Sandpile and SelfOrganized Criticality 20092 POSTECH NCSL n

Sandpile and Self-Organized Criticality 전산물리 입문 2009/2 김승환 POSTECH NCSL

모래탑 n 모래탑 Sandpile – 사태 (Avalanche)가 일어남. n 희귀하지만 큰 규모의 사건. – 스스로짜인 임계현상 self-organized criticality (SOC) P. Bak, How Nature Works: The Science of Self Organized Criticality , Springer-Verlag, 1996 P. Bak, C. Tang and K. Wiesenfeld, PRL 59, 381(1987) POSTECH NCSL

지진 n Gutenberg-Richter frequency-magnitude relationship – 평균적으로 진도 6은 1년에 100번, 진도 7은 10번, 진도 8은 한번 정도 일어남. POSTECH NCSL

모래탑 실험 n Sandpile 실험 - G. A. Held et al – IBM Thomas J. Watson Research Center 참고문헌: G. A. Held et al, PRL 65, 1120 (1990). POSTECH NCSL

1차원 모래탑 모듈 Sandpile 1 POSTECH NCSL

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2차원 모래탑 모형 n Critical-height 모형 – 낟알 더하기: m(i, j) = m (i, j) + 1 – 낟알 넘어가기: m(i, j) = m(i, j) – 4 if m(I, j) > mc m(i+1, j) = m(i+1, j) +1 m(i-1, j) = m(i-1, j) +1 m(i, j+1) = m(i, j+1) +1 m(i, j-1) = m(i, j-1) +1 e. g. mc =3 L. P. Kadanoff et al, Phys. Rev. A 39, 6524, 1989. POSTECH NCSL

이 차원 모래탑 모듈 Sandpile 2 POSTECH NCSL

Bak, Tang, Wiesenfeld Model n Cellular Automata Model – Height Z(x, y) n Dynamcs: n n Addition : z(x, y) -> z(x, y) + 1 Toppling : if z(x, y) > zc =3, neighbors get one grain each Sand is lost at the boundaries SOC & Power-law – System reaches a self-organized critical state – P(s) ~ S-1. 2 – P. Bak, C. Tang and K. Wiesenfeld, PRL 59, 381 (1987) Java Applet : http: //cmth. phy. bnl. gov/~maslov/Sandpile. htm POSTECH NCSL

Avalanches in BTW Model POSTECH NCSL

Identity state Avalanche in a periodic lattice - forever POSTECH NCSL

Forest Fire n. Yellow – Fire, green – trees, black – empty spaces n. Tree grow on empty spaces with probability p. n. A fire burns down to an empty space in one time step but manages to ignite all neighboring trees. POSTECH NCSL

Ising Model Critical state POSTECH NCSL