Theoretical and Numerical Approaches to Critical Natures of A Sandpile

• Main Authors: Chao-shun Yang, 楊朝順
• Other Authors: none
• Format: Others
• Language: zh-TW
• Published: 2005
• Online Access: http://ndltd.ncl.edu.tw/handle/57767432690643987818

博士 === 國立中山大學 === 物理學系研究所 === 93 === A self-organized criticality (SOC) system is driven and maintained by repeatedly adding energy at random, and by dissipating energy in a specified way. The dissipating way is seldom considered, yet it plays an important role in the source of a SOC. Here, we use sandpile models as an example to point out the effects of dissipation on a SOC. First, we study the dissipation through a losing probability $f$ during each toppling process. In such a dissipative system, we find the SOC behavior is broken when $f > 0.1$ and that it is not evident for $0.1>f>0.01$. Numerical simulations of the toppling size exponents for all ($ au_a$), dissipative ($ au_d$), and last ($ au_l$) waves have been investigated for $f le 0.01$. We find that $ au_a=1$ is independent of $f$ and identical to the original sandpile model which dissipates energy at the boundary. However, the values of $ au_d$ and $ au_l$ do indeed depend on $f$. Furthermore, we derive analytic expressions of the exponents of $ au_d$ and $ au_l$, and conjecture $ au_l + au_d = frac{11}{8}$ and the exponent of the dissipative last waves $ au_{ld}=frac{3}{8}$. All of them are well consistent with the numerical study. We conclude that dissipation drives a system from being a non-SOC to a SOC. However, these SOC universality classes consist of three kinds of exponents: overall ($ au_a$), local ($ au_{ld}$), and detailed ($ au_d$ and $ au_l$).