Critical dynamic approach to stationary states in complex systems
- 40 Downloads
A dynamic scaling Ansatz for the approach to stationary states in complex systems is proposed and tested by means of extensive simulations applied to both the Bak-Sneppen (BS) model, which exhibits robust Self-Organised Critical (SOC) behaviour, and the Game of Life (GOL) of J. Conway, whose critical behaviour is under debate. Considering the dynamic scaling behaviour of the density of sites (ρ(t)), it is shown that i) by starting the dynamic measurements with configurations such that ρ(t=0) →0, one observes an initial increase of the density with exponents θ= 0.12(2) and θ= 0.11(2) for the BS and GOL models, respectively; ii) by using initial configurations with ρ(t=0) →1, the density decays with exponents δ= 0.47(2) and δ= 0.28(2) for the BS and GOL models, respectively. It is also shown that the temporal autocorrelation decays with exponents Ca = 0.35(2) (Ca = 0.35(5)) for the BS (GOL) model. By using these dynamically determined critical exponents and suitable scaling relationships, we also obtain the dynamic exponents z = 2.10(5) (z = 2.10(5)) for the BS (GOL) model. Based on this evidence we conclude that the dynamic approach to stationary states of the investigated models can be described by suitable power-law functions of time with well-defined exponents.
KeywordsEuropean Physical Journal Special Topic Initial Density Initial Increase Dynamic Approach Occupied Site
Unable to display preview. Download preview PDF.
- P. Bak, How Nature Works, Copernicus (Springer-Verlag, New York, 1996) Google Scholar
- H.J. Jensen, Self-Organized Criticality (Cambridge University Press, Cambridge, 1998) Google Scholar
- Int. J. Mod. Phys. B. 12, No. 14 (1998); Physica A 283, 80 (2000) Google Scholar
- E.R. Berlekamp, J.H. Conway, R.K. Guy, Winning Ways for your Mathematical Plays, Vol. 2. (Academic, New York, 1982) Google Scholar
- J. Nordfalk, P. Alstrom, Phys. Rev. E 54, R1025 (1996) Google Scholar
- Notice that in the GOL model, high initial density states decay to a different stationary state than that observed during the initial increase for ρ0 < 0.1 (see Fig. [SEE TEXT]b). So, the derived relationship among exponents is not strictly valid and we assumed it as a first approximation Google Scholar
- P. Alstrom, J. Leão, Phys. Rev. E 49, R2507 (1994) Google Scholar
- Notice that in magnetic systems the time scale of the initial increase of the magnetisation is set by tmax ∝m0 -z/x0, where m0 →0 is the initial magnetisation Huse Google Scholar