the absorbing state with the populationor viruses extinct and the active phase where some sitesare populated or infected.Usually the sequential update formalism is used tostudy these models. It is based on the differential ki-netic equation for the probability distribution function.In this framework the generic second order transition intoabsorbing state of the directed percolation (DP) univer-sality class is established for the majority of such contactprocesses. [1] Another approach is the parallel updatescheme when the model is represented as a discrete-timeprobabilistic cellular automaton (PCA). [4, 5, 6] It isknown that these two approaches can give rather differ-ent results [1] and one may reasonably suppose that insuch case parallel update gives more adequate descrip- tion of real kinetics just because in Nature there is noqueue for the fulfillment of the state transformation pro-cesses. So the studies of PCA implementation of replica-tion (epidemic) processes may give a more realistic pic-ture of them and reveal some new types of critical be-havior.In this paper we consider a one-dimensional PCA rep-resenting the contact process of population replication ordisease spreading. Imagine a line of plants which main-tain population by spreading the seeds to the nearby sitesso the new plants can grow on them if these sites areempty. Let q be the probability of such event for emptysite having only one neighboring plant. For the case oftwo plants around the empty one we can choose this prob-ability r > q to be r = 1 − (1 − q ) = q (2 − q ) assumingthe independence of the two plants’ seeds “not spreadingeffects”. Some other choices are also possible but herewe consider only the case r = q (2 − q ). Let also p be theprobability for plant to survive in the one time step. Thuswe have the PCA on the line of two-state sites (emptyand full S i = 0 , i is the intrachain siteindex) with the evolution step probabilities defined forthe local three-site configurations in Table I. TABLE I: Automaton rules S i − , S i , S i +1 S i = 1) p q q (2 − q ) 0 Obviously, this model describes also the epidemic pro-cess among the plants, since instead of the plant seeds wecan consider the virus spreading. Note also that this pro-cess differs essentially from the one of the cell replicationconsidered in Ref.[3].We have performed Monte Carlo simulations of thisPCA starting from several random initial states for thechains of N = 10000 sites for up to 5000 time steps. Theperiodic boundary conditions were implemented. Theresulting p − q diagram (Fig. 1) consists of a single tran-sition line between absorbing (all occupation numbers S i = 0 in the infinite-time limit) and active (some S i = 0)phases. The curve has the following approximate analyti-cal representation q = 0 . − . p − . , determinedfrom a direct fit. FIG. 1: (Color online) Phase diagram.
We assume the second order transition between thephases, since the concentration of active sites defined as n ≡ N N X i =1 S i (1)appears to be continuous. The examples of such behav-ior are presented in Fig. 2, where the (continuous) q -dependence of n is shown for p = 0 . , . , .
8. We foundthat the points obtained in simulations follow closely thepower law n ∝ ( q − q c ) β with the indices β given inTable II. TABLE II: Transition points q c and critical indices β for dif-ferent p values. p q c β One can notice the unusual variation the index β with q instead of its more conventional universality. The indexvaries closely to the directed percolation value β DP =0 . n for different trials (andinitial states) as well as its fluctuations in the nominallysteady state at large times. Not too close to the transition FIG. 2: (Color online) q -dependence of n for p = 0.2 ( × ), 0.5( ◦ ), 0.8 ( (cid:3) ). Solid lines show power law dependencies withthe indices given in Table II. point ( | q − q c | & .
05) these variations of n are of theorder of several percents and could not possibly accountfor the differences in β of the order of ten percents.Therefore we are lead to the assumption that this repli-cation process does not belong to the DP universalityclass, and it is characterized by the variable critical in-dices. To date the signs of the non-universal behaviorare found only in the 1 d pair contact process with dif-fusion [7] but the type of its critical anomalies is stilldebated; see Ref. [8] and references therein. The non-universality of some equilibrium spin models (i.e., thecoupling-dependent critical exponents) is well known.For example, it is established for the classical XY model[9], the eight-vertex model solved by Baxter [10], orfor several two-dimensional Ising models with competingnearest-neighbor (nn) and next-nearest-neighbor (nnn)interactions, see Refs. [11, 12, 13, 14] and more referencestherein. More examples and discussions on the equiva-lence between the two-dimensional classical spin models(Ising, Potts, Ashkin-Teller, XY ) and (1+1) quantummodels (quantum spin chains, Luttinger, Gaussian), theircritical properties and (non)universality can be found inRefs. [15, 16].From the renormalization group (RG) point of view thenon-universality of the PCA implies that the RG flowfor the equivalent equilibrium spin model [5, 6] wouldhave a manifold of fixed points instead of a unique fixedpoint. A paradigmatic example of such behavior is theKosterlitz-Thouless picture of RG flow for the XY model,[9] which appears also in many (1+1) quantum models.[17] According to Kadanoff and Wegner, non-universalitycan be traced back to the presence of an RG marginaloperator. [11]To corroborate our conjecture of non-universality wepoint out that the structure of the active state in thismodel undergoes considerable qualitative changes under p and q variations at constant n . The space-time pat-terns of the process which has reached a steady state with n ≈ . p and q . Theevolution of 100 sites chosen out of 10000 is shown thereduring 100 time steps in the steady state. FIG. 3: (Color online) (a) - p =0.2, q = 0.89, (b) - p =0.5, q = 0.66, (c) - p =0.8, q = 0.32. Blue sites are empty. Thetime direction is horizontal. The pattern difference, quite distinct visually, can beassessed numerically. It appears that along with n , theconcentration of clusters of adjacent active and inactive sites in the steady states n c ≡ N N X i =1 δ ( S i + S i +1 ,
1) (2)stays almost constant with negligible fluctuations, seeFig.4. In the above equation and throughout, we use δ ( m, n ) as a notation for the conventional Kroneckerdelta. FIG. 4: (Color online) Time dependence of n ( ◦ red), n c ( (cid:3) blue), n ( × green), and n , ( ⋄ magenta); (a) p = 0.2, q = 0.89, (b) p = 0.8, q = 0.32. This parameter, as well as n , is almost the same in thedifferent trials. The values of n c for three steady statesin Fig. 3 are presented in Table III. The concentrationof active adjacent pairs (11) in steady configurations n is also shown there with n ≡ N N X i =1 δ ( S i + S i +1 ,
2) (3)Note that an exact relation n = n − n c / TABLE III: Parameters of three steady state configurationsshown in Fig. 3.( p, q ) (0.2, 0.89) (0.5, 0.66) (0.8, 0.32) n n c n n , holds. We have also calculated the average values of thesupposed marginal operator n , ≡ N N X i =1 S i,t S i +1 ,t S i − ,t S i,t +1 (5)This four-spin operator is present in the Hamiltonian ofthe corresponding equilibrium spin model and it can beresponsible for the non-universality in (1+1) dimensions.The average n , is also nearly constant in steady states(see Fig. 4), trial independent and varies with p and q ,as shown in Table III. Its value defines the structuralcharacteristics of emergent active state and, probably,their indices. Our data also show that n (and n c ) isproportional to n near the transition points. Summary & Discussion:
The structural characteristicsof steady states with nearly the same n are considerablydifferent, which is not only seen quite clearly from Fig. 3,but it manifests quantitatively via variations of numbersof clusters per site n c (2) and the four-spin operator n , (5). Along with the proportionality of n (and n c ) to n this suggests a multi-component order parameter. Thelatter violates one of the requirements of the Janssen-Grassberger hypothesis for a model to belong to the DPuniversality class [8]. We conjecture the non-universalityin this model, most likely due to the marginal four-spinterm of the Hamiltonian [cf. Eq. (5)], which manifestsitself in variations of the critical index β .However to confirm that this PCA does exhibit thenon-universal critical behavior, the numerical values ofother critical indices are needed. Although it is a ratherdifficult task for the present model to get them due tostrong fluctuations which other quantities of interest suchas correlation lengths exhibit near the transition point.Strong fluctuations also hinder determination of the in-dex α = 2 − ν k − ν ⊥ [1] from the energy-like behavior of n , ∼ ( q − q c ) − α .From the analytical side, the non-universality canbe probed by analysis of the marginal perturbationaround an integrable fixed point. For instance, the non-universality of the eight-vertex model or the Ising withcompeting nn and nnn couplings, can be demonstratedby the RG analysis of the flow generated by the nn or four-spin interactions which couple two independent (in-tegrable) Ising models. [12, 13, 15, 16] In the present casethe RG analysis of non-universality, i.e., of the marginalperturbations, is more difficult problem, since even the(1+1) DP fixed point presumably controlling the univer-sality, is not integrable. [8] Note that the ǫ -expansionsaround the DP upper critical dimension d c = 4 are notreliable for our case of d = 1. [18] Clearly, an RG study ofthe field theory corresponding to our model is warranted.It can shed more light on the the critical properties ofthe model. In particular, the field theory formulation ofthe present PCA model could help to answer a naturalquestion of how this PCA is distinct from others knownto belong to the DP universality class. [1] We plan toaddress all these issues in the forthcoming paper.We acknowledge financial support from the NaturalScience and Engineering Research Council of Canada(NSERC) and the Laurentian University Research Fund(LURF). We thank V. Oudovenko for useful discussionsand his help with the numerical calculations on the ear-lier version of this project. P.N.T. thanks LaurentianUniversity, where the initial stage of the work was done,for hospitality. ∗ Electronic address: [email protected] † Electronic address: [email protected][1] H.Hinrichsen, Adv. Phys. , 1 (2000).[2] G. Odor, Rev. Mod. Phys. , 663 (2004).[3] S. C. Ferreira, Jr., Phys. Rev. E , 036119 (2004).[4] W. Kinzel, Z. Phys. B , 229 (1985).[5] P. Rujan, J. Stat. Phys. , 139 (1987).[6] A. Georges and P. Le Doussal, J. Stat. Phys. , 1011(1989).[7] J. D. Noh and H. Park, Phys. Rev. E , 016122 (2004).[8] H. Hinrichsen, Physica A , 1 (2006).[9] J. M. Kosterlitz and D. J. Thouless, J. Phys. C , 1181(1973); J. M. Kosterlitz, J. Phys. C , 1046 (1974)[10] R. J. Baxter, Exactly Solved Models in Statistical Me-chanics , (Academic Press, London, 1982).[11] L. P. Kadanoff and F. J. Wegner, Phys. Rev. B , 3989(1971).[12] M. N. Barber, J. Phys. A , 679 (1979).[13] K. Minami and M. Suzuki, J. Phys. A , 7301 (1994).[14] R. Liebmann, Statistical Mechanics of Periodic Frus-trated Ising Systems , (Springer, Berlin, 1986).[15] M. P. M. den Nijs, Phys. Rev. B , 6111 (1981).[16] J. L. Black and V. J. Emery, Phys. Rev. B , 429 (1981).[17] T. Giamarchi, Quantum Physics in One Dimension (Ox-ford University Press, Oxford, 2004).[18] For reviews on the field theory approaches and more ref-erences see, e.g., H.-K. Janssen and U. C. T¨auber, Ann.Phys. , 147 (2005); U. C. T¨auber, M. Howard, andB. P. Vollmayr-Lee, J. Phys. A38