- Email: [email protected]
Dynamic process and variation in the contact process
7 August 2000
Physics Letters A 272 Ž2000. 416–420 www.elsevier.nlrlocaterpla
Dynamic process and variation in the contact process Kei-ichi Tainaka) , Mineo Hoshiyama, Yasuhiro Takeuchi Department of Systems Engineering, Shizuoka UniÕersity, Hamamatsu 432-8561, Japan Received 2 February 2000; received in revised form 6 June 2000; accepted 9 June 2000 Communicated by C.R. Doering
Abstract Dynamic process in one-dimensional contact process ŽCP. is studied by computer simulation. We carry out the experiment of phase transition by two different methods: CP and its mean-field version. It is known that for both methods, stationary state exhibits the critical slowing-down; relaxation time diverges near the phase boundary. In the present article, we find only for CP that the enhancement of fluctuation occurs in the transient state of dynamical process. q 2000 Elsevier Science B.V. All rights reserved.
X
1. Introduction The contact process ŽCP. has been extensively studied from mathematical w1,2x and physical w3–5x aspects. Considerable data on CP have been accumulated, but they are mainly related to stationary state. Recently, extended models of CP have a growing attention in the field of ecology w6–11x; we need the investigation of population dynamics of biospecies with short-range interaction. In the present article, we explore the dynamical process of CP.
2. Perturbation experiment The rule of CP can be defined as follows: XqO
)
™ 2X,
Ž 1a .
Corresponding author. Fax: q81-53-478-1228. E-mail address: [email protected] ŽK. Tainaka..
™ O, a
Ž 1b .
where X denotes a particle Žindividual of biospecies. and O is the vacant site. The reactions Ž1a. and Ž1b. mean reproduction and annihilation processes of X, respectively. The parameter a represents the annihilation rate of X. We carry out perturbation experiments of phase transition. It is known that CP exhibits a nonequilibrium phase transition at a critical point a s ac : In the case of a G a c , particles cannot exist in stationary state Žextinct phase., while for a - a c , they can exist Žexisting phase.. The value of a c depends on the spatial dimension d; a c ; 0.606 for d s 1, a c ; 1.213 for d s 2. The perturbation experiment is performed as follows: Before the perturbation Ž t - 0., our system stays in a stationary state at a s a1 Žexisting phase: a1 - a c .. At t s 0, the annihilation rate a of particle is suddenly increased from a1 to a 2 Ž a 2 ) a c .. After this perturbation, the system eventually reaches the extinct phase.
0375-9601r00r$ - see front matter q 2000 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 5 - 9 6 0 1 Ž 0 0 . 0 0 4 2 1 - 7
K. Tainaka et al.r Physics Letters A 272 (2000) 416–420
417
Simulation is carried out by two different methods; namely, CP and its mean-field version. The method for CP is defined as follows:
1.
Initially, we distribute particles on a one-dimensional lattice; each lattice site is either empty ŽO. or occupied by a particle ŽX.. 2. The reactions in Ž1. are performed in the following two steps: Ži. First, we perform two-body reaction Ž1a.: Choose one lattice site randomly, and then specify one of the nearest-neighbor sites. If these sites are X and O, then O is changed into X. Žii. Next, we perform a single particle reaction Ž1b.. Choose one lattice point randomly; if the site is occupied by X, the site will become O by a probability Žrate. a. 3. Repeat step 2. by L times, where L is the total number of lattice points. This step is called as Monte Carlo step w6,7x. We assume that the value of L is sufficiently large but finite. 4. Repeat step 3. until the system reaches a stationary state. Here we employ periodic boundary conditions.
Fig. 1. Snapshot of typical dynamics of phase transition of CP. The value of annihilation rate a is jumped from 0.3 to 2.0 at time t s 0. Each column represents a state of system; the colors black and white denote X and O, respectively. The column at t s 0 denotes the initial pattern.
calculate the ensemble average AŽ t . and the variance V Ž t . which are defined by AŽ t . s VŽ t. s
1 N
Ý xi Ž t . ,
Ž 2.
i
1
Ý N
2
x i Ž t . y AŽ t . .
Ž 3.
i
Next, we describe the method of mean-field version of CP. Almost all procedures of CP are not changed in the mean-field limit, but the step Ži. of 2. for CP is merely changed as follows: Ži.’ Two lattice sites are randomly and independently chosen: if these sites are X and O, then O is changed into X. In Fig. 1, a typical example of perturbation experiment for CP is illustrated, where black and white denote the lattice sites of X and O, respectively. Each column represents a state of system; the column at t s 0 denotes an initial state. We prepare N kinds of initial patterns Ž ensembles. which are in stationary state at a s a1; each of them has the density x i Ž0. Ž i s 1,2 . . . N .. The value of x i Ž0. is almost equivalent to the steady-state density. We obtain the dynamics x i Ž t . for t ) 0 in order to
In both methods of CP and its mean-field version, the stationary state near the critical point a c exhibits the so-called ‘critical slowing-down’ Ždivergence of the relaxation time.. Since the variance in stationary state increases with the decrease of steady-state density, it is expected that fluctuation enhancement w12–15x Žextreme increase of V Ž t .. occurs in the transient state of dynamic process.
3. Mean-field limit
L
limit, the dynamic equation for ™In`mean-field becomes;
x˙ s Rx Ž 1 y xrr . ,
Ž 4.
K. Tainaka et al.r Physics Letters A 272 (2000) 416–420
418
where the dot represents the derivative with respect to the time t which is measured by the Monte Carlo step, and the quantities R and r are defined by R s 2 y a,
r s Ž 2 y a . r2.
Ž 5.
The steady-state solution of Eq. Ž4. Žlogistic equation. can be obtained by setting the time derivative in Ž4. to be zero. It is well known that non-trivial solution Ž x s r . for a - 2 is stable, and that phase transition occurs at a c s 2: for a G a c Žor a - a c ., particles cannot Žor can. exist. Moreover, the logistic equation explains the critical slowing-down near a s a c . When the difference x Ž t . y r is much smaller than unity, we have from Ž4. that x Ž t . y r A exp Ž yN ac y a N t . .
Ž 6.
™
Hence, we see that the relaxation time 1rN a c y a N diverges in the limit a a c . If a s a c , we obtain power-law decay. Because of critical slowing-down, the variance in stationary state increases with the decrease of r . We carry out perturbation experiments of phase transition in mean-field method. It is found from computer simulation that the fluctuation enhancement never takes place for various values of a1 and a 2 ; especially when a 2 takes a value much larger than a c , the value of V Ž t . rapidly decreases with time. In Fig. 2, an example of perturbation experiment is displayed, where the value of a is jumped from a1 s 1 to a 2 s 2.1; in this case, a 2 takes a
Fig. 2. The result of perturbation experiment in the mean-field limit. The time dependencies of both average AŽ t . and variance V Ž t . defined by Ž4. and Ž5. are displayed, where the value of V Ž t . is multiplied by 200. At time t s 0, the annihilation rate a is jumped from 1.0 to 2.1. We repeat the similar experiment 100 times Ž N s100. with different initial patterns. The system has 10 4 sites Ž Ls10 4 ..
value near the critical point a c s 2. Both values of average density AŽ t . and the variance V Ž t . are obtained by performing the similar experiment 100 times Ž N s 100. with different initial patterns. In the below, we give an explanation for the fact that fluctuation enhancement never occurs in the mean-field limit. From Ž3., we have VŽ t. s
1 N
2
Ý xi Ž t . y i
1 s
N2
Ý
1 N
2
ž
Ý xi Ž t . i
/
2
xi Ž t . yx j Ž t . .
i)j
It follows that V˙ s
2 N2
Ý Ž x i y x j . Ž x˙ i y x˙ j . .
Ž 7.
i)j
The dynamics for t ) 0 is assumed to be determined by the logistic eqquation Ž4., where both parameters R and r are negative Ž a2 ) 2.. Let x i Ž t . and x j Ž t . be two solutions of Ž4. whose initial values are slightly different from each other. Here we put x i Ž0. ) x j Ž0.. From Ž4., we have x˙ i y x˙ j s R Ž x i y x j . 1 y Ž x i q x j . rr ,
Ž 8.
Note that x i Ž t . / x j Ž t . for t ) 0 because of the uniqueness of the solution to Ž4. and x i Ž t . ) x j Ž t . for all t ) 0. Hence, the right-hand side of Ž8. is always negative, and the difference x i Ž t . y x j Ž t . decreases as the time proceeds. It is therefore proved from Ž7. that the variance V Ž t . always decreases. Strictly speaking, this proof is insufficient, since the dynamics for a finite value of L is described by some stochastic differential equation which contains both deterministic Žlogistic equation. and fluctuating parts. Nevertheless, the logistic equation Ž4. well explains that i. critical slowing-down occurs, and ii. enhancement of fluctuation does not occur.
4. Results of CP The basic equation of CP for L x˙ s Ž P XO q POX . y ax
™ ` becomes; Ž 9.
K. Tainaka et al.r Physics Letters A 272 (2000) 416–420
419
where Pi j is the probability density finding a state i at a site and a state j at a nearest neighbor of the former site Ž i, j s X,O.. Note the difference from the conditional probability; the relations Pi j s Pji ,
Ý Pi j s Pi
Ž 10 .
j
Unfortunately, Ž9. cannot be solved; nevertheless, it is known that the phase transition occurs at a s a c Ž a c ; 0.606 for d s 1.. We carry out similar perturbation experiment for CP: the value of annihilation rate a is jumped from a1 to a 2 Ž a1 - a c and a 2 ) a c .. In Fig. 3, the time dependencies of both average density AŽ t . and the variance V Ž t . are plotted, where we put a1 s 0.3 and a 2 s 0.7. It is found from Fig. 3 that the fluctuation enhancement takes place. Even though the average density decreases, the variance V Ž t . increases at intermediate stage of phase transition. Note that such a phenomenon is not always observed for all experiments. Fig. 4 illustrates the time dependencies of both AŽ t . and V Ž t . for the cases of a2 s 2.0 and 5.0. A typical process of Fig. 4Ža. is illustrated in Fig. 1. As the value of a 2 increases, the enhancement of fluctuation disappears. The fluctuation enhancement is not caused by the critical slowing-down, since it is not observed in the mean-field limit. We associate the enhancement with the cluster formation of particles: due to the clump-
Fig. 4. The same as Fig. 3, but the annihilation rate a is jumped from 0.3 to a2 ; Ža. a 2 s 2 and Žb. a 2 s 5.
ing behavior, the variation may become rich. This cause is consistent with the results of Figs. 1 and 4. To explore the degree of clumping, we obtain the ratio R XX ' P XX rx 2
Ž 11 .
in stationary state w16x. When the distribution of particles is just random, we have R XX s 1. When R XX ) 1 Ž R XX - 1., they distribute contagiously Žuniformly.. The basic Eq. Ž9. in stationary state becomes P XO s a rr2. On the other hand, Ž10. leads to PXX s r y PX O . From Eq. Ž11., we have R XX A ry1 . It is well known for CP that near the critical point Ž a a c ., the steady-state density r satisfies r A Ž ac y a. b, where b is a positive constant w3–5x. It follows that
™
Fig. 3. Perturbation experiment for CP. The value of a is jumped from 0.3 to 0.7 Ž N s100 and Ls10 4 .. Both average AŽ t . and variance V Ž t . are depicted against time, where the value of V Ž t . is increased by a factor of 70.
R XX A Ž a c y a .
yb
.
Ž 12 .
K. Tainaka et al.r Physics Letters A 272 (2000) 416–420
420
™
When a approaches a c , the degree of clumping of X becomes rapidly high Ž R XX `.. The enhancement of V Ž t . may be caused by the clumping behavior Ž12. of particles. If a2 is not so large, the system more or less stays in stationary state; the variance thus increases as density decreases.
5. Conclusions Perturbation experiments of phase transition are studied on a one-dimensional lattice. The annihilation rate a of particle is suddenly increased from a1 to a2 , where a1 - a c and a 2 ) a c . The density x of particle Žorder parameter. is thus changed from a positive value to zero. Simulation is carried out by two different methods: CP and its mean-field version. In the latter case, the enhancement of variation Žfluctuation. does not occur. On the other hand, when the method of CP is applied, and when a2 takes a value near the critical point a c , then the fluctuation enhancement in dynamic processes takes place ŽFig. 3.. So far, the enhancement of fluctuation has been observed in some physical systems w12–15x. However, our system has distinct properties never seen in the previous works: i. The fluctuation enhancement is not always originated in the critical slowing-down: it is not observed in the mean-field limit ŽFig. 2.. ii. Previously, the order parameter was increased from zero to a positive value, whereas the present paper reports the just opposite case: the order parameter Ždensity. is decreased. If the order parameter would increase, the variance V Ž t . often increased; examples are the exponential growth and logistic equation. iii. We deal with the one-dimensional system Ž d s 1.. In the case of contact process ŽCP., the phase transition can occur even in d s 1.
Our results suggest that for the fluctuation enhancement, not only the critical slowing-down but also the clumping behavior of particles are essential. The latter property has an important meaning in ecology, since almost all species spatially form a clumped distribution w17x. Especially, when a species becomes endangered, the degree of clumping usually increases. This is due to the inherent nature of biological species: a offspring is produced in the neighborhood of parents. In the system Ž1., we regard X as a biospecies, and a as the death rate. Even if the density x approaches zero, the conditional probability P XX rx never vanishes Ž12.. The fluctuation enhancement ecologically means that there are a variety of processes to extinction of a species. It is not easy to predict the extinction process of species.
References w1x T.E. Harris, Ann. Prob. 2 Ž1974. 969. w2x T.M. Liggett, Interacting Particle Systems, Springer, Berlin, 1985. w3x F. Schlogl, Z. Physik 253 Ž1972. 147. w4x N. Konno, Phase Transition of Interacting Particle Systems, World Scientific, Singapore, 1994. w5x J. Marro, R. Dickman, Nonequilibrium Phase Transition in Lattice Models, Cambridge University Press, Cambridge, 1999. w6x K. Tainaka, J. Phys. Soc. Jpn. 57 Ž1988. 2588. w7x K. Tainaka, Phys. Rev. Lett. 63 Ž1989. 2688. w8x H. Matsuda, N. Ogita, A. Sasaki, K. Sato, Prog. Theor. Phys. 88 Ž1992. 1035. w9x J.E. Satulovsky, T. Tome, Phys. Rev. E 49 Ž1994. 5073. w10x K. Sato, H. Matsuda, A. Sasaki, J. Math. Biol. 32 Ž1994. 251. w11x K. Tainaka, N. Araki, J. Theor. Biol. 197 Ž1999. 1. w12x R. Kubo, K. Matsuo, K. Kitahara, J. Stat. Phys. 9 Ž1973. 51. w13x M. Suzuki, J. Stat. Phys. 16 Ž1977. 11. w14x M. Suzuki, Adv. Chem. Phys. 46 Ž1981. 195. w15x Y. Tsuchiya, S. Horie, J. Phys. Soc. Jpn. 54 Ž1985. 1. w16x K. Tainaka, J. Theor. Biol. 164 Ž1994. 91. w17x P. Stiling, Ecology, Prentice Hall International, London, 1999.
Physics Letters A 272 Ž2000. 416–420 www.elsevier.nlrlocaterpla
Dynamic process and variation in the contact process Kei-ichi Tainaka) , Mineo Hoshiyama, Yasuhiro Takeuchi Department of Systems Engineering, Shizuoka UniÕersity, Hamamatsu 432-8561, Japan Received 2 February 2000; received in revised form 6 June 2000; accepted 9 June 2000 Communicated by C.R. Doering
Abstract Dynamic process in one-dimensional contact process ŽCP. is studied by computer simulation. We carry out the experiment of phase transition by two different methods: CP and its mean-field version. It is known that for both methods, stationary state exhibits the critical slowing-down; relaxation time diverges near the phase boundary. In the present article, we find only for CP that the enhancement of fluctuation occurs in the transient state of dynamical process. q 2000 Elsevier Science B.V. All rights reserved.
X
1. Introduction The contact process ŽCP. has been extensively studied from mathematical w1,2x and physical w3–5x aspects. Considerable data on CP have been accumulated, but they are mainly related to stationary state. Recently, extended models of CP have a growing attention in the field of ecology w6–11x; we need the investigation of population dynamics of biospecies with short-range interaction. In the present article, we explore the dynamical process of CP.
2. Perturbation experiment The rule of CP can be defined as follows: XqO
)
™ 2X,
Ž 1a .
Corresponding author. Fax: q81-53-478-1228. E-mail address: [email protected] ŽK. Tainaka..
™ O, a
Ž 1b .
where X denotes a particle Žindividual of biospecies. and O is the vacant site. The reactions Ž1a. and Ž1b. mean reproduction and annihilation processes of X, respectively. The parameter a represents the annihilation rate of X. We carry out perturbation experiments of phase transition. It is known that CP exhibits a nonequilibrium phase transition at a critical point a s ac : In the case of a G a c , particles cannot exist in stationary state Žextinct phase., while for a - a c , they can exist Žexisting phase.. The value of a c depends on the spatial dimension d; a c ; 0.606 for d s 1, a c ; 1.213 for d s 2. The perturbation experiment is performed as follows: Before the perturbation Ž t - 0., our system stays in a stationary state at a s a1 Žexisting phase: a1 - a c .. At t s 0, the annihilation rate a of particle is suddenly increased from a1 to a 2 Ž a 2 ) a c .. After this perturbation, the system eventually reaches the extinct phase.
0375-9601r00r$ - see front matter q 2000 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 5 - 9 6 0 1 Ž 0 0 . 0 0 4 2 1 - 7
K. Tainaka et al.r Physics Letters A 272 (2000) 416–420
417
Simulation is carried out by two different methods; namely, CP and its mean-field version. The method for CP is defined as follows:
1.
Initially, we distribute particles on a one-dimensional lattice; each lattice site is either empty ŽO. or occupied by a particle ŽX.. 2. The reactions in Ž1. are performed in the following two steps: Ži. First, we perform two-body reaction Ž1a.: Choose one lattice site randomly, and then specify one of the nearest-neighbor sites. If these sites are X and O, then O is changed into X. Žii. Next, we perform a single particle reaction Ž1b.. Choose one lattice point randomly; if the site is occupied by X, the site will become O by a probability Žrate. a. 3. Repeat step 2. by L times, where L is the total number of lattice points. This step is called as Monte Carlo step w6,7x. We assume that the value of L is sufficiently large but finite. 4. Repeat step 3. until the system reaches a stationary state. Here we employ periodic boundary conditions.
Fig. 1. Snapshot of typical dynamics of phase transition of CP. The value of annihilation rate a is jumped from 0.3 to 2.0 at time t s 0. Each column represents a state of system; the colors black and white denote X and O, respectively. The column at t s 0 denotes the initial pattern.
calculate the ensemble average AŽ t . and the variance V Ž t . which are defined by AŽ t . s VŽ t. s
1 N
Ý xi Ž t . ,
Ž 2.
i
1
Ý N
2
x i Ž t . y AŽ t . .
Ž 3.
i
Next, we describe the method of mean-field version of CP. Almost all procedures of CP are not changed in the mean-field limit, but the step Ži. of 2. for CP is merely changed as follows: Ži.’ Two lattice sites are randomly and independently chosen: if these sites are X and O, then O is changed into X. In Fig. 1, a typical example of perturbation experiment for CP is illustrated, where black and white denote the lattice sites of X and O, respectively. Each column represents a state of system; the column at t s 0 denotes an initial state. We prepare N kinds of initial patterns Ž ensembles. which are in stationary state at a s a1; each of them has the density x i Ž0. Ž i s 1,2 . . . N .. The value of x i Ž0. is almost equivalent to the steady-state density. We obtain the dynamics x i Ž t . for t ) 0 in order to
In both methods of CP and its mean-field version, the stationary state near the critical point a c exhibits the so-called ‘critical slowing-down’ Ždivergence of the relaxation time.. Since the variance in stationary state increases with the decrease of steady-state density, it is expected that fluctuation enhancement w12–15x Žextreme increase of V Ž t .. occurs in the transient state of dynamic process.
3. Mean-field limit
L
limit, the dynamic equation for ™In`mean-field becomes;
x˙ s Rx Ž 1 y xrr . ,
Ž 4.
K. Tainaka et al.r Physics Letters A 272 (2000) 416–420
418
where the dot represents the derivative with respect to the time t which is measured by the Monte Carlo step, and the quantities R and r are defined by R s 2 y a,
r s Ž 2 y a . r2.
Ž 5.
The steady-state solution of Eq. Ž4. Žlogistic equation. can be obtained by setting the time derivative in Ž4. to be zero. It is well known that non-trivial solution Ž x s r . for a - 2 is stable, and that phase transition occurs at a c s 2: for a G a c Žor a - a c ., particles cannot Žor can. exist. Moreover, the logistic equation explains the critical slowing-down near a s a c . When the difference x Ž t . y r is much smaller than unity, we have from Ž4. that x Ž t . y r A exp Ž yN ac y a N t . .
Ž 6.
™
Hence, we see that the relaxation time 1rN a c y a N diverges in the limit a a c . If a s a c , we obtain power-law decay. Because of critical slowing-down, the variance in stationary state increases with the decrease of r . We carry out perturbation experiments of phase transition in mean-field method. It is found from computer simulation that the fluctuation enhancement never takes place for various values of a1 and a 2 ; especially when a 2 takes a value much larger than a c , the value of V Ž t . rapidly decreases with time. In Fig. 2, an example of perturbation experiment is displayed, where the value of a is jumped from a1 s 1 to a 2 s 2.1; in this case, a 2 takes a
Fig. 2. The result of perturbation experiment in the mean-field limit. The time dependencies of both average AŽ t . and variance V Ž t . defined by Ž4. and Ž5. are displayed, where the value of V Ž t . is multiplied by 200. At time t s 0, the annihilation rate a is jumped from 1.0 to 2.1. We repeat the similar experiment 100 times Ž N s100. with different initial patterns. The system has 10 4 sites Ž Ls10 4 ..
value near the critical point a c s 2. Both values of average density AŽ t . and the variance V Ž t . are obtained by performing the similar experiment 100 times Ž N s 100. with different initial patterns. In the below, we give an explanation for the fact that fluctuation enhancement never occurs in the mean-field limit. From Ž3., we have VŽ t. s
1 N
2
Ý xi Ž t . y i
1 s
N2
Ý
1 N
2
ž
Ý xi Ž t . i
/
2
xi Ž t . yx j Ž t . .
i)j
It follows that V˙ s
2 N2
Ý Ž x i y x j . Ž x˙ i y x˙ j . .
Ž 7.
i)j
The dynamics for t ) 0 is assumed to be determined by the logistic eqquation Ž4., where both parameters R and r are negative Ž a2 ) 2.. Let x i Ž t . and x j Ž t . be two solutions of Ž4. whose initial values are slightly different from each other. Here we put x i Ž0. ) x j Ž0.. From Ž4., we have x˙ i y x˙ j s R Ž x i y x j . 1 y Ž x i q x j . rr ,
Ž 8.
Note that x i Ž t . / x j Ž t . for t ) 0 because of the uniqueness of the solution to Ž4. and x i Ž t . ) x j Ž t . for all t ) 0. Hence, the right-hand side of Ž8. is always negative, and the difference x i Ž t . y x j Ž t . decreases as the time proceeds. It is therefore proved from Ž7. that the variance V Ž t . always decreases. Strictly speaking, this proof is insufficient, since the dynamics for a finite value of L is described by some stochastic differential equation which contains both deterministic Žlogistic equation. and fluctuating parts. Nevertheless, the logistic equation Ž4. well explains that i. critical slowing-down occurs, and ii. enhancement of fluctuation does not occur.
4. Results of CP The basic equation of CP for L x˙ s Ž P XO q POX . y ax
™ ` becomes; Ž 9.
K. Tainaka et al.r Physics Letters A 272 (2000) 416–420
419
where Pi j is the probability density finding a state i at a site and a state j at a nearest neighbor of the former site Ž i, j s X,O.. Note the difference from the conditional probability; the relations Pi j s Pji ,
Ý Pi j s Pi
Ž 10 .
j
Unfortunately, Ž9. cannot be solved; nevertheless, it is known that the phase transition occurs at a s a c Ž a c ; 0.606 for d s 1.. We carry out similar perturbation experiment for CP: the value of annihilation rate a is jumped from a1 to a 2 Ž a1 - a c and a 2 ) a c .. In Fig. 3, the time dependencies of both average density AŽ t . and the variance V Ž t . are plotted, where we put a1 s 0.3 and a 2 s 0.7. It is found from Fig. 3 that the fluctuation enhancement takes place. Even though the average density decreases, the variance V Ž t . increases at intermediate stage of phase transition. Note that such a phenomenon is not always observed for all experiments. Fig. 4 illustrates the time dependencies of both AŽ t . and V Ž t . for the cases of a2 s 2.0 and 5.0. A typical process of Fig. 4Ža. is illustrated in Fig. 1. As the value of a 2 increases, the enhancement of fluctuation disappears. The fluctuation enhancement is not caused by the critical slowing-down, since it is not observed in the mean-field limit. We associate the enhancement with the cluster formation of particles: due to the clump-
Fig. 4. The same as Fig. 3, but the annihilation rate a is jumped from 0.3 to a2 ; Ža. a 2 s 2 and Žb. a 2 s 5.
ing behavior, the variation may become rich. This cause is consistent with the results of Figs. 1 and 4. To explore the degree of clumping, we obtain the ratio R XX ' P XX rx 2
Ž 11 .
in stationary state w16x. When the distribution of particles is just random, we have R XX s 1. When R XX ) 1 Ž R XX - 1., they distribute contagiously Žuniformly.. The basic Eq. Ž9. in stationary state becomes P XO s a rr2. On the other hand, Ž10. leads to PXX s r y PX O . From Eq. Ž11., we have R XX A ry1 . It is well known for CP that near the critical point Ž a a c ., the steady-state density r satisfies r A Ž ac y a. b, where b is a positive constant w3–5x. It follows that
™
Fig. 3. Perturbation experiment for CP. The value of a is jumped from 0.3 to 0.7 Ž N s100 and Ls10 4 .. Both average AŽ t . and variance V Ž t . are depicted against time, where the value of V Ž t . is increased by a factor of 70.
R XX A Ž a c y a .
yb
.
Ž 12 .
K. Tainaka et al.r Physics Letters A 272 (2000) 416–420
420
™
When a approaches a c , the degree of clumping of X becomes rapidly high Ž R XX `.. The enhancement of V Ž t . may be caused by the clumping behavior Ž12. of particles. If a2 is not so large, the system more or less stays in stationary state; the variance thus increases as density decreases.
5. Conclusions Perturbation experiments of phase transition are studied on a one-dimensional lattice. The annihilation rate a of particle is suddenly increased from a1 to a2 , where a1 - a c and a 2 ) a c . The density x of particle Žorder parameter. is thus changed from a positive value to zero. Simulation is carried out by two different methods: CP and its mean-field version. In the latter case, the enhancement of variation Žfluctuation. does not occur. On the other hand, when the method of CP is applied, and when a2 takes a value near the critical point a c , then the fluctuation enhancement in dynamic processes takes place ŽFig. 3.. So far, the enhancement of fluctuation has been observed in some physical systems w12–15x. However, our system has distinct properties never seen in the previous works: i. The fluctuation enhancement is not always originated in the critical slowing-down: it is not observed in the mean-field limit ŽFig. 2.. ii. Previously, the order parameter was increased from zero to a positive value, whereas the present paper reports the just opposite case: the order parameter Ždensity. is decreased. If the order parameter would increase, the variance V Ž t . often increased; examples are the exponential growth and logistic equation. iii. We deal with the one-dimensional system Ž d s 1.. In the case of contact process ŽCP., the phase transition can occur even in d s 1.
Our results suggest that for the fluctuation enhancement, not only the critical slowing-down but also the clumping behavior of particles are essential. The latter property has an important meaning in ecology, since almost all species spatially form a clumped distribution w17x. Especially, when a species becomes endangered, the degree of clumping usually increases. This is due to the inherent nature of biological species: a offspring is produced in the neighborhood of parents. In the system Ž1., we regard X as a biospecies, and a as the death rate. Even if the density x approaches zero, the conditional probability P XX rx never vanishes Ž12.. The fluctuation enhancement ecologically means that there are a variety of processes to extinction of a species. It is not easy to predict the extinction process of species.
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