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^{1/\beta}</math> of degree <math>\beta \delta\ .</math>  The scaling function <math>j(x)</math> vanishes proportionally to <math>x+b</math> as <math>x</math> approaches <math>-b\ ,</math> with <math>b</math> a positive constant; it diverges proportionally to <math>x^{\beta(\delta-1)}</math> as <math>x\rightarrow \infty\ ;</math> and <math>j(0) = c\ ,</math> another positive constant (Fig. 1). Although ({{EquationNote|7}}) is confined to the immediate neighborhood of the critical point <math>(t, M, H</math> all near 0), the scaling variable <math>x = t/\mid M\mid ^{1/\beta}</math> nevertheless traverses the infinite range <math>-b < x < \infty\ .</math>

^{1/\beta}</math> of degree <math>\beta \delta\ .</math>  The scaling function <math>j(x)</math> vanishes proportionally to <math>x+b</math> as <math>x</math> approaches <math>-b\ ,</math> with <math>b</math> a positive constant; it diverges proportionally to <math>x^{\beta(\delta-1)}</math> as <math>x\rightarrow \infty\ ;</math> and <math>j(0) = c\ ,</math> another positive constant (Fig. 1). Although ({{EquationNote|7}}) is confined to the immediate neighborhood of the critical point <math>(t, M, H</math> all near 0), the scaling variable <math>x = t/\mid M\mid ^{1/\beta}</math> nevertheless traverses the infinite range <math>-b < x < \infty\ .</math>

其中<math>j(x)</math>是“标度”函数，<math>\beta</math>和<math>\delta</math>是临界点指数。因此由({{EquationNote|2}})和({{EquationNote|7}})，当铁磁物质趋近于临界点时<math>(H\rightarrow 0</math>且<math>t\rightarrow 0)\ </math>，<math>\mid H\mid</math>是 <math> t </math> 和<math>\mid M\mid

其中 <math>j(x)</math> 是“标度”函数，<math>\beta</math> 和<math>\delta</math> 是临界点指数。因此由({{EquationNote|2}})和({{EquationNote|7}})，当铁磁物质趋近于临界点时<math>(H\rightarrow 0</math> 且 <math>t\rightarrow 0)\ </math>，<math>\mid H\mid</math> 是 <math> t </math> 和 <math>\mid M\mid

^{1/\beta}</math> 的<math>\beta \delta\ </math>次齐次函数。当<math> x </math>趋近于<math>-b\</math>（正常数）时，标度函数<math>j(x)</math>趋近于零；当<math>x\rightarrow \infty\ </math>时，它依<math>x^{\beta(\delta-1)}</math>成比例发散（如图一），且<math>j(0) = c\ </math>（正常数）。尽管({{EquationNote|7}})局限在临界点<math>(t, M, H</math>都接近零）附近的极小范围内，但标度变量<math>x = t/\mid M\mid ^{1/\beta}</math>却遍历<math>-b < x < \infty\</math>的无穷范围。

^{1/\beta}</math> 的 <math>\beta \delta\ </math> 次齐次函数。当 <math> x </math> 趋近于<math>-b\</math>（正常数）时，标度函数 <math>j(x)</math> 趋近于零；当 <math>x\rightarrow \infty\ </math>时，它依 <math>x^{\beta(\delta-1)}</math> 成比例发散（如图一），且 <math>j(0) = c\ </math>（正常数）。尽管({{EquationNote|7}})局限在临界点<math>(t, M, H</math> 都接近零）附近的极小范围内，但标度变量 <math>x = t/\mid M\mid ^{1/\beta}</math>却遍历<math>-b < x < \infty\</math>的无穷范围。

[[Image:scaling_laws_widom_nocaption_Fig1.png|thumb|300px|right|Scaling function  

[[Image:scaling_laws_widom_nocaption_Fig1.png|thumb|300px|right|Scaling function  

({{EquationNote|8}}) are examples of scaling laws, Eq.({{EquationNote|7}}) being a statement of homogeneity and the exponent relation ({{EquationNote|8}}) a consequence of that homogeneity.

({{EquationNote|8}}) are examples of scaling laws, Eq.({{EquationNote|7}}) being a statement of homogeneity and the exponent relation ({{EquationNote|8}}) a consequence of that homogeneity.

当<math>\mid H\mid = 0+</math>且<math>t<0\ </math>，<math>M</math>是自发磁化率，由({{EquationNote|7}})可得<math>\mid M\mid = (-\frac{t}{b})^\beta\ </math>，其中<math>\beta</math>对应这一临界指数。在临界等温线<math>(t=0)\ </math>，当<math>M\rightarrow 0</math>时，我们有<math>H \sim cM\mid

当<math>\mid H\mid = 0+</math> 且 <math>t<0\ </math>，<math>M</math> 是自发磁化率，由({{EquationNote|7}})可得<math>\mid M\mid = (-\frac{t}{b})^\beta\ </math>，其中 <math>\beta</math> 为临界指数。在临界等温线<math>(t=0)\ </math>，当 <math>M\rightarrow 0</math> 时，我们有<math>H \sim cM\mid

M\mid ^{\delta-1}\ </math>，其中<math>\delta</math>为此时的临界指数。由前文中<math>j(x)</math>的第一个性质和({{EquationNote|7}})式，我们可以计算磁化率<math>(\partial

M\mid ^{\delta-1}\ </math>，其中 <math>\delta</math> 为此时的临界指数。由前文中 <math>j(x)</math> 的第一个性质和({{EquationNote|7}})式，我们可以计算磁化率 <math>(\partial

M/\partial H)_T\ </math>，它在<math>\mid

M/\partial H)_T\ </math>，它在<math>\mid

H\mid = 0+</math>且<math>t<0</math>，以及在<math>H=0</math>且<math>t>0</math>时依<math>\mid t\mid ^{-\beta(\delta-1)}\</math>成比例发散（尽管系数不同）。磁化率指数的常用符号是<math>\gamma\ </math>，因此有{{NumBlk|:|<math>\gamma =

H\mid = 0+</math> 且 <math>t<0</math>，以及在 <math>H=0</math>且<math>t>0</math> 时依 <math>\mid t\mid ^{-\beta(\delta-1)}\</math>成比例发散（尽管系数不同）。磁化率指数的常用符号是 <math>\gamma\ </math>，因此有{{NumBlk|:|<math>\gamma =

\beta(\delta-1). </math>|{{EquationRef|8}}}}

\beta(\delta-1). </math>|{{EquationRef|8}}}}

\alpha +2\beta +\gamma=2.  </math>

\alpha +2\beta +\gamma=2.  </math>

由于<math>M = -(\partial

由于<math>M = -(\partial

F/\partial H)_T\ </math>，等温条件下自由能<math>F</math>可以通过积分由({{EquationNote|7}})式得出，且相应的热容<math>C_H = -(\partial ^2

F/\partial H)_T\ </math>，等温条件下自由能 <math>F</math> 可以通过积分由({{EquationNote|7}})式得出，且相应的热容 <math>C_H = -(\partial ^2

F/\partial T^2)_H\ </math>。由({{EquationNote|7}})式可知，在<math>H=0</math>时<math>C_H</math>在临界点处依<math>\mid t\mid ^{-\alpha}</math>比例发散（其中<math>t\rightarrow 0-</math>和<math>t\rightarrow

F/\partial T^2)_H\ </math>。由({{EquationNote|7}})式可知，在<math>H=0</math> 时 <math>C_H</math> 在临界点处依<math>\mid t\mid ^{-\alpha}</math>比例发散（其中 <math>t\rightarrow 0-</math> 和 <math>t\rightarrow

0+\ </math>各有不同的系数），临界点指数<math>\alpha</math>与<math>\beta</math>和<math>\gamma</math>满足以下标度律：{{NumBlk|:|<math>\alpha +2\beta +\gamma=2. </math>|{{EquationRef|9}}}}When <math>2\beta+\gamma=2</math> the resulting <math>\alpha =0</math> means, generally, a logarithmic rather than power-law divergence together with a superimposed finite discontinuity occurring between <math>t=0+</math> and <math>t=0-</math> [4].  In the 2-dimensional Ising model the discontinuity is absent and only the logarithm remains, while in mean-field (van der Waals, Curie-Weiss, Bragg-Williams) approximation the logarithm is absent but the discontinuity is still present.

0+ </math>各有不同的系数），临界点指数 <math>\alpha</math> 与<math>\beta</math> 和 <math>\gamma</math> 满足以下标度律：{{NumBlk|:|<math>\alpha +2\beta +\gamma=2. </math>|{{EquationRef|9}}}}When <math>2\beta+\gamma=2</math> the resulting <math>\alpha =0</math> means, generally, a logarithmic rather than power-law divergence together with a superimposed finite discontinuity occurring between <math>t=0+</math> and <math>t=0-</math> [4].  In the 2-dimensional Ising model the discontinuity is absent and only the logarithm remains, while in mean-field (van der Waals, Curie-Weiss, Bragg-Williams) approximation the logarithm is absent but the discontinuity is still present.

当<math>2\beta+\gamma=2</math>，则有<math>\alpha =0</math>，这通常意味着对数发散而不是幂律发散，并且在<math>t=0+</math>和<math>t=0-</math>之间存在叠加有限不连续。在二维伊辛模型中，仅有对数关系而这种不连续是不存在的；而在平均场近似中情形相反。

当<math>2\beta+\gamma=2</math>，则有<math>\alpha =0</math>，这通常意味着对数发散而不是幂律发散，并且在<math>t=0+</math>和<math>t=0-</math>之间存在叠加有限不连续。在二维伊辛模型中，仅有对数关系而这种不连续是不存在的；而在平均场近似中情形相反。