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Eur. Phys. J. B (2012) 85: 134 DOI: 10.1140/epjb/e2012-20979-3 Regular Article T HE EUROPEAN P HYSICAL JOURNAL B Autonomous models solvable through the full interval method M. Khorrami a and A. Aghamohammadi Department of Physics, Alzahra University, 19938-93973 Tehran, Iran Received 1st December 2011 / Received in final form 16 February 2012 Published online 26 April 2012 – c EDP Sciences, Societ`a Italiana di Fisica, Springer-Verlag 2012 Abstract. The most general exclusion single species one dimensional reaction-diffusion models with nearest- neighbor interactions which are both autonomous and can be solved exactly through full interval method are introduced. Using a generating function method, the general solution for Fn, the probability that n consecutive sites be full, is obtained. Some other correlation functions of number operators at nonadjacent sites are also explicitly obtained. It is shown that for a special choice of initial conditions some correlation functions of number operators called full intervals remain uncorrelated. 1 Introduction Most of the analytical studies on non-equilibrium statisti- cal models belong to the low-dimensional (specially one dimensional) models [113]. Analyzing one dimensional models, which are usually easier to investigate, helps us gaining knowledge on systems far from equilibrium. One of the techniques used to obtain exact results is the empty interval method (EIM), or its equivalent the full interval method (FIM). Among other things, it has been used to analyze the one dimensional dynamics of diffusion-limited coalescence [1417]. In these, one dimensional diffusion- limited processes have been studied using EIM. There, some of the reaction rates have been taken infinite, and the models have been worked out on continuum. For the cases of finite reaction-rates, some approximate solutions have been obtained. Using this method E n (the proba- bility that n consecutive sites be empty) has been calcu- lated. (Alternatively, F n is the probability that n con- secutive sites be full.) This method has been used to study a reaction-diffusion process with three-site interac- tions [18]. EIM has been also generalized to study the kinetics of the q-state one dimensional Potts model in the zero-temperature limit [19]. In [20], all the one dimen- sional reaction-diffusion models with nearest neighbor in- teractions which can be exactly solved by EIM have been found and studied. Conditions have been obtained for the systems with finite reaction rates to be solvable via EIM, and then the equations of EIM have been solved. There solvability means that the evolution equation for E n be closed. It turned out there, that certain relations between the reaction rates are needed, so that the system is solv- able via EIM. When these conditions between reaction rates are met, the time derivative of E n ’s would be lin- ear combinations of E n ’s. It was shown that if certain a e-mail: [email protected] reactions are absent, namely reactions that produce par- ticles in two adjacent empty sites, the coefficients of the empty intervals in the evolution equation of the empty in- tervals are n-independent, so that the evolution equation can be easily solved. The criteria for solvability, and the solution of the empty-interval equation were generalized to cases of multi-species systems and multi-site interactions in [2123]. In [24], models were studied which were solvable through EIM, but did include interaction which produce particles in two adjacent empty sites. There these models were investigated in continuum, although some terms in the evolution equation were missed, as will be discussed in the present paper. In [25], conventional EIM has been extended to a more generalized form. Using this extended version, a model has been studied which can not be solved by conventional EIM. Recently, the coagulation-diffusion process on a one dimensional chain has been studied us- ing the empty-interval method [26]. There the behavior of the time-dependent double-empty-interval probability has been studied. In [27], the exact two-time correlation and response functions for a one dimensional coagulation- diffusion process has been studied using EIM. In [28], a ten-parameter family of reaction-diffusion processes was introduced for which the evolution equa- tion of n-point functions contains only n- or less-point functions. We call such systems autonomous. The expec- tation value of the particle-number in each site has been obtained exactly for these models. In order to be an au- tonomous model there should be some constraints on the reaction rates. Here we study the most general exclusion single species one dimensional reaction-diffusion model with nearest- neighbor interactions, which can be solved exactly through the full interval method, and is autonomous. Apart from corrections to [24], here lattice models are studied, while in [24] such models on continuum were investigated. The change from the empty interval to the full interval is, of

Autonomous models solvable through the full interval method

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Page 1: Autonomous models solvable through the full interval method

Eur. Phys. J. B (2012) 85: 134DOI: 10.1140/epjb/e2012-20979-3

Regular Article

THE EUROPEANPHYSICAL JOURNAL B

Autonomous models solvable through the full interval method

M. Khorramia and A. Aghamohammadi

Department of Physics, Alzahra University, 19938-93973 Tehran, Iran

Received 1st December 2011 / Received in final form 16 February 2012Published online 26 April 2012 – c© EDP Sciences, Societa Italiana di Fisica, Springer-Verlag 2012

Abstract. The most general exclusion single species one dimensional reaction-diffusion models with nearest-neighbor interactions which are both autonomous and can be solved exactly through full interval methodare introduced. Using a generating function method, the general solution for Fn, the probability that nconsecutive sites be full, is obtained. Some other correlation functions of number operators at nonadjacentsites are also explicitly obtained. It is shown that for a special choice of initial conditions some correlationfunctions of number operators called full intervals remain uncorrelated.

1 Introduction

Most of the analytical studies on non-equilibrium statisti-cal models belong to the low-dimensional (specially onedimensional) models [1–13]. Analyzing one dimensionalmodels, which are usually easier to investigate, helps usgaining knowledge on systems far from equilibrium. Oneof the techniques used to obtain exact results is the emptyinterval method (EIM), or its equivalent the full intervalmethod (FIM). Among other things, it has been used toanalyze the one dimensional dynamics of diffusion-limitedcoalescence [14–17]. In these, one dimensional diffusion-limited processes have been studied using EIM. There,some of the reaction rates have been taken infinite, andthe models have been worked out on continuum. For thecases of finite reaction-rates, some approximate solutionshave been obtained. Using this method En (the proba-bility that n consecutive sites be empty) has been calcu-lated. (Alternatively, Fn is the probability that n con-secutive sites be full.) This method has been used tostudy a reaction-diffusion process with three-site interac-tions [18]. EIM has been also generalized to study thekinetics of the q-state one dimensional Potts model in thezero-temperature limit [19]. In [20], all the one dimen-sional reaction-diffusion models with nearest neighbor in-teractions which can be exactly solved by EIM have beenfound and studied. Conditions have been obtained for thesystems with finite reaction rates to be solvable via EIM,and then the equations of EIM have been solved. Theresolvability means that the evolution equation for En beclosed. It turned out there, that certain relations betweenthe reaction rates are needed, so that the system is solv-able via EIM. When these conditions between reactionrates are met, the time derivative of En’s would be lin-ear combinations of En’s. It was shown that if certain

a e-mail: [email protected]

reactions are absent, namely reactions that produce par-ticles in two adjacent empty sites, the coefficients of theempty intervals in the evolution equation of the empty in-tervals are n-independent, so that the evolution equationcan be easily solved. The criteria for solvability, and thesolution of the empty-interval equation were generalized tocases of multi-species systems and multi-site interactionsin [21–23]. In [24], models were studied which were solvablethrough EIM, but did include interaction which produceparticles in two adjacent empty sites. There these modelswere investigated in continuum, although some terms inthe evolution equation were missed, as will be discussedin the present paper. In [25], conventional EIM has beenextended to a more generalized form. Using this extendedversion, a model has been studied which can not be solvedby conventional EIM. Recently, the coagulation-diffusionprocess on a one dimensional chain has been studied us-ing the empty-interval method [26]. There the behaviorof the time-dependent double-empty-interval probabilityhas been studied. In [27], the exact two-time correlationand response functions for a one dimensional coagulation-diffusion process has been studied using EIM.

In [28], a ten-parameter family of reaction-diffusionprocesses was introduced for which the evolution equa-tion of n-point functions contains only n- or less-pointfunctions. We call such systems autonomous. The expec-tation value of the particle-number in each site has beenobtained exactly for these models. In order to be an au-tonomous model there should be some constraints on thereaction rates.

Here we study the most general exclusion single speciesone dimensional reaction-diffusion model with nearest-neighbor interactions, which can be solved exactly throughthe full interval method, and is autonomous. Apart fromcorrections to [24], here lattice models are studied, whilein [24] such models on continuum were investigated. Thechange from the empty interval to the full interval is, of

Page 2: Autonomous models solvable through the full interval method

Page 2 of 5 Eur. Phys. J. B (2012) 85: 134

course, not important; as a simple interchange of parti-cles and holes would do that. The scheme of the paperis as follows. In Section 2, the most general exclusion sin-gle species one dimensional reaction-diffusion models withnearest-neighbor interactions are introduced, which areautonomous and can be solved exactly through FIM. InSection 3, the evolution equation for the full interval prob-abilities, Fn’s are obtained. In Section 4, the steady statesolutions for these probabilities are obtained, and then us-ing a generating function method, the general solution forFn’s is calculated. Correlation functions of number opera-tors at nonadjacent sites are obtained in Section 5. Finally,in Section 6 some correlation functions of number opera-tors are explicitly calculated for some special choice ofinitial conditions. These are probabilities of some disjointparts of the lattice be full.

2 Full interval and autonomy

Consider a one dimensional lattice, any site of which iseither occupied by a single particle or empty, and assumethat the reactions, as well as the state of the system, aretranslationally invariant. Implicit in this, is that the latticehas no boundaries. But the lattice can still be finite, if it iscircular. Defining Fn as the probability that n consecutivesites be full

Fn := P (n

︷ ︸︸ ︷• • · · · •), (1)where an empty (occupied) site is denoted by by ◦ (•), it isfound (for example similar to [24]), that the most generalsingle species nearest-neighbor interactions for which theevolution equations governing Fn’s are closed are

◦• →{

◦◦, q1

•◦, r1,

•◦ →{

◦◦, q2

◦•, r2,

◦◦ →{

•◦, r1

◦•, r2,

•• →

⎪⎨

⎪⎩

•◦, w1

◦•, w2

◦◦, w

, (2)

where r1, r2, q1, q2, w, w1 and w2 are reaction rates. Thenthe Hamiltonian for a two site interaction for models solv-able through FIM is

H =

−r1 − r2 q1 q2 wr2 −q1 − r1 r2 w2

r1 r1 −q1 − r1 w1

0 0 0 −w − w1 − w2

⎠ .

(3)The autonomy criteria leads to two more constraints ([28],for example)

r1 + q1 = w1 + w,

r2 + q2 = w2 + w. (4)

So an autonomous model solvable through FIM has fivefree parameters.

3 The full interval evolution

The full interval equation is

dFn

dt= (r1 + r2) (Fn−1 + Fn+1 − 2 Fn)

− (q1 + q2)(Fn − Fn+1)−(n − 1)(w1 + w2 + w)Fn

− (w1 + w2 + 2 w)Fn+1, (5)

whereF0 := 1. (6)

The difference of this with what obtained in [24], apartfrom the obvious interchange of particles and vacancies, isthat the last term had been missed in [24]. Using (4), itis seen that the coefficient of Fn+1 in the right-hand sideof (5) vanishes. So,

dFn

dt= (r1 + r2)Fn−1 − [2 (r1 + r2) + q1 + q2 + (n − 1)

× (w1 + w2 + w)] Fn. (7)

Rescaling the time by (w1 + w2 + w):

t := (w1 + w2 + w) t, (8)

the evolution equation becomes

dFn

dt= b Fn−1 − (a + n − 1)Fn, (9)

where

a :=2 (r1 + r2) + q1 + q2

w + w1 + w2,

b :=r1 + r2

w + w1 + w2. (10)

Equation (4), and the fact that the rates are nonegativeguarantee that (w1 + w2 + w) is positive, unless all of therates are zero. These also show that

a ≥ 1 + b, (11)b ≥ 0. (12)

From now on, the rescaled time t is denoted by t, so thatthe evolution equation is written as

Fn = b Fn−1 − (a + n − 1)Fn, (13)

where dot means differentiation with respect to therescaled time.

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Eur. Phys. J. B (2012) 85: 134 Page 3 of 5

4 The general solution for the full interval

Let’s first consider the large time values for Fn or thesteady state solution. Denoting the time independentequation for the full interval by F st

n , one has

F stn =

b

a + n − 1F st

n−1, (14)

which combined with (6) results in

F stn =

bn Γ (a)Γ (a + n)

. (15)

The general solution to (13) is of the form

Fn(t) = F stn +

n∑

m=1

cn m exp[−(a + m − 1) t], (16)

where cn m’s are constants. Putting (16) in (13), one ar-rives at

(n − m) cn m = b cn−1m, m < n. (17)

This allows one to determine cn m’s in terms of cm m’s,which are denoted by dm:

cn m =bn−m

(n − m)!cm m,

=:bn−m

(n − m)!dm. (18)

So,

Fn(t) = F stn +

n∑

m=1

bn−m

(n − m)!dm exp[−(a+m−1) t], (19)

showing that the (largest) relaxation time τ is obtainedfrom

τ =1a. (20)

The constants dm are to be obtained from the initial con-ditions. One way is to define the generating functions Gand Gst through

G :=∞∑

n=0

Fn(0)xn,

Gst :=∞∑

n=0

F stn xn. (21)

Using these and (19),

G(x) = Gst(x) +∞∑

n=0

n∑

m=1

bn−m

(n − m)!dm xn,

= Gst(x) +∞∑

m=1

∞∑

n=m

bn−m

(n − m)!dm xn,

= Gst(x) +∞∑

m=1

(

∂b

)m ∞∑

n=0

bn

n!dm xn,

= Gst(x) +∞∑

m=1

dm

(

∂b

)m

exp(b x),

= Gst(x) + exp(b x)∞∑

m=1

dm xm. (22)

So,∞∑

m=1

dm xm = exp(−b x) [G(x) − Gst(x)], (23)

which results in

dm =m−1∑

k=0

(−b)k

k!

[

Fm−k(0) − bm−k Γ (a)Γ (a + m − k)

]

. (24)

For example, the occupation probability of any site is

F1(t) =b

a+

[

F1(0) − b

a

]

exp(−a t). (25)

5 Correlation functions of number operatorsat nonadjacent site

Denoting the number operator in the site i by ni, it is seenthat

〈ni〉 = b − a 〈ni〉. (26)

This is in fact the same as the evolution equation for F1,as it should be. The solution to (26) is

〈ni〉(t) = 〈ni〉(0) exp(−a t) +b

a[1 − exp(−a t)]. (27)

Defining the correlation Ci1···ikas

Ci0···ik:= 〈ni0 · · ·nik

〉, (28)

where no two indices are adjacent, it is seen that

Ci0···ik= b

k∑

j=0

Ci0···ij ···ik− (k + 1) a Ci0···ik

, (29)

where i means that the index i has been omitted.A simple change of variable makes the above equations

simpler. Defining

Ci0···ik:=

⟨(

ni0 −b

a

)

· · ·(

nik− b

a

)⟩

, (30)

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Page 4 of 5 Eur. Phys. J. B (2012) 85: 134

(again for the case no two indices are adjacent) one arrivesat

˙Ci0···ik= −(k + 1) a Ci0···ik

. (31)

The so called connected correlations Cc are defined induc-tively through

Ci =: Cci , (32)

Ci0···ik=:

PCc

P1(i0···ik) · · · CcPα(i0···ik), (33)

where the summation runs over all partitions P of(i0 · · · ik). Such a partition divides the indices (i0 · · · ik)into α parts, where the β’th part is denoted byPβ(i0 · · · ik). A simple induction shows that

˙Cci0···ik

= −(k + 1) a Cci0···ik

. (34)

Another induction shows that for k > 0, adding a con-stant to any of nij ’s does not change the connected corre-lations. The reason is that, denoting the correlations andconnected correlations corresponding to (nik

+ Δ) by asuperscript Δ, one has

CΔ ci0···ik

= CΔi0···ik

− CΔi0···ik−1

CΔik

+ R, (35)

where R contains terms in which the index ik enters con-nected correlations of (m + 1) operators, where m is pos-itive but less than k. Assuming that the independence ofthe connected correlations of (m + 1) operators is true form positive and less than k, it is seen that R does not de-pend on Δ. The sum of the first two terms of the righthand side is also obviously independent of Δ. So the lefthand side is independent of Δ as well. To complete the in-duction, one should prove that the connected correlationof two number operators does not change upon adding aconstant term to each of them. This is obvious since inthat case, R in the right hand side of (35) is zero.

In particular, one arrives at

Cci0···ik

= Cci0···ik

, k > 0, (36)

showing that

Cci0···ik

(t) = Cci0···ik

(0) exp[−(k + 1) a t], k > 0. (37)

The simplest example of this, is the connected two pointfunction:

(〈ninj〉−〈ni〉〈nj〉)(t) = (〈ninj〉−〈ni〉〈nj〉)(0) exp(−2at),|j − i| > 1. (38)

To consider correlations with possibly adjacent sites, letus first obtain the evolution equation for the full interval,without the assumption of translational invariance. Defin-ing Fi j as the probability that the sites beginning from iending in j are full, one arrives at

Fi j = b1 Fi j−1 + b2 Fi+1 j − (a + j − i)Fi j , (39)

where

b1 :=r1

w + w1 + w2,

b2 :=r2

w + w1 + w2. (40)

Then, reassuming translational invariance define D� as

D� :=

k∏

j=0

�2 j∏

m=1

n�0+···+�2 j−1+m

, (41)

where� := (�0, . . . , �2 k), (42)

and �j ’s are positive integers. This is the probability that�0 consecutive sites beginning from 1 be full, �2 consecu-tive sites beginning from (�0 + �1 + 1) be full, and so on.In the special case k = 0, this correlation is the same asthe full interval:

D� = F�. (43)

In the special case that all �j ’s with even j are one, Dis reduced to the (k + 1)-point function with nonadjacentsites:

D1,�1 1,...,�2 k−1,1 = Ci0···ik, (44)

whereim+1 − im = �2 m+1 + 1. (45)

Using (39) and (41), it is then seen that

D� =k

m=0

[b2 D�−e2 m+e2 m−1 + b1 D�−e2 m+e2 m+1

− (a + �2 m − 1)D�], (46)

where em is a (2 k+1)-tuple, the only nonzero componentof which is the m-th component being equal to one, ande−1 and e2 k+1 are zero.

A special case is the two-point function. For nonadja-cent sites, one has

Ci j = b (Ci + Cj) − 2 a Ci j , (47)

the solution to which is

Ci j(t) = 〈ni nj〉(0) exp(−2 a t) +b2

a2[1 − exp(−2 a t)]

+b

a

[

〈ni〉(0) + 〈nj〉(0) − 2 b

a

]

× [exp(−a t) − exp(−2 a t)], (48)

where (27) has been used, and the fact that

Ci(t) = 〈ni〉(t). (49)

Using (39), one has

Fi j = b1 Ci + b2 Cj − (a + 1)Fi j , j − i = 1, (50)

Page 5: Autonomous models solvable through the full interval method

Eur. Phys. J. B (2012) 85: 134 Page 5 of 5

the solution to which is

Fi j(t) = 〈ni nj〉(0) exp[−(a + 1) t] +b2

a (a + 1)

× {1 − exp[−(a + 1) t]}

+[

b1 〈ni〉(0) + b2 〈nj〉(0) − b2

a

]

× {exp(−a t) − exp[−(a + 1) t]}, j − i = 1.(51)

One thus arrives for an expression for the two point func-tions:

〈ni nj〉(t) =

⎪⎨

⎪⎩

Ci j(t), j − i > 1Fi j(t), j − i = 1〈ni〉(t), j − i = 0

. (52)

6 Uncorrelated full intervals

A special case of the initial conditions is when

D�(0) =k

j=0

⟨ �2 j∏

m=1

n�0+···+�2 j−1+m

(0), ∀ �, (53)

which can be written as

D�(0) =k

j=0

F�2 j (0), ∀ �. (54)

Using (13), it is seen that the ansatz

D�(t) =k

j=0

F�2 j (t), (55)

does satisfy (46). This does not mean that the system iscompletely uncorrelated, as the ansatz

Fn(t) = (F1)n(t), (56)

does not satisfy (13). But it means that the full intervalsare uncorrelated to each other. Among other things, itdoes mean that the correlators C satisfy

Ci0···ik= (F1)k+1, (57)

if this holds initially.

This work was supported by the research council of the AlzahraUniversity.

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