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 reactiondiﬀusion 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, 240610435, USA
(Dated: February 24, 2005)We present the exact solution for the full dynamics of a nonequilibrium spin chain and its dualreactiondiﬀusion 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 reactiondiﬀusion 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 timedependence are found: if both temperatures are positive, the magnetization,density and correlation functions decay exponentially to their steadystate 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 meanﬁeld 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 manybody 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 framework is still lacking: As yet, there is no equivalent of Gibbs ensemble theory for nonequilibrium systems. As aconsequence, in contrast to equilibrium statistical mechanics, macroscopic observables cannot be computedwithout explicit reference to the imposed dynamics, generally 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 testing 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 siteindependent 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 endowed with Glauberlike dynamics. Our analysis of thismodel provides a full description of its dual counterpart,namely a reactiondiﬀusion system (RDS), characterizedby spatially alternating annihilation and creation rates.Members of these two classes – i.e., kinetic Ising andreactiondiﬀusion models – are prototypical nonequilib
∗
Electronic address: mmobilia, schmittm, rkpzia @vt.edu
rium systems which have been thoroughly studied on homogeneous lattices [2, 3, 4, 5, 6, 7]. Yet, they still oﬀer
surprises and novel behaviors, when nontrivial spatialrates are investigated.Our model was ﬁrst introduced by R`acz and Zia [8]
who recognized that (stationary) twopoint 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 stationary solution, its representation as exp(
−H
eﬀ
) is nontrivial, involving a proliferation of longerranged multispin couplings [10] . Finally, we recently reported the exact solution for all
dynamic
correlation functions, starting from a very simple initial condition, i.e., zero magnetization and vanishing correlations [11].In this article, we complete these earlier studies bydemonstrating how competing sitedependent rates maydramatically aﬀect the dynamics by giving rise to an
oscillatory
approach toward the nonequilibrium steadystate. We use a generating functional approach to obtain the complete solution for all correlation functionswith arbitrary initial conditions. We focus speciﬁcallyon the dynamical magnetization and the spinspin correlations and explore their longtime behavior. We will alsoconsider the dynamics of domain walls in the spin chainwhich can be mapped onto a reactiondiﬀusion system.Interpreting our results in the language of particle annihilation and creation, negative “temperatures” acquire anatural physical meaning, leading to unexpected oscillatory dynamics. From a more technical point of view, weare able to obtain a complete solution for two nontrivialnonequilibrium manybody systems which provides someinsight into the solvability of two whole classes of relatedmodels.The mapping to a reactiondiﬀusion 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 apparent until one recognizes the equivalent spin chain model.Also, from an experimental perspective, it is well knownthat diﬀusionlimited reactions with annihilation and creation of pairs of particles are good models for the photogrowthproperties 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 reactiondiﬀusion system. Section III presents the complete solution of the spin chain. Some technical details are relegated to two Appendices. In Section IV, we map thetwotemperature spin chain onto a reactiondiﬀusion system with alternating rates, whose density and correlationfunctions are computed. We analyze the conditions under which damped oscillations characterize the approachto the steady state, and we compare our exact resultsto a simple meanﬁeld 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 manyparticle systems on a onedimensional lattice: (
i
) a kinetic Ising spin chain (KISC) endowed with a generalizedGlauberlike dynamics; and (
ii
) a reactiondiﬀusion system (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 results 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. Nearestneighbor 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 timedependent 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 compute the timedependent 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 relationship [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
equaltime correlation functions is equivalent to the complete knowledge of the distribution function
P
(
{
σ
}
,t
).Recently, this implication was exploited for the steadystate [9], and for the timedependent situation yet restricted to a particularly simple initial condition [11].The spinﬂip dynamics of this Ising chain can be expressed in terms of the creation, annihilation and diffusion of
domain walls
, i.e., pairs of spins with opposite sign. For example, ﬂipping
σ
j
in the local conﬁguration
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 domain wall diﬀusion. By identifying a domain wall with a“particle”,
A
, our spinﬂip dynamics can be recast as areactiondiﬀusion 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 annihilation 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 unexpected, beneﬁt from this mapping: Assigning
negative
values for the temperatures
T
e
and/or
T
o
may appear artiﬁcial 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 parameter 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 generating 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 magnetization,
m
j
(
t
) =
σ
j
t
for
arbitrary
initial condition, and(
ii
) a
particular
twopoint equaltime correlation function,
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 information about these two observables.
A. The general
t
dependent magnetization.
From our earlier work [11], we recall that the magnetization
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 equation 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 modiﬁed 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 oscillatory asymptotic behavior [16]. This translates into anoscillatory decay of the magnetization [11].
B. A special twopoint equaltime correlationfunction.
The second fundamental quantity, i.e., the equaltimespinspin correlation function
c
k,j
(
t
), with
k > j
, is already 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 values is exponential and monotonic, as
e
−
2(1
−
α
)
t
t
−
3
/
2
, provided
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 dependence 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 timedependent correlations
only for a system with no initial magnetization and twospin correlations (e.g., a random distribution). In particular, 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 multispincorrelations.
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 technical 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 Grassmann 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 functions 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 equaltime spinspin 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 second 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 stationary spinspin correlation function becomes independentof the initial values.Higher order correlations are can also be evaluated but are rather complex for general initial conditions. For uncorrelated, nonmagnetized initial conditions, however, they simplify signiﬁcantly [11]. For example, the 4point function
σ
j
1
σ
j
2
σ
j
3
σ
j
4
t
factorizesinto twopoint 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 steadystate behavior can be computed directly from the master equation [9] or from the stationary limit of the generating function, Ψ(
{
η
}
,
∞
) = exp
k>j
η
j
η
k
c
k,j
(
∞
)
.Thanks to this simple form, the 2
n
point correlations factorize into a product of 2point 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 spinspin correlation functions
c
k,j
(
t
′
;
t
) describing 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 spinspin correlation functions for general translationally invariant initial conditions.
IV. CONSEQUENCES FOR AREACTIONDIFFUSION MODEL WITHALTERNATING RATES
In this section, our exact results will be translatedinto the language of the corresponding reactiondiﬀusionmodel. 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 lattice. Each site ˆ
can be occupied by at most one particle, 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 mapping from spin to particle language is nonlinear, namely,
n
ˆ
=
12
[1
−
σ
j
−
1
σ
j
]. As before, we seek the probability, 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 perfectly clear from the context which distribution is relevant. The dynamics of our model is characterized bysymmetric diﬀusion of particles (with rate 1
/
2) and pairannihilation/creation of particles with spatially alternating rates (1
±
γ
j
)
/
2. In this case, the two particles arecreated on the (dual lattice) sites ˆ
and ˆ
+ 1, by ﬂipping 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 process 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 displays a
mild sitedependent 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 photoexcitedsolitons in conjugated polymers or linear chain compounds. MX chain compounds, [Pt(
en
)
2
][Pt(
en
)Cl
2
]
Y
4
,where
Y
stands for ClO
4
or BF
4
and (
en
) for enthylenediamine, are of particular experimental interest [12, 13].
In these compounds, photogenerated solitons are so longlived that they can be experimentally studied. Irradiation with continuous wave (nonpulsed) blue light generates solitonantisoliton 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 spatially 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 meanﬁeld theory which is widely used for the homogeneous (
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 average 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 lowerordercorrelations to higherorder ones. In general, such hierarchies 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 correlations,
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 steadystate 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 longtime 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 relaxationtime 2(1
−
α
) ,and a subdominant powerlaw 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 longtime behavior is very similar to that found inthe usual (
γ
e
=
γ
o
=
±
1) pair diﬀusion, annihilation,and creation process
AA
⇄
∅∅
[4, 18].