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Exact dynamics of a reaction-diffusion model with spatially alternating rates

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We present the exact solution for the full dynamics of a nonequilibrium spin chain and its dual reaction-diffusion model, for arbitrary initial conditions. The spin chain is driven out of equilibrium by coupling alternating spins to two thermal baths
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    a  r   X   i  v  :  c  o  n   d  -  m  a   t   /   0   4   1   2   5   7   6  v   3   [  c  o  n   d  -  m  a   t .  s   t  a   t  -  m  e  c   h   ]   3   1   M  a  y   2   0   0   5 Exact dynamics of a reaction-diffusion model with spatially alternating rates M. Mobilia, B. Schmittmann and R. K. P. Zia 1,  ∗ 1 Center for Stochastic Processes in Science and Engineering,Department of Physics, Virginia Tech, Blacksburg, VA, 24061-0435, USA (Dated: February 24, 2005)We present the exact solution for the full dynamics of a nonequilibrium spin chain and its dualreaction-diffusion model, for arbitrary initial conditions. The spin chain is driven out of equilibriumby coupling alternating spins to two thermal baths at different temperatures. In the reaction-diffusion model, this translates into spatially alternating rates for particle creation and annihilation,and even negative “temperatures” have a perfectly natural interpretation. Observables of interestinclude the magnetization, the particle density, and all correlation functions for both models. Twogeneric types of time-dependence are found: if both temperatures are positive, the magnetization,density and correlation functions decay exponentially to their steady-state values. In contrast, if one of the temperatures is negative, damped oscillations are observed in all quantities. They canbe traced to a subtle competition of pair creation and annihilation on the two sublattices. Wecomment on the limitations of mean-field theory and propose an experimental realization of ourmodel in certain conjugated polymers and linear chain compounds. PACS numbers: 02.50.-r, 75.10.-b, 05.50.+q, 05.70.Ln I. INTRODUCTION Nonequilibrium many-body systems abound in thephysical and life sciences and have recently received muchattention (see e.g. [1, 2, 3] and references therein). De- spite these efforts, a comprehensive theoretical frame-work is still lacking: As yet, there is no equivalent of Gibbs ensemble theory for nonequilibrium systems. As aconsequence, in contrast to equilibrium statistical me-chanics, macroscopic observables cannot be computedwithout explicit reference to the imposed dynamics, gen-erally described by a master equation, and most progressin the field is made by studying paradigmatic models[2]. In this context,  exact   solutions of simple models arescarce, but very precious, since they can serve as test-ing grounds for approximate and/or numerical schemesand shed light on general properties of whole classes of related models. Not surprisingly,  nontrivial   solutions arealmost entirely restricted to one dimension (1D; see e.g.[2, 3] ), and have focused on completely uniform lattices with site-independent rates. Clearly, however, one wouldlike to take into account more complex situations, e.g.,those with spatially varying coupling constants or rates.Arguably, one of the simplest generalizations beyond acompletely uniform system is one with alternating rates.In the following, we consider a 1D kinetic Ising chain(KISC), coupled to two alternating temperatures and en-dowed with Glauber-like dynamics. Our analysis of thismodel provides a full description of its dual counterpart,namely a reaction-diffusion system (RDS), characterizedby spatially alternating annihilation and creation rates.Members of these two classes – i.e., kinetic Ising andreaction-diffusion models – are prototypical nonequilib- ∗ Electronic address: mmobilia, schmittm, rkpzia @vt.edu rium systems which have been thoroughly studied on ho-mogeneous lattices [2, 3, 4, 5, 6, 7]. Yet, they still offer surprises and novel behaviors, when non-trivial spatialrates are investigated.Our model was first introduced by R`acz and Zia [8] who recognized that (stationary) two-point correlationfunctions are easily found exactly, even though spins onalternating sites are coupled to  different   temperatures.Schmittmann and Schm¨user subsequently realized that all stationary   correlation functions are exactly calculable[9]. While this information is equivalent to the full sta-tionary solution, its representation as exp( −H eff  ) is non-trivial, involving a proliferation of longer-ranged multi-spin couplings [10] . Finally, we recently reported the ex-act solution for all  dynamic   correlation functions, start-ing from a very simple initial condition, i.e., zero magne-tization and vanishing correlations [11].In this article, we complete these earlier studies bydemonstrating how competing site-dependent rates maydramatically affect the dynamics by giving rise to an oscillatory   approach toward the nonequilibrium steadystate. We use a generating functional approach to ob-tain the complete solution for all correlation functionswith arbitrary initial conditions. We focus specificallyon the dynamical magnetization and the spin-spin corre-lations and explore their long-time behavior. We will alsoconsider the dynamics of domain walls in the spin chainwhich can be mapped onto a reaction-diffusion system.Interpreting our results in the language of particle anni-hilation and creation, negative “temperatures” acquire anatural physical meaning, leading to unexpected oscilla-tory dynamics. From a more technical point of view, weare able to obtain a complete solution for two nontrivialnonequilibrium many-body systems which provides someinsight into the solvability of two whole classes of relatedmodels.The mapping to a reaction-diffusion system is of inter-  2est for two reasons. On the theoretical side, the equationsfor densities and correlation functions in the RDS forman  infinite hierarchy   whose solution is not at all appar-ent until one recognizes the equivalent spin chain model.Also, from an experimental perspective, it is well knownthat diffusion-limited reactions with annihilation and cre-ation of pairs of particles are good models for the pho-togrowthproperties of excited states (solitons/antisolitonpairs) in certain conjugated polymers and linear chaincompounds [12, 13, 14]. We propose that spatially al- ternating creation/annihilation rates in these systems –especially in MX chain compounds – can be generatedwith the help of a laser with spatially modulated poweroutput.This article is organized as follows: In the next sectionwe introduce the kinetic spin chain and its dual reaction-diffusion system. Section III presents the complete so-lution of the spin chain. Some technical details are rel-egated to two Appendices. In Section IV, we map thetwo-temperature spin chain onto a reaction-diffusion sys-tem with alternating rates, whose density and correlationfunctions are computed. We analyze the conditions un-der which damped oscillations characterize the approachto the steady state, and we compare our exact resultsto a simple mean-field description. Section V is devotedto a brief discussion of the solvability of related models,with Section VI reserved for our conclusions. II. THE MODELS We consider two closely related nonequilibrium many-particle systems on a one-dimensional lattice: ( i  ) a ki-netic Ising spin chain (KISC) endowed with a generalizedGlauber-like dynamics; and ( ii  ) a reaction-diffusion sys-tem (RDS), with spatially periodic pair annihilation andcreation rates. For convenience, we restrict ourselves to aperiodic lattice (a ring) with an even number of sites andstudy the thermodynamic limit. We expect our exact re-sults to be valid for the general cases of odd number of sites and/or arbitrary boundary conditions, apart fromthe usual caveats.Since the RDS follows from the spin chain via a dualityrelationship, we focus mainly on the detailed descriptionof model ( i  ). A spin variable,  σ j  = ± 1, denotes the valueof the spin at site  j , with  j  = 1 , 2 ,...L , and  L  an eveninteger. Nearest-neighbor spins interact according to theusual Ising Hamiltonian:  H  =  − J   j  σ j σ j +1 , where J >  0 ( J <  0) is the (anti-) ferromagnetic exchangecoupling. Our model is endowed with a  nonequilibrium  generalization of the usual Glauber [5] dynamics: spinson even and odd sites experience different temperatures, T  e  and  T  o , which implies the violation of detailed balance[8, 9, 10]. To be specific, a configuration { σ 1 ,σ 2 ,...,σ L } evolves into a new one by random sequential spin flips:A spin  σ j  flips to  − σ j  with rate w j ( { σ } )  ≡  w j ( σ j  →− σ j )= 12  −  γ  j 4  σ j  ( σ j − 1  + σ j +1 ) (1)where  γ  2 i  =  γ  e  = tanh(2 J/k b T  e ) and  γ  2 i +1  =  γ  o  =tanh(2 J/k b T  o ), on even (  j  = 2 i ) and odd (  j  = 2 i  +1) sites. The time-dependent probability distribution P  ( { σ } ,t ) obeys the master equation: ∂  t P  ( { σ } ,t ) ==  j  w j ( { σ } j ) P  ( { σ } j ,t ) − w j ( { σ } ) P  ( { σ } ,t )  (2)where the state  { σ } j differs from  { σ }  only by the spinflip of   σ j . Our main goal in this work is to com-pute the time-dependent distribution  P  ( { σ } ,t ). To doso, we compute  all   correlation functions   σ j 1  ...σ j n  t  ≡  { σ }  σ j 1  ...σ j n P  ( { σ } ,t ) and invoke the following rela-tionship [5]:2 L P  ( { σ } ,t ) = 1 +  i σ i  σ i  t  +  j>k σ j σ k  σ j σ k  t  ++  j>k>l σ j σ k σ l  σ j σ k σ l  t  + ...  (3)This expression illustrates that the knowledge of   all  equal-time correlation functions is equivalent to the com-plete knowledge of the distribution function  P  ( { σ } ,t ).Recently, this implication was exploited for the steadystate [9], and for the time-dependent situation yet re-stricted to a particularly simple initial condition [11].The spin-flip dynamics of this Ising chain can be ex-pressed in terms of the creation, annihilation and dif-fusion of   domain walls  , i.e., pairs of spins with oppo-site sign. For example, flipping  σ j  in the local config-uration  σ j − 1  =  σ j  =  σ j +1  = +1 creates two domainwalls:  σ j − 1  =  − σ j  and  σ j  =  − σ j +1 , located on the bonds   (  j  − 1 ,j ) and (  j,j  + 1). Similarly, flipping  σ j  inthe local configuration  σ j − 1  =  σ j  =  − σ j +1  = +1 hasthe effect of moving the domain wall on bond (  j,j  + 1)by one lattice constant to the left, corresponding to do-main wall diffusion. By identifying a domain wall with a“particle”,  A , our spin-flip dynamics can be recast as areaction-diffusion model, and the two examples translateinto  ∅∅→ AA  and  ∅ A → A ∅ , respectively. The mappingfrom the KISC into its dual RDS is described in detail inTable 1.Clearly, the presence of alternating temperatures  T  e , T  o  in spin language translates into alternating pair anni-hilation and creation rates (1 ± γ  e,o ) / 2 in the RDS. Wecan see easily that letting  T  e  or T  o  vanish simply prohibitspair creation entirely at even or odd sites. Remarkably,we can derive an additional, and possibly rather unex-pected, benefit from this mapping: Assigning  negative  values for the temperatures  T  e  and/or  T  o  may appear ar-tificial in the KISC, but is  perfectly natural   in the RDS:For example,  T  e  <  0 simply corresponds to a creationrate (1 − γ  e ) / 2  >  1 / 2 which is easily implemented in asimulation. In other words, the RDS version is physicallymeaningful, and readily accessible, on a much wider pa-rameter space.  3 TABLE I: Basic processes underlying the KISC (left) and RDS (middle) dynamics Spin flip of site   j  Reactions at bonds next to site   j  Rates  + −−−→ + + −  and  −− + −→− + +  A ∅−→∅ A  and  ∅ A −→ A ∅  1 / 2+ − + −→ + + + (  j  even)  AA −→∅∅  (  j  even) (1 +  γ  e ) / 2+ − + −→ + + + (  j  odd)  AA −→∅∅  (  j  odd) (1 +  γ  o ) / 2+ + + −→ + − + (  j  even)  ∅∅−→ AA  (  j  even) (1 − γ  e ) / 2+ + + −→ + − + (  j  odd)  ∅∅−→ AA  (  j  odd) (1 − γ  o ) / 2 III. COMPLETE SOLUTION OF THE KINETICSPIN CHAIN In this section, we completely solve the dynamics of the KISC. It was shown previously [15] that the gener-ating function, and hence the full distribution  P  ( { σ } ,t ),of a broad class of Ising models can be computed fromtwo very basic observables, namely: ( i  ) the magnetiza-tion,  m j ( t ) =   σ j  t  for  arbitrary   initial condition, and( ii  ) a  particular   two-point equal-time correlation func-tion,  c j,k ( t ) =   σ j σ k  t , the resultant from the specialinitial conditions:  m j (0) =  c j,k (0) = 0 (see Appendix Afor a more detailed discussion of this statement). Here, · t  ≡   { σ } · P  ( { σ } ,t ) denotes the usual configurationalaverage. In the following, we assemble the necessary in-formation about these two observables. A. The general  t -dependent magnetization. From our earlier work [11], we recall that the magneti-zation  m j ( t ) =   σ j  t  of the KISC obeys the equation of motion,  ddt m j ( t ) =  γ  j 2  [ m j − 1 ( t ) + m j +1 ( t )] − m j ( t ) whichis easily derived from the master equation, Eqn. (2). Asshown in [11], the general solution of this linear equa-tion takes the form  m j ( t ) =  k  M  j,k  ( t ) m k (0) ,  wherethe “propagator” M  j,k ( t ) can be written in term of mod-ified Bessel functions of first kind  I  n ( t ) [16]: M  j,k  ( t ) =  e − t   γ  j γ  k I  k − j ( αt ) ,  with  α ≡ ( γ  e γ  o ) 1 / 2 (4)If   γ  e γ  o  <  0, the propagator becomes  M  j,k  ( t ) = i ( − 1) ( k − j ) / 2 | γ  j /γ  k | 1 / 2 e − t J  k − j ( | α | t ) [11], where  J  n ( t ) isa Bessel function of the first kind, with damped oscilla-tory asymptotic behavior [16]. This translates into anoscillatory decay of the magnetization [11]. B. A special two-point equal-time correlationfunction. The second fundamental quantity, i.e., the equal-timespin-spin correlation function  c k,j ( t ), with  k > j , is al-ready known from [11]. For our purposes, it suffices toconsider an initial condition with zero magnetization andzero initial correlations. With the boundary condition  σ j σ k  t  = 1 for  j  =  k , this basic correlation depends onlyon the distance between the two sites and their parity, µ ( k ) ,µ (  j ) ∈{ e,o }  [11]: c k,j ( t )  ≡  c µ ( k ) ,µ ( j ) k − j  ( t )= ¯ γ α 2 √  γ  j γ  k  ( k −  j )    2 t 0 dτ τ  e − τ  I  k − j ( ατ  ) (5)where¯ γ   ≡ ( γ  e  + γ  o ) / 2 .  (6)For long times, these settle into their stationary values[8, 9], independent of initial conditions:  σ j σ k  ∞  ≡ c k,j ( ∞ ) = ¯ γ  √  γ  j − 1 γ  k − 1 ω k − j ,  (7)where ω  ≡  α 1 + √  1 − α 2 ,  (8)a quantity that reduces to the familiar tanh( J/k b T  ) inthe equilibrium Ising chain. The approach to these val-ues is exponential and monotonic, as  e − 2(1 − α ) t t − 3 / 2 , pro-vided  γ  e γ  o  >  0. However, for  γ  e γ  o  <  0, the approachis oscillatory and damped by  e − 2 t t − 3 / 2 [11]. For laterreference, it is convenient to display the parity depen-dence explicitly. Since translation invariance ensures c oek − j  ( t ) =  c eok − j  ( t ), we need to distinguish three typesof correlations. The simplest display, which manifestlyshows the underlying symmetries, is  c eek − j  ( t ) c eok − j  ( t ) c ook − j  ( t )  =  ¯ γ/γ  o ¯ γ/α ¯ γ/γ  e  ( k −  j )    2 t 0 dτ τ  e − τ  I  k − j  ( ατ  ) . (9)Note that the last factor is of exactly the same form asin the ordinary Ising chain coupled to a single thermalbath, the only difference being the geometric mean of thetwo  γ  ’s here plays the role of   γ   = tanh(2 J/k b T  ). Beforeturning to the general case, let us remind the reader thatEqns. (5) and (9) give the time-dependent correlations only for a system with no initial magnetization and two-spin correlations (e.g., a random distribution). In partic-ular, these forms, also used in the next sections, shouldnot be confused with the more general cases consideredin Appendix B.  4 C. Generating function and general multi-spincorrelations. In this section, starting from our knowledge of   m j ( t )and  c k,j ( t ), we compute the generating function of theKISC, following [15]. By construction, this generatingfunction allows us to find  all   correlation functions, sub- ject to  arbitrary   initial conditions. A few additional tech-nical details are provided in Appendix A.The generating function is defined via Ψ( { η } ,t )  ≡  j  (1 + η j σ j )  t , where the  { η j }  are standard Grass-mann variables [15, 17]. In the thermodynamic limit, L →∞ , it simplifies toΨ( { η } ,t ) =  j  1 +  σ j  k η k M  k,j ( t )  0 ×  exp  j 2 >j 1 η j 1 η j 2  c j 2 ,j 1 ( t )  ,  (10)If the initial magnetization and all initial correlationsvanish, the average   ...  0  on the right hand side of Eqn.(10) reduces to unity, and one recovers the bilinear formfor Ψ( { η } ,t ) which we already reported in [11]. Eqn.(10) is one of the key results of this paper.Given the generating function, all correlation func-tions can be obtained by simple differentiation [11, 15]:  σ j 1  ...σ j n  t  =  ∂  n Ψ( { η } ,t ) ∂η jn ...∂η j 1  { η } =0 . As an illustration, wecompute the equal-time spin-spin correlation functions,for  k > j :  σ j σ k  t  =  ∂  2 Ψ( { η } ,t ) ∂η k ∂η j  { η } =0 =  c k,j ( t ) + (11)+  ℓ<m  σ ℓ σ m  0  [ M  ℓ,j ( t ) M  m,k ( t ) − M  ℓ,k ( t ) M  m,j ( t )]We emphasize that this is a completely  general result  ,valid for  any   initial conditions, whether homogeneous orinhomogeneous, translationally invariant or not. The twoterms in (11) have simple interpretations. While the sec-ond term reflects the decay of the  initial   correlations, thefirst provides the buildup to the final stationary valuesgiven above (7). Thus, we see explicitly how the station-ary spin-spin correlation function becomes independentof the initial values.Higher order correlations are can also be evalu-ated but are rather complex for general initial condi-tions. For uncorrelated, non-magnetized initial condi-tions, however, they simplify significantly [11]. For ex-ample, the 4-point function   σ j 1 σ j 2 σ j 3 σ j 4  t  factorizesinto two-point functions, according to   σ j 1 σ j 2 σ j 3 σ j 4  t  = c j 2 ,j 1 ( t ) c j 4 ,j 3 ( t )  −  c j 3 ,j 1 ( t ) c j 4 ,j 2 ( t ) +  c j 4 ,j 1 ( t ) c j 3 ,j 2 ( t ) for  j 4  ≥  j 3  ≥  j 2  ≥  j 1  [11]. Similar factorizationshold for all correlations. Their steady-state behav-ior can be computed directly from the master equa-tion [9] or from the stationary limit of the generat-ing function, Ψ( { η } , ∞ ) = exp  k>j  η j η k c k,j ( ∞ )  .Thanks to this simple form, the 2 n -point correla-tions factorize into a product of 2-point correlations:  σ j 1 σ j 2  ...σ j 2 n − 1 σ j 2 n  ∞  =   σ j 1 σ j 2  ∞ ...  σ j 2 n − 1 σ j 2 n  ∞ ,where  j 2 n  > j 2 n − 1  > ... > j 2  > j 1 .Finally, following Refs [5, 11], we can also derive the unequal  -time spin-spin correlation functions  c k,j ( t ′ ; t ) de-scribing how a spin on site  k  at time  t  is correlated withthe spin on site  j  at a later time  t + t ′ : c k,j ( t ′ ; t ) =  ℓ M  jℓ  ( t ′ )  σ k σ ℓ  t =  ℓ M  j,ℓ ( t ′ ) c k,ℓ ( t ) +  ℓ  k 1 <ℓ 1  σ k 1 σ ℓ 1  0 M  j,ℓ ( t ′ )[ M  k 1 ,k ( t ) M  ℓ 1 ,ℓ ( t ) − M  k 1 ,ℓ ( t ) M  ℓ 1 ,k ( t )] (12)As an illustration of these general results, in AppendixB we specifically compute the spin-spin correlation func-tions for general translationally invariant initial condi-tions. IV. CONSEQUENCES FOR AREACTION-DIFFUSION MODEL WITHALTERNATING RATES In this section, our exact results will be translatedinto the language of the corresponding reaction-diffusionmodel. We first associate a site ˆ    on the dual latticewith every bond (  j  − 1 ,j ) of the srcinal chain. Sincethe particles of the RDS are identified with domain wallsin the spin chain, they obviously reside on the dual lat-tice. Each site ˆ    can be occupied by at most one parti-cle, described by an occupation variable  n ˆ   which takesthe value 0 (1) if the site is empty (occupied). Sincea domain wall involves two neighboring spins, the map-ping from spin to particle language is nonlinear, namely, n ˆ   =  12  [1 − σ j − 1 σ j ]. As before, we seek the probabil-ity, ˆ P  ( { n } ,t ), to find configuration  { n }  at time  t , andits averages: the local particle density  ρ ˆ  ( t )  ≡  n ˆ   t  ≡  { n } n ˆ   ˆ P  ( { n } ,t ) and the  m -point correlation functions,  5  n ˆ  1  ...n ˆ  m  t  ≡   { n } n ˆ  1  ...n ˆ  m ˆ P  ( { n } ,t ). To simplifynotation, we continue to denote averages by · t  for bothspins and occupation variables, even though they arecontrolled by different statistical weights,  P  ( { σ } ,t ) andˆ P  ( { n } ,t ), respectively. In each case, it should be per-fectly clear from the context which distribution is rel-evant. The dynamics of our model is characterized bysymmetric diffusion of particles (with rate 1 / 2) and pairannihilation/creation of particles with spatially alternat-ing rates (1 ± γ  j ) / 2. In this case, the two particles arecreated on the (dual lattice) sites ˆ    and ˆ    + 1, by flip-ping a spin on the (srcinal lattice) site  j . Since  γ  j  canbe positive or negative, subject only to  − 1  ≤  γ  j  ≤  1for all  j  , two very distinct behaviors emerge: ( i  ) whenboth  γ  e  and  γ  o  are positive (corresponding to positive“temperatures” in the spin model), the annihilation pro-cess always occurs with a  larger   rate than the creationprocess, irrespective of whether  j  is even or odd; ( ii  )when, e.g.,  γ  o  is negative and  γ  e  positive, the system dis-plays a  mild site-dependent frustration  : at even sites  j (i.e., ˆ    even and ˆ    + 1 odd) annihilation is more likelythan creation, whereas the situation is reversed on oddsites (where ˆ    odd and ˆ   +1 even). As we will see shortly,this gives rise to  oscillatory   dynamics.Before diving into the details, some further remarkson physical realizations of this model are in order. Whenthe rates are uniform ( γ  e  =  γ  o ), it is well known thatsuch an RDS describes the dynamics of photo-excitedsolitons in conjugated polymers or linear chain com-pounds. MX chain compounds, [Pt( en ) 2 ][Pt( en )Cl 2 ] Y  4 ,where  Y   stands for ClO 4  or BF 4  and ( en ) for enthylene-diamine, are of particular experimental interest [12, 13]. In these compounds, photogenerated solitons are so long-lived that they can be experimentally studied. Irradia-tion with continuous wave (non-pulsed) blue light gener-ates soliton-antisoliton pairs which can diffuse apart orannihilate. Their static and dynamic properties are inquantitative agreement with theoretical models [4, 18]. Since creation, annihilation, and hopping rates can becontrolled by tuning the laser power, we believe that spa-tially alternating rates such as ours will be generated if anMX chain compound is exposed to a spatially modulatedlight intensity.Returning to our model, our goal in this section is first,to derive all correlation functions from our exact solutionof the KISC. We will also comment on the validity of asimple mean-field theory which is widely used for the ho-mogeneous ( γ  e  =  γ  o ) case [18, 19]. Further, we show that particle hops in the RDS develop a peculiar directionalpreference in the steady state, even though there is noexplicit bias in the rates, boundary or initial conditions.Finally, we illustrate how oscillatory behaviors may resultfrom a competition of the underlying processes. A. Density of particles in the RDS The observable of most immediate interest is the aver-age density of particles,  ρ ˆ  ( t ), in the RDS. Its equation of motion can be derived easily from the associated masterequation, resulting in:2  ddtρ ˆ  ( t ) = (2 − γ  j  − γ  j − 1 ) + ( γ  j − 1 ρ ˆ  − 1 ( t )+  γ  j ρ ˆ  +1 ( t )) − (4 − γ  j  − γ  j − 1 ) ρ ˆ  ( t ) −  2[ γ  j  n ˆ  n ˆ  − 1  t  + γ  j +1  n ˆ  n ˆ  +1  t ] (13)It is worthwhile noting that this equation is the firstmember of an infinite hierarchy, connecting lower-ordercorrelations to higher-order ones. In general, such hi-erarchies cannot be solved directly, without recourse tocrude approximations. Here, the mapping to the spinchain develops its full power, allowing us to compute allcorrelation functions for the RDS.The mapping from spins to particles implies that ρ ˆ  ( t ) ≡ n ˆ   =  12 [1 − σ j − 1 σ j  t ], so that we can just turnto Eqn. (11) to read off the answer. To express it fully inRDS language, we also need to translate the initial cor-relations,  σ k σ ℓ  0 . For  k < ℓ  and any  t  (including  t  = 0),we may write   σ k σ ℓ  t  =   σ k σ k +1 σ k +1 σ k +2 ...σ ℓ − 1 σ ℓ  t  =  (1 − 2 n ˆ k +1 )(1 − 2 n ˆ k +2 ) ... (1 − 2 n ˆ ℓ )  t  [18, 20] whence we obtain, for arbitrary initial condition: ρ ˆ  ( t ) = 12 { 1 − c j,j − 1 ( t ) }−  12  ˆ k< ˆ ℓ  (1 − 2 n ˆ k +1 )(1 − 2 n ˆ k +2 ) ... (1 − 2 n ˆ ℓ )  0 ×  [ M  k,j − 1 ( t ) M  ℓ,j ( t ) − M  k,j ( t ) M  ℓ,j − 1 ( t )] (14)Since the “propagators”  M  i,j ( t ) decay exponentially as t →∞ , the steady-state density is independent of initialconditions and spatially uniform: ρ ( ∞ ) ≡ ρ j ( ∞ ) = 12  1 −  ¯ γ  √  γ  e γ  o ω  .  (15)In Appendix B, we explicitly evaluate Eqn. (14) fora generic but simple initial condition, characterized bya uniform, uncorrelated initial distribution of particles,with density  ρ (0). For simplicity, we discuss only its long-time limit here, for  ρ (0) = 1 / 2. We observe two distinctkinds of behaviors:( i  ) When  γ  e γ  o  >  0, the stationary density of particles isapproached exponentially fast [except when  γ  e  =  γ  o  = ± 1, see (B15)], with inverse relaxation-time 2(1 − α ) ,and a subdominant power-law prefactor  t − 3 / 2 : ρ ( t ) = 12  1 −  ¯ γ α    2 t 0 dτ τ  e − τ  I  1 ( ατ  )  ≃  ρ ( ∞ ) +  t − 3 / 2 e − 2(1 − α ) t 2 √  2 πα (1 − α ) .  (16)This long-time behavior is very similar to that found inthe usual ( γ  e  =  γ  o   =  ± 1) pair diffusion, annihilation,and creation process  AA ⇄ ∅∅  [4, 18].
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