- Email: [email protected]
Slow dynamics in self-organizing systems
Physica A 314 (2002) 567 – 574
www.elsevier.com/locate/physa
Slow dynamics in self-organizing systems J'anos Kert'esza;∗ , J'anos T+or+oka , Supriya Krishnamurthyb , St'ephane Rouxc a Department
of Theoretical Physics, Institute of Physics, Budapest University of Technology, 8 Budafoki u t, H-1111 Budapest, Hungary b Santa Fe Institute, 1399 Hyde Park Road, Santa Fe, NM 87501, USA c Surface du Verre et Interfaces, UMR CNRS=Saint-Gobain, 39 Quai Lucien Lefranc, 93303 Aubervilliers Cedex, France
Abstract Based on a recent model of shearing granular media we propose a new mechanism for slow dynamics. It consists of two main steps: A global optimization and a restructuring. We present numerical results of the model and its analytical treatment on the hierarchical lattice. A simpli5ed local version can be mapped to the true self-avoiding random walk and can therefore be solved c 2002 Elsevier Science B.V. All rights reserved. exactly. PACS: 05.65.+k; 81.05.Rm Keywords: Self-organization; Granular systems
1. Introduction Slow dynamics with no separation of time scales represent a major challenge of statistical physics. Experimental or simulation approaches are extremely di=cult, so in most cases new ideas and models are needed for the understanding of this kind of problems. There can be di>erent roots of slow dynamics: Systems close to the critical point slow down enormously due to the increasing characteristic time. Phase separation is often accompanied by a slow coarsening process. In glasses the free energy landscape is so complicated and structured that the system never 5nds the global minimum and shows a history-dependent behaviour called ageing [1]. Slow dynamics may also occur ∗
Corresponding author. Tel.: +36-1-463-3568; fax: +36-1-463-3567. E-mail address: [email protected] (J. Kert'esz).
c 2002 Elsevier Science B.V. All rights reserved. 0378-4371/02/$ - see front matter PII: S 0 3 7 8 - 4 3 7 1 ( 0 2 ) 0 1 0 5 1 - 8
568
J. Kert esz et al. / Physica A 314 (2002) 567 – 574
in intrinsically dynamic, driven systems leading to scale-free fractal structures. The name of self-organized criticality covers a whole family of related models [2]. Motivated by our study of shearing granular materials [3], we report in this paper about a new mechanism leading to slow dynamics. 2. Global optimization and restructuring Let us introduce the following model: We de5ne a potential %(x; y) placed on the bonds of a two-dimensional square lattice. Initially, this potential 5eld is assigned randomly from the interval [%0 ; 1] with uniform distribution. In every timestep we look for the minimal path P∗ that spawns the system in the x direction. The minimal path is a directed path along which the sum of the potential is minimal among all possible paths. Formally, if the weight of a path is denoted by S: S(P) = %(x; y) (1) (x;y)∈P
the minimal path is P∗ for which S(P∗ ) = min. Once the minimal path is found the potential of the bonds belonging to this path is modi5ed randomly from the range of [0 : 1]. Let us note that the initial distribution of the potential can be di>erent from the refreshing one if %0 ¿ 0. The selection of the minimal path is the same process as 5nding the ground state of the directed polymer in a random 5eld [5]. The concept of 5nding the minimum and refreshing it makes an allusion to the Bak–Sneppen model of evolution [6]; however, here we do not update any neighbours which has a crucial di>erence [7]. The model was originally motivated by the large strain simple shear of granular materials. If this is done in an annular cell where the material is sheared by the opposite motion of the bottom and top cover, due to the periodicity in the annular direction we may assume that the whole system is independent of it, so we average along it. The potential refers to a parameter, like density and=or strength on a mesoscopic scale and the minimal path plays the role of the shear band where all the relative movement of the particles and thus the restructuring of the material takes place. 3. Numerical results The dynamics consists of choosing the minimal path and replacing it with random numbers. It is natural to de5ne the quantity that we call average potential and denote by % which is the mean value of the potential outside the minimal path. The value of this parameter is a monotonously increasing quantity approaching the asymptotic value of unity. So, it is a good candidate to measure the distance from the steady state. The value of the average potential at t = 0 determined by the initial parameter: %(t = 0) = (1 − %0 )=2. In Fig. 1 we show the time dependence of the distance of the average value of the potential from its asymptotic value with %0 = 0. One can see that the time dependence of this decay is slower than any power law, we veri5ed it but no logarithmic 5t
J. Kert esz et al. / Physica A 314 (2002) 567 – 574
569
1
1 − 〈
〈 0.1
0.01
10-3
10 -1
10 1 t/L
10 3
10 5
Fig. 1. The mean value of the potential subtracted from its asymptotic value for square systems of size L = 32; 64; 128; 256; 512 with initial condition %0 = 0. 1
d/L
10 10 10 10
(a)
-1
10
-1
-2
P (d)
10
1
-3
10 -3
-4
-5
10 -3
10 -2
10 -1
10 1
t/L
10
3
10
10 5
(b)
-4
0
10
20
30
40
50
60
d
Fig. 2. (a) The average Hamming distance versus time. The same system sizes were scaled together as on Fig. 1. In larger systems the decrease is steeper indicating relatively faster localization. (b) The distribution of the Hamming distance (d) at times ( ) t = 0:16L, (+) t = 1:5L and (♦) t = 20L. Let us note the complete reversal of the distribution function.
was satisfactory either. After about ∼ 10t=L timesteps we observe a non-trivial size dependence that cannot be scaled together. The Hamming distance (d) is de5ned as the number of sites two successive minimal paths share in common. In Fig. 2(a) the average value of the Hamming distance is plotted. For early times it takes the maximal value (d = L) then it decreases gradually. The t=L scaling works much better than in the case of the average potential. The distribution of d (Fig. 2(b)) shows a complete reversal. At the beginning it is peaked around the maximal value d=L, while for late times it becomes peaked at the minimum value d = 0. The early time behaviour where d is large can be interpreted as a repulsive regime where the minimal path have the tendency to avoid its former position. This behaviour
570
J. Kert esz et al. / Physica A 314 (2002) 567 – 574
is continuously altered into an attractive one, a localization, where large parts of the minimal path remains unchanged. This reversal of the behaviour is followed by the breakdown of the t=L scaling of the average potential. It is easy to understand the main reason behind these behaviours. The 5rst minimal path has an average potential of %SB 0:22, known from the directed polymer analogy [5]. It is much lower than the refreshing potential which has the mean of 0.5. So, it has very low probability that the next path will share bonds with the previous one. This reasoning holds until the potential is increased so much that there is no path with mean potential less than 0.5. At this point the last minimal path becomes the almost unique candidate for the next minimal path with sharing only a few bonds with the rest of the system. This con5nes the activity to a small portion of the lattice. With the activity the increase of the potential is also limited to this small domain of the system which in turn intensi5es the localization by making less likely the change in the position of the minimal path. The escape from this trap can only be done by an unprobable full jump to other part of the system (out of the much visited domain) where the same procedure is repeated. Thus, the system is cut into subsystems in a self-organized way. However, the time spent in these subsystems is far from being equally distributed which gives rise to the breakdown of the ergodicity which leads to the breakdown of the t=L scaling on the average potential plots. 4. Aging: dependence on preparation After the 5rst step the average potential always increases. This feature is obvious from the de5nition of the model. What is more striking is that if the initial potential is small enough %0 ¡ 12 the asymptotic behaviour of the mean potential % is always the same, independently from the initial condition and without any time rescaling. On the other hand, if %0 ¿ 12 for all values of %0 a di>erent evolution is observed as can be seen in Fig. 3. The reason behind it is that at early times when the average potential is not too high the minimal path is mobile enough to visit all bonds with potential less than 0.5. This was veri5ed on the potential distribution functions. The high-density regime is di>erent because the probability of the big jumps, i.e., the possibility of escaping from the potential traps is basically eliminated. Thus, the whole system is further slowed down by the very limited di>using capability of the path in high-density regimes. The behaviour that the fate of the system strongly depends on the initial preparation is called aging in glassy systems; however, the mapping to this non-equilibrium model is not entirely straightforward. There is no such quantity here as temperature, the only control that we have of the system is the initial preparation. Let us note that this feature is also known to be important in granular materials [8]. 5. Analytic solution of the model There exist an analytic asymptotic solution of the above-de5ned model on a special hierarchical diamond lattice [4]: The hierarchical lattice is generated iteratively from a bond. At each level, every bond is replaced by a diamond as shown in Fig. 4.
J. Kert esz et al. / Physica A 314 (2002) 567 – 574
571
0.4 0.3
0.2
1 − 〈
〈 0.1
0.06
1
102
10
10 3
10 4
t/L Fig. 3. Time dependence of the di>erence of the average potential from its asymptotic value for starting potentials with initial potential ranges [%0 : 1] from top to bottom, respectively: %0 = 0:2; 0:3; 0:4; 0:45; 0:5; 0:55; 0:6; 0:7; 0:8. The system size is 64.
A
B
A
(a)
B (b)
A
B (c)
Fig. 4. The 0th (a), 1st (b) and 2nd (c) generations of the hierarchical diamond lattice.
The numerical results of our model on this lattice show a striking similarity to those on the square lattice (see Fig. 5(a) and (b), respectively, Figs. 1 and 2). Indeed comparison of the quantities like average potential %, Hamming distance d and potential distribution and the average potential of the actual minimal path shows a very good quantitative matching of the two lattices. The big advantage of such a geometry that the minimal path at each level determined separately. The level n system consists of two couplets coupled in parallel. The couplets are composed of two n − 1 level subsystems in series where the minimal path is simply the prolongation of the one to the other. For parallel coupling of these couplets one has to choose the smaller out of the two minimal paths. The above reasoning can be put into recursive equations that can be solved exactly in the asymptotic limit. The result is consistent with the 5ndings on the square lattice. The average potential behaves in a non-trivial way: A level n system has an asymptotic behaviour of n
%n (t → ∞) ∼ 1 − t −2 :
(2)
572
J. Kert esz et al. / Physica A 314 (2002) 567 – 574 1
1
10
d/ 2 N
1 − 〈
〈 0.1
10 10 10
0.01 10 -2
1
(a)
10 2
t/N
10 4
10
10 6
(b)
-1
-2
-3
-4
-5
10 -3
10 -1
10 1
t/ 2
10 3
10 5
N
Fig. 5. (a) The time evolution of the di>erence of the average potential from its asymptotic value 1 on the hierarchical lattice for generation N = 2 to 7 from bottom to top, respectively. (b) The Hamming distance versus time plot on the hierarchical lattice for the same system sizes.
In the thermodynamic limit the asymptotic behaviour is found to be slower than any power law. Let us mention here that both in the square and in the hierarchical lattice case the reason for the non-trivial dependence is the strong localization in a subsystem and the coupling of the subsystems coupled in parallel. The subsystems are clearly de5ned in the hierarchical case from its de5nition. However, on the regular square lattice the subsystems are created in a self-organized way by the dynamics itself. The similarity of the results on these two lattices is due to this similar nature di>erences arise from the inhomogeneity of the square lattice case. This can be observed on the nicer data collapse in Fig. 5(b) as compared to Fig. 2(a) [4]. 6. Self-quenching walk The above model inspired another that we call self-quenching walk (SQW) [10]. It shows non-trivial slow behaviour while capturing the essential features of the above presented model. The model is de5ned as follows: A walker moves in a d-dimensional random landscape %(r) towards the local maximum at the next-nearest neighborhood, or stays if she is at the local maximum. Let us denote this maximum by . After each move the walker changes the potential randomly at its position. Here, we always take the initial and the refreshing random distribution to be a uniform distribution between 0 and 1. The model belongs to the class of active walkers [9]. It is natural to expect that as before the walker will spend more and more time at a given place. So the natural variables of the model should be rede5ned: As the walker spends most of the steps by doing nothing so instead of the time (t) it is worth counting by the number of moves (n) the walker makes. We can estimate the time spent waiting by a Poisson process with a characteristic time of 1=. The potential
J. Kert esz et al. / Physica A 314 (2002) 567 – 574
573
% soon gets very low everywhere. It is also better from numerical point of view to calculate with the logarithm of the potential: V = −log(%) which should be taken from the distribution exp[ − (V − V ())]. In these variables the model is governed by the same equations as the “True” Self-Avoiding Walk [11]. This correspondence reveals that the average distance of the walker at time t from its starting point is R(t) ∼ log(t) , with = 23 in one dimension and 12 for higher dimensions. The dynamics in all cases is logarithmically slow. Furthermore, the potential landscape V (r) gets rough with the width scaling, the usual way w(n) ˙ L %(n=Lz ):
(3)
However, the scaling exponents and does not follow the Family–Vicsek scaling, but instead given by the formula = ( + 12 )=(1 + ) typical for extremal growth models though the SQW does neither contain a global extremum criterion nor a long-range-order interaction. 7. Conclusion In this paper we presented a model that shows slow dynamics due to global optimization and restructuring. The time evolution of the model is characterized by an unusual system size dependence that can be understood from the analytic solution on the hierarchical lattice. The strange behaviour of the system is due to the localization of the minimal path into self-organized subsystems. This localization is realized though a complete reversal of the behaviour of the minimal path from strong repulsion to attractions. A simpli5ed active walker model is also presented. It captures the essence of the previous model featuring slow dynamics. This model is found to be part of the class of extremal models in spite that it does not contain any global criterion. Acknowledgements Support by OTKA T029985 and T035028 is acknowledged. References [1] M. Mezard, G. Parisi, M.A. Virasoro, Spin Glass Theory and Beyond, World Scienti5c, Singapore, 1987. [2] P. Bak, How Nature Works: The Science of Self-Organized Criticality, Copernicus, New York, 1996. [3] J. T+or+ok, S. Krishnamurthy, J. Kert'esz, S. Roux, Phys. Rev. Lett. 84 (2000) 3851. [4] J. T+or+ok, S. Krishnamurthy, J. Kert'esz, S. Roux, preprint. [5] T. Halpin-Healy, Y.-C. Zhang, Phys. Rep. 254 (1995) 215. [6] P. Bak, K. Sneppen, Phys. Rev. Lett. 71 (1993) 4083. [7] J. T+or+ok, J. Kert'esz, in preparation.
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J. Kert esz et al. / Physica A 314 (2002) 567 – 574
[8] L. Vanel, D. Howell, D. Clark, R.P. Behringer, E. Cl'ement, Phys. Rev. E 60 (1999) R5040; C.-H. Liu, S.R. Nagel, Phys. Rev. B 48 (1993) 15 646. [9] V.B. Priezzhev, D. Dhar, A. Dhar, S. Krishnamurthy, Phys. Rev. Lett. 77 (1996) 5079; D. Helbing, F. Schweitzer, J. Keltsch, P. Moln'ar, Phys. Rev. E 56 (1997) 2527 and references therein. [10] J. T+or+ok, S. Krishnamurthy, J. Kert'esz, S. Roux, Eur. Phys. J. B 18 (2000) 697. [11] B. T'oth, J. Stat. Phys. 77 (1994) 17; B. T'oth, Ann. Probab. 23 (1995) 1523.
www.elsevier.com/locate/physa
Slow dynamics in self-organizing systems J'anos Kert'esza;∗ , J'anos T+or+oka , Supriya Krishnamurthyb , St'ephane Rouxc a Department
of Theoretical Physics, Institute of Physics, Budapest University of Technology, 8 Budafoki u t, H-1111 Budapest, Hungary b Santa Fe Institute, 1399 Hyde Park Road, Santa Fe, NM 87501, USA c Surface du Verre et Interfaces, UMR CNRS=Saint-Gobain, 39 Quai Lucien Lefranc, 93303 Aubervilliers Cedex, France
Abstract Based on a recent model of shearing granular media we propose a new mechanism for slow dynamics. It consists of two main steps: A global optimization and a restructuring. We present numerical results of the model and its analytical treatment on the hierarchical lattice. A simpli5ed local version can be mapped to the true self-avoiding random walk and can therefore be solved c 2002 Elsevier Science B.V. All rights reserved. exactly. PACS: 05.65.+k; 81.05.Rm Keywords: Self-organization; Granular systems
1. Introduction Slow dynamics with no separation of time scales represent a major challenge of statistical physics. Experimental or simulation approaches are extremely di=cult, so in most cases new ideas and models are needed for the understanding of this kind of problems. There can be di>erent roots of slow dynamics: Systems close to the critical point slow down enormously due to the increasing characteristic time. Phase separation is often accompanied by a slow coarsening process. In glasses the free energy landscape is so complicated and structured that the system never 5nds the global minimum and shows a history-dependent behaviour called ageing [1]. Slow dynamics may also occur ∗
Corresponding author. Tel.: +36-1-463-3568; fax: +36-1-463-3567. E-mail address: [email protected] (J. Kert'esz).
c 2002 Elsevier Science B.V. All rights reserved. 0378-4371/02/$ - see front matter PII: S 0 3 7 8 - 4 3 7 1 ( 0 2 ) 0 1 0 5 1 - 8
568
J. Kert esz et al. / Physica A 314 (2002) 567 – 574
in intrinsically dynamic, driven systems leading to scale-free fractal structures. The name of self-organized criticality covers a whole family of related models [2]. Motivated by our study of shearing granular materials [3], we report in this paper about a new mechanism leading to slow dynamics. 2. Global optimization and restructuring Let us introduce the following model: We de5ne a potential %(x; y) placed on the bonds of a two-dimensional square lattice. Initially, this potential 5eld is assigned randomly from the interval [%0 ; 1] with uniform distribution. In every timestep we look for the minimal path P∗ that spawns the system in the x direction. The minimal path is a directed path along which the sum of the potential is minimal among all possible paths. Formally, if the weight of a path is denoted by S: S(P) = %(x; y) (1) (x;y)∈P
the minimal path is P∗ for which S(P∗ ) = min. Once the minimal path is found the potential of the bonds belonging to this path is modi5ed randomly from the range of [0 : 1]. Let us note that the initial distribution of the potential can be di>erent from the refreshing one if %0 ¿ 0. The selection of the minimal path is the same process as 5nding the ground state of the directed polymer in a random 5eld [5]. The concept of 5nding the minimum and refreshing it makes an allusion to the Bak–Sneppen model of evolution [6]; however, here we do not update any neighbours which has a crucial di>erence [7]. The model was originally motivated by the large strain simple shear of granular materials. If this is done in an annular cell where the material is sheared by the opposite motion of the bottom and top cover, due to the periodicity in the annular direction we may assume that the whole system is independent of it, so we average along it. The potential refers to a parameter, like density and=or strength on a mesoscopic scale and the minimal path plays the role of the shear band where all the relative movement of the particles and thus the restructuring of the material takes place. 3. Numerical results The dynamics consists of choosing the minimal path and replacing it with random numbers. It is natural to de5ne the quantity that we call average potential and denote by % which is the mean value of the potential outside the minimal path. The value of this parameter is a monotonously increasing quantity approaching the asymptotic value of unity. So, it is a good candidate to measure the distance from the steady state. The value of the average potential at t = 0 determined by the initial parameter: %(t = 0) = (1 − %0 )=2. In Fig. 1 we show the time dependence of the distance of the average value of the potential from its asymptotic value with %0 = 0. One can see that the time dependence of this decay is slower than any power law, we veri5ed it but no logarithmic 5t
J. Kert esz et al. / Physica A 314 (2002) 567 – 574
569
1
1 − 〈
〈 0.1
0.01
10-3
10 -1
10 1 t/L
10 3
10 5
Fig. 1. The mean value of the potential subtracted from its asymptotic value for square systems of size L = 32; 64; 128; 256; 512 with initial condition %0 = 0. 1
d/L
10 10 10 10
(a)
-1
10
-1
-2
P (d)
10
1
-3
10 -3
-4
-5
10 -3
10 -2
10 -1
10 1
t/L
10
3
10
10 5
(b)
-4
0
10
20
30
40
50
60
d
Fig. 2. (a) The average Hamming distance versus time. The same system sizes were scaled together as on Fig. 1. In larger systems the decrease is steeper indicating relatively faster localization. (b) The distribution of the Hamming distance (d) at times ( ) t = 0:16L, (+) t = 1:5L and (♦) t = 20L. Let us note the complete reversal of the distribution function.
was satisfactory either. After about ∼ 10t=L timesteps we observe a non-trivial size dependence that cannot be scaled together. The Hamming distance (d) is de5ned as the number of sites two successive minimal paths share in common. In Fig. 2(a) the average value of the Hamming distance is plotted. For early times it takes the maximal value (d = L) then it decreases gradually. The t=L scaling works much better than in the case of the average potential. The distribution of d (Fig. 2(b)) shows a complete reversal. At the beginning it is peaked around the maximal value d=L, while for late times it becomes peaked at the minimum value d = 0. The early time behaviour where d is large can be interpreted as a repulsive regime where the minimal path have the tendency to avoid its former position. This behaviour
570
J. Kert esz et al. / Physica A 314 (2002) 567 – 574
is continuously altered into an attractive one, a localization, where large parts of the minimal path remains unchanged. This reversal of the behaviour is followed by the breakdown of the t=L scaling of the average potential. It is easy to understand the main reason behind these behaviours. The 5rst minimal path has an average potential of %SB 0:22, known from the directed polymer analogy [5]. It is much lower than the refreshing potential which has the mean of 0.5. So, it has very low probability that the next path will share bonds with the previous one. This reasoning holds until the potential is increased so much that there is no path with mean potential less than 0.5. At this point the last minimal path becomes the almost unique candidate for the next minimal path with sharing only a few bonds with the rest of the system. This con5nes the activity to a small portion of the lattice. With the activity the increase of the potential is also limited to this small domain of the system which in turn intensi5es the localization by making less likely the change in the position of the minimal path. The escape from this trap can only be done by an unprobable full jump to other part of the system (out of the much visited domain) where the same procedure is repeated. Thus, the system is cut into subsystems in a self-organized way. However, the time spent in these subsystems is far from being equally distributed which gives rise to the breakdown of the ergodicity which leads to the breakdown of the t=L scaling on the average potential plots. 4. Aging: dependence on preparation After the 5rst step the average potential always increases. This feature is obvious from the de5nition of the model. What is more striking is that if the initial potential is small enough %0 ¡ 12 the asymptotic behaviour of the mean potential % is always the same, independently from the initial condition and without any time rescaling. On the other hand, if %0 ¿ 12 for all values of %0 a di>erent evolution is observed as can be seen in Fig. 3. The reason behind it is that at early times when the average potential is not too high the minimal path is mobile enough to visit all bonds with potential less than 0.5. This was veri5ed on the potential distribution functions. The high-density regime is di>erent because the probability of the big jumps, i.e., the possibility of escaping from the potential traps is basically eliminated. Thus, the whole system is further slowed down by the very limited di>using capability of the path in high-density regimes. The behaviour that the fate of the system strongly depends on the initial preparation is called aging in glassy systems; however, the mapping to this non-equilibrium model is not entirely straightforward. There is no such quantity here as temperature, the only control that we have of the system is the initial preparation. Let us note that this feature is also known to be important in granular materials [8]. 5. Analytic solution of the model There exist an analytic asymptotic solution of the above-de5ned model on a special hierarchical diamond lattice [4]: The hierarchical lattice is generated iteratively from a bond. At each level, every bond is replaced by a diamond as shown in Fig. 4.
J. Kert esz et al. / Physica A 314 (2002) 567 – 574
571
0.4 0.3
0.2
1 − 〈
〈 0.1
0.06
1
102
10
10 3
10 4
t/L Fig. 3. Time dependence of the di>erence of the average potential from its asymptotic value for starting potentials with initial potential ranges [%0 : 1] from top to bottom, respectively: %0 = 0:2; 0:3; 0:4; 0:45; 0:5; 0:55; 0:6; 0:7; 0:8. The system size is 64.
A
B
A
(a)
B (b)
A
B (c)
Fig. 4. The 0th (a), 1st (b) and 2nd (c) generations of the hierarchical diamond lattice.
The numerical results of our model on this lattice show a striking similarity to those on the square lattice (see Fig. 5(a) and (b), respectively, Figs. 1 and 2). Indeed comparison of the quantities like average potential %, Hamming distance d and potential distribution and the average potential of the actual minimal path shows a very good quantitative matching of the two lattices. The big advantage of such a geometry that the minimal path at each level determined separately. The level n system consists of two couplets coupled in parallel. The couplets are composed of two n − 1 level subsystems in series where the minimal path is simply the prolongation of the one to the other. For parallel coupling of these couplets one has to choose the smaller out of the two minimal paths. The above reasoning can be put into recursive equations that can be solved exactly in the asymptotic limit. The result is consistent with the 5ndings on the square lattice. The average potential behaves in a non-trivial way: A level n system has an asymptotic behaviour of n
%n (t → ∞) ∼ 1 − t −2 :
(2)
572
J. Kert esz et al. / Physica A 314 (2002) 567 – 574 1
1
10
d/ 2 N
1 − 〈
〈 0.1
10 10 10
0.01 10 -2
1
(a)
10 2
t/N
10 4
10
10 6
(b)
-1
-2
-3
-4
-5
10 -3
10 -1
10 1
t/ 2
10 3
10 5
N
Fig. 5. (a) The time evolution of the di>erence of the average potential from its asymptotic value 1 on the hierarchical lattice for generation N = 2 to 7 from bottom to top, respectively. (b) The Hamming distance versus time plot on the hierarchical lattice for the same system sizes.
In the thermodynamic limit the asymptotic behaviour is found to be slower than any power law. Let us mention here that both in the square and in the hierarchical lattice case the reason for the non-trivial dependence is the strong localization in a subsystem and the coupling of the subsystems coupled in parallel. The subsystems are clearly de5ned in the hierarchical case from its de5nition. However, on the regular square lattice the subsystems are created in a self-organized way by the dynamics itself. The similarity of the results on these two lattices is due to this similar nature di>erences arise from the inhomogeneity of the square lattice case. This can be observed on the nicer data collapse in Fig. 5(b) as compared to Fig. 2(a) [4]. 6. Self-quenching walk The above model inspired another that we call self-quenching walk (SQW) [10]. It shows non-trivial slow behaviour while capturing the essential features of the above presented model. The model is de5ned as follows: A walker moves in a d-dimensional random landscape %(r) towards the local maximum at the next-nearest neighborhood, or stays if she is at the local maximum. Let us denote this maximum by . After each move the walker changes the potential randomly at its position. Here, we always take the initial and the refreshing random distribution to be a uniform distribution between 0 and 1. The model belongs to the class of active walkers [9]. It is natural to expect that as before the walker will spend more and more time at a given place. So the natural variables of the model should be rede5ned: As the walker spends most of the steps by doing nothing so instead of the time (t) it is worth counting by the number of moves (n) the walker makes. We can estimate the time spent waiting by a Poisson process with a characteristic time of 1=. The potential
J. Kert esz et al. / Physica A 314 (2002) 567 – 574
573
% soon gets very low everywhere. It is also better from numerical point of view to calculate with the logarithm of the potential: V = −log(%) which should be taken from the distribution exp[ − (V − V ())]. In these variables the model is governed by the same equations as the “True” Self-Avoiding Walk [11]. This correspondence reveals that the average distance of the walker at time t from its starting point is R(t) ∼ log(t) , with = 23 in one dimension and 12 for higher dimensions. The dynamics in all cases is logarithmically slow. Furthermore, the potential landscape V (r) gets rough with the width scaling, the usual way w(n) ˙ L %(n=Lz ):
(3)
However, the scaling exponents and does not follow the Family–Vicsek scaling, but instead given by the formula = ( + 12 )=(1 + ) typical for extremal growth models though the SQW does neither contain a global extremum criterion nor a long-range-order interaction. 7. Conclusion In this paper we presented a model that shows slow dynamics due to global optimization and restructuring. The time evolution of the model is characterized by an unusual system size dependence that can be understood from the analytic solution on the hierarchical lattice. The strange behaviour of the system is due to the localization of the minimal path into self-organized subsystems. This localization is realized though a complete reversal of the behaviour of the minimal path from strong repulsion to attractions. A simpli5ed active walker model is also presented. It captures the essence of the previous model featuring slow dynamics. This model is found to be part of the class of extremal models in spite that it does not contain any global criterion. Acknowledgements Support by OTKA T029985 and T035028 is acknowledged. References [1] M. Mezard, G. Parisi, M.A. Virasoro, Spin Glass Theory and Beyond, World Scienti5c, Singapore, 1987. [2] P. Bak, How Nature Works: The Science of Self-Organized Criticality, Copernicus, New York, 1996. [3] J. T+or+ok, S. Krishnamurthy, J. Kert'esz, S. Roux, Phys. Rev. Lett. 84 (2000) 3851. [4] J. T+or+ok, S. Krishnamurthy, J. Kert'esz, S. Roux, preprint. [5] T. Halpin-Healy, Y.-C. Zhang, Phys. Rep. 254 (1995) 215. [6] P. Bak, K. Sneppen, Phys. Rev. Lett. 71 (1993) 4083. [7] J. T+or+ok, J. Kert'esz, in preparation.
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