更改

\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>自由能<math>F</math>可以通过固定温度下积分由({{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>在临界点处成比例发散至<math>\mid t\mid ^{-\alpha}</math>（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。

({{EquationNote|8}})和({{EquationNote|9}})分别来自里斯和斯考特的贡献。它们大概是历史上最早版本的临界指数关系。在此之后，Domb和Sykes以及Fisher注意到指数<math>\gamma</math>实际上比平均场值<math>\gamma =1</math>大。而在更早之前，Guggenheim的对应状态分析就清楚地表明<math>\beta</math>值更靠近1/3而非平均场值的1/2。之后在<math>\gamma

({{EquationNote|8}})和({{EquationNote|9}})分别来自里斯和斯考特的贡献。它们大概是历史上最早版本的临界指数关系。在此之后，Domb和Sykes以及Fisher注意到指数<math>\gamma</math>实际上比平均场值<math>\gamma =1</math>大。而在更早之前，Guggenheim的对应状态分析就清楚地表明<math>\beta</math>值更靠近1/3而非平均场值的1/2。之后在<math>\gamma

=1</math>和<math>\beta \simeq 1/3\ ,</math>的假设下，里斯由({{EquationNote|8}})式总结出<math>\delta = 1+1/\beta

=1</math>和<math>\beta \simeq 1/3\ </math>的假设下，里斯由({{EquationNote|8}})式总结出<math>\delta = 1+1/\beta

\simeq 4</math>（如今已知正确值接近5）。同时斯考特由({{EquationNote|9}})式得出<math>\alpha =1-2\beta \simeq 1/3</math>（正确值接近1/10）.另外平均场值<math>\delta

\simeq 4</math>（如今已知正确值接近5）。同时斯考特由({{EquationNote|9}})式得出<math>\alpha =1-2\beta \simeq 1/3</math>（正确值接近1/10）。另外平均场值<math>\delta

=3</math>，<math>\alpha =0\ </math>。

=3</math>，<math>\alpha =0\ </math>。

Thus, as <math>r\rightarrow \infty</math> in any fixed thermodynamic state (fixed t) near the critical point, <math>h</math> decays with increasing <math>r</math> proportionally to <math>r^{-\frac{1}{2}(d-1)}e^{-r/\xi}\ ,</math> as in the [https://en.wikipedia.org/wiki/Ornstein%E2%80%93Zernike_equation?oldformat=true '''Ornstein-Zernike theory'''].  If, instead, the critical point is approached <math>(\xi \rightarrow \infty)</math> with a fixed, large <math>r\ ,</math> we have <math>h(r)</math> decaying with <math>r</math> only as an inverse power, <math>r^{-(d-2+\eta)}\ ,</math> which corrects the <math>r^{-(d-2)}</math> that appears in the Ornstein-Zernike theory in that limit. The scaling law({{EquationNote|1=10}}) with scaling function <math>G(x)</math> interpolates between these extremes.

Thus, as <math>r\rightarrow \infty</math> in any fixed thermodynamic state (fixed t) near the critical point, <math>h</math> decays with increasing <math>r</math> proportionally to <math>r^{-\frac{1}{2}(d-1)}e^{-r/\xi}\ ,</math> as in the [https://en.wikipedia.org/wiki/Ornstein%E2%80%93Zernike_equation?oldformat=true '''Ornstein-Zernike theory'''].  If, instead, the critical point is approached <math>(\xi \rightarrow \infty)</math> with a fixed, large <math>r\ ,</math> we have <math>h(r)</math> decaying with <math>r</math> only as an inverse power, <math>r^{-(d-2+\eta)}\ ,</math> which corrects the <math>r^{-(d-2)}</math> that appears in the Ornstein-Zernike theory in that limit. The scaling law({{EquationNote|1=10}}) with scaling function <math>G(x)</math> interpolates between these extremes.

因此，在任何靠近临界点的恒温热力学状态下，当<math>r\rightarrow \infty</math>时，<math>h</math>随<math>r</math>的增加而衰减至<math>r^{-\frac{1}{2}(d-1)}e^{-r/\xi}\ </math>（参见'''[https://en.wikipedia.org/wiki/Ornstein%E2%80%93Zernike_equation? 奥恩斯泰因-泽尔尼克理论 Ornstein-Zernike theory]'''）。

因此，在任何靠近临界点的恒温热力学状态下，当<math>r\rightarrow \infty</math>时，<math>h</math>随<math>r</math>的增加依<math>r^{-\frac{1}{2}(d-1)}e^{-r/\xi}\ </math>成比例衰减（参见'''[https://en.wikipedia.org/wiki/Ornstein%E2%80%93Zernike_equation? 奥恩斯泰因-泽尔尼克理论 Ornstein-Zernike theory]'''）。

In the language of fluids, with <math>\rho</math> the number density and <math>\chi</math> the isothermal compressibility, we have as an exact relation in the Ornstein-Zernike theory  

In the language of fluids, with <math>\rho</math> the number density and <math>\chi</math> the isothermal compressibility, we have as an exact relation in the Ornstein-Zernike theory  

\rho kT

\rho kT

\chi =1+\rho \int h(r) \rm{d}\tau </math>

\chi =1+\rho \int h(r) \rm{d}\tau </math>

在流体语境中，有数密度<math>\rho</math>和等温压缩率<math>\chi</math>，我们可以得到一个'''[https://en.wikipedia.org/wiki/Ornstein%E2%80%93Zernike_equation? 奥恩斯泰因-泽尔尼克理论]'''的精确表达式：{{NumBlk|:|<math>\rho kT

在流体研究中，有数密度<math>\rho</math>和等温压缩率<math>\chi</math>，我们可以得到一个'''[https://en.wikipedia.org/wiki/Ornstein%E2%80%93Zernike_equation? 奥恩斯泰因-泽尔尼克理论]'''的精确表达式：{{NumBlk|:|<math>\rho kT

\chi =1+\rho \int h(r) \rm{d}\tau</math>|{{EquationRef|13}}}}

\chi =1+\rho \int h(r) \rm{d}\tau</math>|{{EquationRef|13}}}}

:<math>\label{eq:14}

:<math>\label{eq:14}

(2-\eta)\nu = \gamma .  </math>

(2-\eta)\nu = \gamma .  </math>

其中<math>k</math>是'''[[玻尔兹曼常数]]'''，<math>\rm{d} \tau</math>是体积元，积分区域是整个空间。对铁磁体也有相同的关系成立，包含磁化率<math>\chi</math>，<math>\rho</math>与临界密度<math>\rho_c</math>的差值以及磁化强度<math>M\ </math>。在临界点处，<math>\chi</math>无穷大，且对应积分式也发散，因为衰减长度<math>\xi</math>也是无穷大的。而密度<math>\rho</math>为有限正常数<math>\rho_c</math>，<math>T</math>为<math>T_c\ </math>。{{NumBlk|:|<math>(2-\eta)\nu = \gamma . </math>|{{EquationRef|14}}}}The surface tension <math>\sigma</math> in liquid-vapor equilibrium, or the analogous excess free energy per unit area of the interface between coexisting, oppositely magnetized domains, vanishes at the critical point (Curie point) proportionally to <math>(-t)^\mu</math> with <math>\mu</math> another critical-point exponent. The interfacial region has a thickness of the order of the correlation

其中<math>k</math>是'''[[玻尔兹曼常数]]'''，<math>\rm{d} \tau</math>是体积元，积分区域是整个空间。对铁磁体也有相同的关系成立，包含磁化率<math>\chi</math>，<math>\rho</math>与临界密度<math>\rho_c</math>的差值，以及磁化强度<math>M\ </math>。在临界点处，<math>\chi</math>无穷大，且对应积分式也发散，因为衰减长度<math>\xi</math>也是无穷大的。而密度<math>\rho</math>为有限正常数<math>\rho_c</math>，<math>T</math>为<math>T_c\ </math>。{{NumBlk|:|<math>(2-\eta)\nu = \gamma . </math>|{{EquationRef|14}}}}The surface tension <math>\sigma</math> in liquid-vapor equilibrium, or the analogous excess free energy per unit area of the interface between coexisting, oppositely magnetized domains, vanishes at the critical point (Curie point) proportionally to <math>(-t)^\mu</math> with <math>\mu</math> another critical-point exponent. The interfacial region has a thickness of the order of the correlation

length <math>\xi</math> so <math>\sigma/\xi</math> is the free energy per unit volume associated with the interfacial region. That is in its magnitude and in its singular critical-point behavior the same free energy per unit volume as in the bulk phases, from which the heat capacity follows by two differentiations with respect to the temperature. Thus, <math>\sigma/\xi</math> vanishes proportionally to <math>(-t)^{2-\alpha}\ ;</math> so, together with ({{EquationNote|1=9}}),

length <math>\xi</math> so <math>\sigma/\xi</math> is the free energy per unit volume associated with the interfacial region. That is in its magnitude and in its singular critical-point behavior the same free energy per unit volume as in the bulk phases, from which the heat capacity follows by two differentiations with respect to the temperature. Thus, <math>\sigma/\xi</math> vanishes proportionally to <math>(-t)^{2-\alpha}\ ;</math> so, together with ({{EquationNote|1=9}}),