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临界指数 .
临界指数 (英语:critical exponent )是物理学 中用来描述物理量在临界点 附近行为的指数。尽管没有得到严格证明,实验表明临界指数具有普适性,与具体的物理系统无关,仅和系统维度、关联长度与自旋 维度有关。
对于四维及以上的系统,可以通过平均场理论 计算得到临界指数。但对于低维系统而言平均场理论不再适用,需使用重整化群 方法进行研究。
假设相变 出现在临界温度T c 处。为研究临界温度附近物理量 f 的行为,我们引入约化温度
τ
:=
T
−
T
c
T
c
{\displaystyle \tau :={\frac {T-T_{\mathrm {c} )){T_{\mathrm {c} ))))
相变即发生于约化温度为0时。于是,可以定义临界指数
k
{\displaystyle k}
:
k
=
def
lim
τ
→
0
log
|
f
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τ
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{\displaystyle k\,{\stackrel {\text{def)){=))\,\lim _{\tau \to 0}{\frac {\log |f(\tau )|}{\log |\tau |))}
相应的幂律关系为
f
(
τ
)
∝
τ
k
,
τ
≈
0
{\displaystyle f(\tau )\propto \tau ^{k}\,,\quad \tau \approx 0}
这代表了τ → 0 时函数f (τ ) 的渐近行为。更加普遍地,我们有
f
(
τ
)
=
A
τ
k
(
1
+
b
τ
k
1
+
⋯
)
{\displaystyle f(\tau )=A\tau ^{k}\left(1+b\tau ^{k_{1))+\cdots \right)}
Hagen Kleinert and Verena Schulte-Frohlinde, Critical Properties of φ4 -Theories (页面存档备份 ,存于互联网档案馆 ) , World Scientific (Singapore, 2001) ; Paperback ISBN 981-02-4658-7
Toda, M., Kubo, R., N. Saito, Statistical Physics I , Springer-Verlag (Berlin, 1983); Hardcover ISBN 3-540-11460-2
J.M.Yeomans, Statistical Mechanics of Phase Transitions , Oxford Clarendon Press
H. E. Stanley Introduction to Phase Transitions and Critical Phenomena , Oxford University Press, 1971
A. Bunde and S. Havlin (editors), Fractals in Science (页面存档备份 ,存于互联网档案馆 ) , Springer, 1995
A. Bunde and S. Havlin (editors), Fractals and Disordered Systems (页面存档备份 ,存于互联网档案馆 ) , Springer, 1996
Universality classes from Sklogwiki
Zinn-Justin, Jean (2002). Quantum field theory and critical phenomena , Oxford, Clarendon Press (2002), ISBN 0-19-850923-5
Zinn-Justin, J. (2010). "Critical phenomena: field theoretical approach" (页面存档备份 ,存于互联网档案馆 ) Scholarpedia article Scholarpedia, 5(5):8346.
F. Leonard and B. Delamotte Critical exponents can be different on the two sides of a transition: A generic mechanism https://arxiv.org/abs/1508.07852( 页面存档备份 ,存于互联网档案馆 )]
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