Multispecies reaction–diffusion systems
arXiv:cond-mat/0007093v1 6 Jul 2000
A. Aghamohammadi1,3,a , A. H. Fatollahi2,3,b , M. Khorrami2,3,c, A. Shariati2,3,d 1
Department of Physics, Alzahra University, Tehran 19834, Iran. Institute for Advanced Studies in Basic Sciences, P.O.Box 159, Gava Zang, Zanjan 45195, Iran. 3 Institute for Studies in Theoretical Physics and Mathematics, P.O.Box 5531, Tehran 19395, Iran. 2
a
[email protected] [email protected] c
[email protected] d
[email protected] b
PACS numbers: 82.20.Mj, 02.50.Ga, 05.40.+j Keywords: reaction–diffusion, multispecies
Abstract Multispecies reaction–diffusion systems, for which the time evolution equation of correlation functions become a closed set, are considered. A formal solution for the average densities is found. Some special interactions and the exact time dependence of the average densities in these cases are also studied. For the general case, the large time behaviour of the average densities has also been obtained.
1
Introduction
In recent years, reaction–diffusion systems have been studied by many people. As mean field techniques, generally do not give correct results for low dimensional systems, people are motivated to study stochastic models in low dimensions. Moreover, solving one dimensional systems should in principle be easier. Exact results for some models in a one–dimensional lattice have been obtained, for example in [1–10]. Different methods have been used to study these models, including analytical and asymptotic methods, mean field methods, and large-scale numerical methods. Systems with more than one species have also been studied [11–23]. Most of the arguments are based on simulation results. There are, however, some exact results as well ( [18, 20, 22] for example). In [23], a 10–parameter family of stochastic models has been studied. In these models, the k–point equal time correlation functions hni nj · · · nk i satisfy linear differential equations involving no higher–order correlations. These linear equations for the average density hni i has been solved. But these set of equations can not be solved easily for higher order correlation functions. We have generalized the same idea to multi-species models. We have considered general reaction diffusion processes of multi-species in one dimension with two-site interaction. We have obtained the conditions the Hamiltonian should satisfy in order to give rise to closed set of time evolution equation for correlation functions. The set of equations for average densities can be written in terms of four matrices. The time evolution equation for more-point functions, besides these four matrices, generally depend explicitly on the elements of the Hamiltonian , and generally can not be solved easily. These matrices are not determined uniquely from the Hamiltonian: there is a kind of gauge transformation one can apply on them which of course, does not change the evolution equation. A formal solution for average densities of different species is found. For some special choices of the four matrices we also give the explicit form of interactions and the exact time dependence of average densities. At the end, we study the large time behaviour of the average densities of different species for the general case.
2
A brief review of linear stochastic systems
To fix the notation used in this article, here we briefly review the already well known formalism of linear stochastic systems. The master equation for P (σ, t) is X ∂ P (σ, t) = ω(τ → σ)P (τ, t) − ω(σ → τ )P (σ, t) , (1) ∂t τ 6=σ
where ω(τ → σ) is the transition rate from the configuration τ to σ. Introducing the state vector X |P i = P (σ, t)|σi, (2) σ
1
where the summation runs over all possible states of the system, one can write the above equation in the form d |P i = H|P i, dt
(3)
where the matrix elements of H are hσ|H|τ i = ω(τ → σ), τ 6= σ, X hσ|H|σi = − ω(σ → τ ).
(4)
τ 6=σ
The basis {hσ|} is dual to {|σi}, that is hσ|τ i = δστ .
(5)
The operator H is called a Hamiltonian, and it is not necessarily hermitian. Conservation of probability, X P (σ, t) = 1, (6) σ
shows that hS|H = 0, where hS| =
X β
hβ|.
(7) (8)
So, the sum of each column of H, as a matrix, should be zero. As hS| is a left eigenvector of H with zero eigenvalue, H has at least one right eigenvector with zero eigenvalue. This state corresponds to the steady state distribution of the system and it does not evolve in time. If the zero eigenvalue is degenerate, the steady state is not unique. The transition rates are non–negative, so the off–diagonal elements of the matrix H are non–negative. Therefore, if a matrix H has the following properties, hS|H = 0, hσ|H|τ i ≥ 0,
(9)
then it can be considered as the generator of a stochastic process. It can be proved that the real part of the eigenvalues of any matrix with the above conditions is less than or equal to zero. The dynamics of the state vectors (3) is given by |P (t)i = exp(tH)|P (0)i, and the expectation value of an observable O is X hOi(t) = O(σ)P (σ, t) = hS|O exp(tH)|P (0)i. σ
2
(10)
(11)
3
Models leading to closed set of evolution equations
The models which we address are multispecies reaction–diffusion models. That is, each site is a vacancy or has one particle. There are several kinds of particles, but at any time at most one kind can be present at each site. Suppose the interaction is between nearest neighbors, and the system is translationally invariant. L X Hi,i+1 . (12) H= i=1
The number of sites is L and the number of possible states in a site is N ; different states of each site are denoted by Aα , α = 1, · · · N , where one of the states is vacancy. Introducing nα i as the number operator of Aα particle in the site i, we have N X (13) nα i = 1. α=1
The average number density of the particle Aα in the site i at the time t is α hnα i i = hS|ni |P (t)i
(14)
where |P (t)i := exp(tH) |P (0)i represents the state of the system at the time t, hS| = hs| ⊗ · · · ⊗ hs|, | {z }
(15)
hs| := (1 1 · · · 1) . | {z }
(16)
L
and
N
So, the time evolution of hnα i i is given by
d α hn i = hS|nα i H|P (t)i. dt i
(17)
The only terms of the Hamiltonian H which are relevant in the above equation are Hi,i+1 and Hi−1,i . The result of acting any matrix Q on the ket hs| is ˜ on the same ket, provided each equivalent to acting the diagonal matrix Q ˜ diagonal element of the matrix Q is the sum of all elements of the corresponding column in the matrix Q. So, the action of (1 ⊗ nα )H and (nα ⊗ 1)H on hs| ⊗ hs| are equivalent to the action of two diagonal matrices on hs| ⊗ hs|. We use the notation ∼, for the equivalent action on hs| ⊗ hs|. (1 ⊗ nα )H
∼
X βγ
3
β γ Aα βγ n ⊗ n
(nα ⊗ 1)H
∼
X βγ
β γ A¯α βγ n ⊗ n .
(18)
¯α where Aα βγ and Aβγ are as the following Aα βγ
:=
A¯α βγ
:=
X
λ X
λα Hβγ αλ Hβγ .
(19)
λ
Then, equation (17) takes the following form X β γ ¯α β γ hn˙ α Aα i i = βγ hni−1 ni i + Aβγ hni ni+1 i.
(20)
βγ
Generally, in the time evolution equation of hnα i the two–point functions hnβ nγ i appear. Using (13), one can see that iff A and A¯ satisfy the following equations, then the right hand side of the (20) can be expressed in terms of only one–point functions. α α α Aα βγ + AN N − AN γ − AβN α α α α ¯ ¯ ¯ ¯ Aβγ + AN N − AN γ − AβN
= =
0, 0.
(21)
These equations give 2(N − 1)3 constraints on the Hamiltonian, so adding the condition of stochasticity of H, we have 2(N − 1)3 + N 2 relations between the elements of H. The constraints (21) mean Aα βγ A¯α
βγ
= =
Cβα − Bγα ¯α + D ¯ α. −B β γ
(22)
So, (17) takes the form hn˙ α i i
=
N X
β=1
¯ α )hnβ i + C α hnβ i + D ¯ α hnβ i . − (Bβα + B β β β i i−1 i+1
(23)
In the simplest case, the one-species, each site is vacant or occupied by only ¯ and D ¯ are two-dimensional. one kind of particles. Then, the matrices B, C, B, 1 Using (13), The equation for hn˙ i i is ¯11 + B21 + B ¯21 hn1i i + C11 − C21 hn1i−1 i hn˙ 1i i = − B11 − B ¯1 + C1 + D ¯1 . ¯1 − D ¯ 1 hn1 i + − B 1 − B (24) + D i+1 2 2 2 2 1 2
This is a linear difference equation, of the kind obtained [23], and its solution can be expressed in terms of modified Bessel functions. The time evolution equation for two-point functions also can be obtained. d α β hn n i = dt i j
N X γ
¯ γα hnγ nβ i ¯γα )hnγ nβ i + Cγα hnγ nβ i + D − (Bγα + B i j i−1 j i+1 j 4
¯ β )hnα nγ i + C β hnα nγ i −(Bγβ + B γ γ i j−1 i j ¯ β hnα nγ i , +D |i − j| > 1, γ i j+1 d α β hn n i = dt i i+1
N X γ
(25)
¯ β hnα nγ i − Bγα hnγi nβi+1 i − B γ i i+1
X αβ γ λ γ ¯ γβ hnα +Cγα hnγi−1 nβi+1 i + D Hγλ hni ni+1 i.(26) i ni+2 i + γλ
For more–point functions, one can deduce similar results. In fact, it is easy to show that if the evolution equations of one–point functions are closed, the evolution equation of n–point functions contain only n- and less-point functions. However, generally these set of equations can not be solved easily.
4
Equivalent Hamiltonians regarding one-point functions, and gauge transformations
¯ and D ¯ does not determine the Hamiltonian uniquely, but as Knowing B, C, B, it is seen from (23), the time evolution of one–point functions depends only on ¯ and D. ¯ The two– and more–point functions depend explicitly on the B, C, B, elements of H. So, different Hamiltonians may give same evolutions for hnα i i. Take two Hamiltonians H and H ′ . Defining R := H − H ′ , if
X
αβ Rγλ =
α
X
αβ Rγλ = 0,
(27) (28)
β
¯ Regarding one–point these two Hamiltonians give rise to the same A and A. α functions hni i, these models are the same. So, we call these models, regarding one–point functions, equivalent. ¯ and D ¯ uniquely. The stochasHowever, A and A¯ do not determine B, C, B, tic condition X αβ Hγλ = 0, (29) αβ
¯ and D: ¯ results in some constraints on B, C, B, X (Cβα − Bγα ) = 0 α X ¯α + D ¯ α ) = 0, (−B β γ
(30)
α
So, the sum of all elements of any column of B (C) should be the same X X Cβα = Bβα = f. α
α
5
(31)
Then, the state hs| is the left eigenvector of B and C, with the same eigenvalue ¯ and D ¯ have also the same property, of course with different eigenvalue g. f. B Changing B and C according to the gauge transformation, α
Cβα → C ′ β = Cβα − f α α Bβα → B ′ β = Bβα − f α
or C ′ = C − |f ihs| or B ′ = B − |f ihs|
does not change A . With a suitable choice of f α : X f α = f,
(32)
(33)
α
the sum of the elements of any column of B or C can be set to zero. In this gauge, the eigenvalues of B and C for the eigenvector hs| will be zero.
5
One-point functions
To solve (23), we introduce the vector Nk 1 hnk i hn2k i · Nk := . · · hnN k i
(34)
Equation (23) can then be written as
¯ k + CNk−1 + DN ¯ k+1 . N˙ k = −(B + B)N
(35)
Introducing the generating function G(z, t), G(z, t) =
∞ X −∞
Nk (t)z k ,
(36)
one arrives at, ˙ ¯ + z C + z −1 D ¯ G(z, t), G(z, t) = − (B + B)
(37)
¯ + z C + z −1 D] ¯ G(z, 0). G(z, t) = exp t[−(B + B)
(38)
the solution to which is
Nk (t)’s are the coefficients of the Laurent expansion of G(z, t), so I ∞ 1 X ¯ + z C + z −1 D] ¯ Nm (0). (39) Nk (t) = dz z m−k−1 exp t[−(B + B) 2πi m=−∞ This is the formal solution of the problem, which is of the form X Nk (t) = Γkm (t)Nm (0). m
6
(40)
5.1
Some special cases
¯ and D. ¯ We now consider special choices for B, C, B, 5.1.1
¯ and D ¯ are two–dimensional (the single– The matrices B, C, B, species case)
We can use the gauge transformation to make hs| the simultaneous null left– ¯ and D. ¯ In this gauge, one has eigenvector of B, C, B, B C ¯ B D
|uihb|, |uihc|, |uih¯b|, |uihd|,
= = = =
where |ui :=
1 −1
.
(41)
(42)
This means that it is orthogonal to hs|, and is a simultaneous right–eigenvector ¯ and D. ¯ Using (41), one can easily calculate the exponential in (39): of B, C, B,
where
tg(z) −1 ¯ + z C + z −1 D] ¯ =1+ e |uihg(z)|, exp t[−(B + B) g(z)
¯ hg(z)| := −hb| − h¯b| + zhc| + z −1 hd|,
(43)
(44)
and g(z) := hg(z)|ui.
(45) ¯ ¯ Now take hv| and |wi to be the left–eigenvector of −B − B + C + D dual to |ui, ¯ +C +D ¯ dual to hs|, respectively. One can and the right–eigenvector of −B − B normalize these, so that hv|ui = hs|wi =
1, 1.
(46)
Of course, hv| is orthogonal to |wi. Then, tg(z) −1 ¯ ¯ = etg(z) |uihv|+|wihs|+hg(z)|wi e |uihs|. exp t[−(B+ B)+z C +z −1 D] g(z) (47) Acting this on Nm (0), and noting that
hs|Nm (0) = 1, it is seen that Nk (t)
=
I ∞ 1 X dz z m−k−1 etg(z) |uihv|Nm (0) |wihs|Nk (0) + 2πi m=−∞ 7
(48)
=
I ∞ 1 X dz z m−k−1 etg(z) |uihv|Nm (0), |wi + 2πi m=−∞
(49)
I ∞ 1 X dz z m−k−1 etg(z) |uihv|Nm (0). 2πi m=−∞
(50)
or hv|Nk (t) =
This is equivalent to (24). 5.1.2
C = pB,
Using (22)
¯ = qB ¯ D ¯ = 0. (1 − p)hs|B = (1 − q)hs|B
(51)
¯ = 0. If hs| is not the left null means that p = 1 or hs|B = 0, and q = 1 or hs|B ¯ ¯ = B. ¯ eigenvector of B and B, then p = q = 1. So, we will have B = C, and D Now, using the definition of A X α α λα Aα Hβγ βγ = Cβ − Cγ = α α Aα γβ = Cγ − Cβ =
λ X
λα Hγβ .
(52)
λ
For α 6= β, and α 6= γ, all the terms in the right hand side summations in the above equations are reaction rates and should be non–negative, but the sum of the left hand sides is zero. So, Cβα = Cγα = f α ,
for
γ 6= α 6= β.
(53)
All the elements of each row except the diagonal elements of C ( or B ) are the same. That is, C = |f ihs| + C ′ , (54) where C ′ is some diagonal matrix. The fact that |si is a left–eigenvector of C, shows that it should be a left–eigenvector of C ′ as well. And this demands C ′ ¯ to be proportional to the unit matrix. One can do the same arguments for B ¯ and D. So, after gauge transformation, ¯ =B ¯ = v1. D
C = B = u1,
(55)
Although, the time evolution of average densities can be written in terms of ¯ and D ¯ , the Hamiltonian H is not uniquely be determined by these B, C, B, matrices. There exist different Hamiltonians which are equivalent, regarding one–point functions. X λα α α Hβγ = Aα βγ = u(δβ − δγ ) λ X λ
αλ α α Hβγ = A¯α βγ = v(δγ − δβ ).
8
(56)
All the elements of the ββ column of H are zero. For α 6= β, the elements of H satisfy X βα λα αα Hαβ + Hαβ + Hαβ =u λ6=α,β
X
λ6=α,β
X
λβ αβ ββ Hαβ + Hαβ + Hαβ = −u βλ βα ββ Hαβ + Hαβ + Hαβ =v
λ6=α,β
X
λ6=α,β
αβ αλ αα Hαβ + Hαβ + Hαβ = −v.
(57)
In general, these sets of equations have several solutions, but for the one–species case, the reaction rates are the following ( ∅A Λ12 A∅ → AA (58) u − Λ12 ∅∅ v − Λ12 ( A∅ Λ21 ∅A → AA (59) v − Λ21 ∅∅ u − Λ21
The above system, with no diffusion, has been studied in [24]. There, the n– point functions have been investigated. This solution can be generalized to the multispecies case. For α 6= β Aα Aβ → Aβ Aa , Aα Aβ → Aα Aα , Aα Aβ → Aβ Aβ ,
Λαβ α, β = 1 · · · N u − Λαβ v − Λαβ .
(60)
The only constraint is the non–negativeness of the reaction rates: u ≥ Λαβ ≥ 0
v ≥ Λαβ ≥ 0.
(61)
This model has N (N −1)+2 free parameters. However, only the two parameters u and v appear in the time evolution equation of average densities: α α α hn˙ α i i = −(u + v)hni i + uhni−1 i + vhni+1 i.
(62)
As it is seen, dynamics of average densities of different particles decouple, and despite the complex interactions of the model, hn˙ α i i’s can be easily calculated. But in the time evolution of two-point functions Λαβ ’s appear as well. So, although models with different exchanging rates (Λαβ ) and same initial conditions have the same average densities, their two–point functions generally are not the same.
9
5.1.3
¯ C, D ¯ commute B, B,
Generally, the gauge transformation do not preserve the commutation relation ¯ and D ¯ ). But if B and C commute, there is a gauge of B and C ( and that of B transformation which leaves the transformed B and C commuting. If we choose |f i to be a right eigenvector of B and C dual to hs|, that is B|f i = C|f i = f |f i,
(63)
then B ′ := B − |f ihs| and C ′ := C − |f ihs| commute. If hs|f i = f,
(64)
¯ and D ¯ commute with each then hs| times B ′ and C ′ will be zero. So, if B, C, B, other, there exists a suitable gauge transformation that makes their eigenvalue corresponding to hs| zero, while they remain commuting: ¯ = hs|D ¯ = 0, hs|B = hs|C = hs|B
(65)
Denote, the matrix which simultaneously diagonalize these four matrices by U , diagonalized matrices by primes, and their eigenvalues by bα , cα , ¯bα , and d¯α , respectively. We have ¯ ′ = hΩ|D ¯′ = 0 hΩ|B ′ = hΩ|C ′ = hΩ|B hΩ| = hs|U.
(66)
We take bN = cN = ¯bN = d¯N = 0, and normalize hΩ| and U so that hΩ| = (0 0 · · · 0 0), and
X
Uαβ = δN β .
(67) (68)
α
U will also diagonalize the exponential in (39). So we have I ∞ 1 X ¯ ′ +z C ′ +z −1 D ¯ ′ ] N ′ (0), (69) Nk′ (t) = dz z m−k−1 exp t[−B ′ − B m 2πi m=−∞ where Nk′ (t) := U −1 Nk (t).
(70)
The matrix in the argument of the exponential in (69) is diagonal, so the integral can be easily calculated: I 1 (71) dz z m−k−1 exp t[−bα − ¯bα + z cα + z −1 d¯α ] . I := 2πi q ¯α Introducing w := dcα z, one arrives at α ¯α I p d¯α m−k e−t(b +b ) 2 I := ( α ) cα d¯α t(w + w−1 ) , dw wm−k−1 exp c 2πi
10
(72)
which can be written in terms of modified Bessel functions I := ( Then,
p d¯α m−k −t(bα +¯bα ) 2 e I (2 ) cα d¯α t). k−m cα
(73)
∞ X
¯β p d m−k −t(bβ +¯bβ ) β β ¯ Nk (t) = U diag ( β ) 2 e Ik−m (2 c d t) U −1 Nm (0). (74) c m=−∞ Note that the right–hand side of (73) is δk,m for α = N , since the N -th eigen¯ and D ¯ is zero. value of B, C, B, One can start with four special diagonal matrices, and then construct the Hamiltonians with different reaction–diffusion rates. Not all diagonal matrices lead to physical stochastic models: negative reaction rates may be obtained. Considering the large time behaviour of average number densities, one can show that p |Re( cα d¯α )| ≤ Re(bα + ¯bα ), (75) which also shows that
Re(bα + ¯bα ) ≥ 0.
(76)
Now, we consider a special choice for U : α α Uβα = δN − (1 − δN )δβα .
(77)
Then Bβα Cβα ¯βα B ¯ βα D
= = = =
α bβ (δβα − δN ) α α cβ (δβ − δN ) ¯bβ (δ α − δ α ) β N α d¯β (δβα − δN ).
(78)
Now, consider Aα βγ =
X λ
λα α Hβγ = −bα δγα + cα δβα + (−bγ + cβ )δN .
(79)
For α 6= γ and α 6= β, X λ
X
λα Hβγ ≥0
λ
λα Hγβ ≥ 0.
(80)
So, taking β, γ 6= N and α = N , b γ ≥ cβ .
(81)
The same reasoning is true for ¯bγ and d¯β : ¯bγ ≥ d¯β . 11
(82)
¯ and Here too, similar to the previous example, the above choices for B, C, B, ¯ do not determine H uniquely. One particular solution for the reaction rates D is For α 6= N ( Aα AN , ΛN α d¯α − ΛN α AN Aα → Aα Aα , (83) AN AN , bα − ΛN α ( AN Aα , ΛαN cα − ΛαN Aα AN → Aα Aα , (84) ¯bα − ΛαN AN AN , and, for α, β 6= N bβ − cα − Λαβ Aα AN , ¯bα − d¯β − Λαβ (85) Aα Aβ → AN Aβ , AN AN , Λαβ For α 6= β, the following reactions may also occur. For α < β Aβ Aα , cα Aα Aβ → Aβ Aβ , −cα + d¯β
and for α > β
Aα Aβ →
Aβ Aα , Aα Aα ,
d¯β cα − d¯β
(86)
(87)
The constraint of non–negativeness of the reaction rates leads to cα ≤ dβ ≤ cγ 0 ≤ Λαβ ≤ bβ − cα Λαβ ≤ ¯bα − d¯β 0 ≤ ΛN α ≤ d¯a 0 ≤ ΛN α ≤ ba 0 ≤ ΛαN ≤ ¯ba 0 ≤ ΛαN ≤ ca 5.1.4
α
