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The exponent is computed for all q via mappings to an exactly solvable one-species coagulation model and also by transforming to a free-fermion description, recovering known special cases.",{"@graph":14,"@context":73},[15,34,56],{"@type":16,"itemListElement":17},"BreadcrumbList",[18,23,27,31],{"item":19,"name":20,"@type":21,"position":22},"https://docshare.wps.com","Home","ListItem",1,{"item":24,"name":25,"@type":21,"position":26},"https://docshare.wps.com/document/","Document",2,{"item":28,"name":29,"@type":21,"position":30},"https://docshare.wps.com/document/research-report/","Research & 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quantity is studied in the paper for the 1D q-state Potts model?","Question",{"text":63,"@type":64},"The paper studies the fraction r(q,t) of persistent spins that never flip up to time t under zero-temperature Glauber dynamics.","Answer",{"name":66,"@type":61,"acceptedAnswer":67},"How does r(q,t) behave at long times?",{"text":68,"@type":64},"It decays as a power law r(q,t) ~ t^{-θ(q)}, where the persistence exponent θ(q) depends on q in a complicated way.",{"name":70,"@type":61,"acceptedAnswer":71},"How is the persistence exponent θ(q) obtained?",{"text":72,"@type":64},"The derivation uses an exact mapping to a one-species coagulation model A+A→A, and also relates the problem to a free-fermion description to obtain an exact expression for 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theOne-Dimensional Potts Model\n\nBernard Derrida,1.2 Vincent Hakim,'and Vincent Pasquier²  \nReceived January 30.1996  \nFor the zero-temperature Glauber dynamics of the q-state Potts model,thefraction r(q.1)of spins which never flip up to time t decays like a power lawr(q,1)～1-09)when the initial condition is random.By mapping the problemonto an exactly soluble one-species coagulation model(A+A→A)or alter-natively by transforming the problem into a ree-fermion model,we obtain theexact expression of (q)for allvalues of qThe exponent 0(q)is in generalirrational,θ(3)=0.53795082...0(4)=0.63151575.…….with the exception ofq=2 and q=c,for which θ(2)=3/8 and θ(c∞)=1.  \nKEY WORDS:Glauber dynamics;coarsening;free fermions;Toeplitz deter-  \nminant;Potts model;reaction-diffusion problems  \n## 1.INTRODUCTION\n\nOne of the simplest systems for which coarsening(1)associated to phaseordering can be studied is the one-dimensional Ising or q-state Potts modelevolving according to zero-temperature Glauber dynamics.(²)At zero tem-perature,the evolution tends to align all the spins.Consequently,for arandom initial condition,one observes domains of parallel spins growingwith time.  \nOne way to implement the zero-temperature dynamics of these one-dimensional systems is to update every spin during each infinitesimal timeinterval △t according to the following dynamics:  \nS;(t+△t)=S;(t)with probability 1-24t(1)with probability △t=S;_i(t)=S;+(1)with probability 4t  \nTherefore,updating a spin in one dimension consists in choosing for itsnew value the value of one of its two neighbors at random.When the valueS;(t)of the spin i at time t is traced back in time,a random walk isobtained which connects site i through various ancestors to a particularsite io in the initial configuration,so that S;(t)=S.(0).The dynamics isnothing but the dynamics of a voter model³)and this makes many proper-ties computable via random walk methods.  \nFor example,the equal-time correlation function ,(i,j)=\u003Cδs(.S;(>evolves according to  \n(2)  \n+Φ,(i,j-1)-4Φ,(i,j)  \nFor a random initial condition,Φa(i,j)=1/q+[(q-1)/q]δ;.;and it iseasy to verify that  \n(3)  \nwhere C,(i,j)is the probability that two random walkers starting at sitesi and j at t=0 do not meet up to time t [according to(2),during eachinfinitesimal time interval △t,each walker hops to its right with probability4t,to its left with probability △t,and does not move with probability1-24t].Indeed,C,(i,j)satisfies the same equation(2)as Φ,withCo(i,j)=1-δ:.j.The solution is,for i\u003Cj,  \nFor large t,C,(i,j)takes a scaling form  \n(4)  \nand one finds that the size of the domains increases like t¹/²as expectedwhen the order parameter is not conserved.One recovers in that way theexact scaling form (3),(4)of the equal-time pair correlation function.(4-81  \nPersistent Spins in 1D Potts Model  \nRandom walks can also be used to calculate the autocorrelationfunction.From the definition of the dynamics(1),one can relate theautocorrelation to the probability that a random walk returns to itsstarting point at time t and show that  \n(5)  \nand this gives the exponent characterizing the decay of the autocorrelationfunction.9-11)  \nFrom these two well-known facts(³)-(5)one could believe that all theexponents appearing in the zero-temperature growth of domains in onedimension are simple and well understood.However,by studying thefraction of spins r(q,t)which have never flipped up to time t,it was found,first numerically(1²2-14)and then analytically,(15)that this fraction r(q,t)ofpersistent spins decreases with time like a power law  \nr(q,t)～t-B(q)  \n(6)  \nwhere the exponent θ(q)has a complicated q dependence.In an earlierletter(15)we gave for the 1D Potts model the exact expression of θ(q),  \n(7)  \nwhich agrees with all the previous numerical estimates based on MonteCarlo simulations and finite-size scaling.(1²-14)Our derivation of(7)in ref.15was based on a mapping ","cbCairiq9Bk1oDNL","https://ap.wps.com/l/cbCairiq9Bk1oDNL","pdf",1182502,35,"English","# Introduction\n## Relation to a Reaction-Diffusion Model\n## Persistent Spins in 1D Potts Model","[{\"question\":\"What quantity is studied in the paper for the 1D q-state Potts model?\",\"answer\":\"The paper studies the fraction r(q,t) of persistent spins that never flip up to time t under zero-temperature Glauber dynamics.\"},{\"question\":\"How does r(q,t) behave at long times?\",\"answer\":\"It decays as a power law r(q,t) ~ t^{-θ(q)}, where the persistence exponent θ(q) depends on q in a complicated way.\"},{\"question\":\"How is the persistence exponent θ(q) obtained?\",\"answer\":\"The derivation uses an exact mapping to a one-species coagulation model A+A→A, and also relates the problem to a free-fermion description to obtain an exact expression for θ(q).\"}]","Exact Exponent for the Number of Persistent Spins in the Zero-Temperature Dynamics of the One-Dimensional Potts Model | PDF",88]