自組臨界系統的驅策與維持來自於隨機加入能量與特定的耗散形式。
然耗散形式的討論鮮為人所注意，但其在自組臨界系統的源起?埵?有著重要的成分。
我們藉由沙堆模型的研究，釐清耗散形式在自組臨界系統中的效應。
首先，我們研究的耗散形式為在每一次的崩塌過程中，粒子的消失機率為 $f$ 。
在這樣的耗散系統中，發現當 $f>0.1$ 時，自組臨界的現象遭到破壞。而當 $0.1>f>0.01$ 時，
其自組臨界的行為亦不明顯。
本文以數值模擬探討在 $fleq0.01$ 時，總波($ au_a$),耗散波 ($ au_d$),和最終波($ au_l$)的崩塌指數。
吾人發現 $ au_a=1$ 與 $f$ 無關，等價於初始沙堆模型在邊界耗散能量的情形。而 $ au_d$ 和 $ au_l$ 則取決於 $f$ ，
且吾人更進而推導出其解析式，並推測 $ au_l + au_d = frac{11}{8}$ 和最終耗散波($ au_{ld}$)的指數為$frac{3}{8}$。
此結果皆與數值模擬相符。
故吾人推論耗散機制可驅策系統，由非自組臨界的狀態變成自組臨界的狀態。

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$).

自組臨界現象與沙堆模型-15
臨界現象]-45
{沙堆模型}-65
chapter[結果與討論]{結果與討論}
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