更改

Unlike the scaling laws ({{EquationNote|1=8}}), ({{EquationNote|1=9}}), ({{EquationNote|1=14}}), and ({{EquationNote|1=15}}), which make no explicit reference to the dimensionality, the <math>d</math>-dependent exponent relations ({{EquationNote|1=16}})-({{EquationNote|1=18}}) hold only for <math>d<4\ .</math>  At <math>d=4</math> the exponents assume the values they have in the mean-field theories but logarithmic factors are then appended to the simple power laws. Then for <math>d>4\ ,</math> the terms in the thermodynamic functions and correlation-function parameters that have as their exponents those given by the mean-field theories are the leading terms. The terms with the original <math>d</math>-dependent exponents, which for <math>d<4</math> were the leading terms, have been overtaken, and, while still present, are now sub-dominant.

Unlike the scaling laws ({{EquationNote|1=8}}), ({{EquationNote|1=9}}), ({{EquationNote|1=14}}), and ({{EquationNote|1=15}}), which make no explicit reference to the dimensionality, the <math>d</math>-dependent exponent relations ({{EquationNote|1=16}})-({{EquationNote|1=18}}) hold only for <math>d<4\ .</math>  At <math>d=4</math> the exponents assume the values they have in the mean-field theories but logarithmic factors are then appended to the simple power laws. Then for <math>d>4\ ,</math> the terms in the thermodynamic functions and correlation-function parameters that have as their exponents those given by the mean-field theories are the leading terms. The terms with the original <math>d</math>-dependent exponents, which for <math>d<4</math> were the leading terms, have been overtaken, and, while still present, are now sub-dominant.

标度律({{EquationNote|1=8}})，({{EquationNote|1=9}})，({{EquationNote|1=14}})和({{EquationNote|1=15}})没有明显的和空间维数相联系，而({{EquationNote|1=16}})-({{EquationNote|1=18}})则是依赖于 <math>d</math> 的指数关系式，且仅对 <math>d<4\ </math>成立。对于 <math>d=4</math>，热力学函数中依据平均场理论给出的以相关函数参数为指数的项是主导项。而本身在<math>d<4</math>时，包含依赖于 <math>d</math> 的指数的主导项，虽然依然存在，但是已经被取代而变成次要项。  

标度律({{EquationNote|1=8}})，({{EquationNote|1=9}})，({{EquationNote|1=14}})和({{EquationNote|1=15}})没有明显的和空间维数相联系，而({{EquationNote|1=16}})-({{EquationNote|1=18}})则是依赖于 <math>d</math> 的指数关系式，且仅对 <math>d<4\ </math>成立。对于 <math>d=4</math>，热力学函数中依据平均场理论给出的以相关函数参数为指数的项是主导项。而本身在 <math>d<4</math> 时，包含依赖于 <math>d</math> 的指数的主导项，虽然依然存在，但是已经被取代而变成次要项。  

This progression in critical-point properties from <math>d<4</math> to <math>d=4</math> to <math>d>4</math> is seen clearly in the phase transition that occurs in the analytically soluble model of the ideal Bose gas. There is no phase transition or critical point in it for <math>d \le 2\ .</math>  When <math>d>2</math> the chemical potential

This progression in critical-point properties from <math>d<4</math> to <math>d=4</math> to <math>d>4</math> is seen clearly in the phase transition that occurs in the analytically soluble model of the ideal Bose gas. There is no phase transition or critical point in it for <math>d \le 2\ .</math>  When <math>d>2</math> the chemical potential

2\beta + \gamma =2</math> are known to follow.

2\beta + \gamma =2</math> are known to follow.

因此，由({{EquationNote|1=2}})得 <math>f(t,H)=t^{d/y}

因此，由({{EquationNote|1=2}})得  

\phi(H^{y/x}/t)=H^{d/x}\psi(t/H^{y/x})</math>，其中 <math>\phi</math> 和 <math>\psi</math> 仅仅是 <math>H^{y/x}/t\ </math> 的函数。当 <math>H=0</math>，由第一个关系式可得 <math>f(t,0)=\phi(0)t^{d/y}\ </math>。但是 <math>f(t,0)</math> 的两个温度导数对单位自旋热容有贡献，且以 <math>t^{-\alpha}\ </math>发散，所以有<math>d/y=2-\alpha\ </math>。另外，在临界等温线<math>(t=0)\ </math>上，由第二个关系式可得<math>f(0,H)=\psi(0)H^{d/x}\ </math>。但单位自旋磁化强度<math>-(\partial f/\partial H)_T\ </math>随 <math>H^{d/x-1}\ </math>衰减，因此 <math>d/x-1=1/\delta\ </math>。指数 <math>d/x</math> 与 <math>d/y</math> 可以由热容指数 <math>\alpha</math> 和临界等温线指数 <math>\delta\ </math>定义。同时再有单位自旋磁化强度<math>-(\partial

 

<math>f(t,H)=t^{d/y}

\phi(H^{y/x}/t)=H^{d/x}\psi(t/H^{y/x})</math>，

 

其中 <math>\phi</math> 和 <math>\psi</math> 仅仅是 <math>H^{y/x}/t\ </math> 的函数。当 <math>H=0</math>，由第一个关系式可得 <math>f(t,0)=\phi(0)t^{d/y}\ </math>。但是 <math>f(t,0)</math> 的两个温度导数对单位自旋热容有贡献，且以 <math>t^{-\alpha}\ </math>发散，所以有<math>d/y=2-\alpha\ </math>。另外，在临界等温线<math>(t=0)\ </math>上，由第二个关系式可得<math>f(0,H)=\psi(0)H^{d/x}\ </math>。但单位自旋磁化强度<math>-(\partial f/\partial H)_T\ </math>随 <math>H^{d/x-1}\ </math>衰减，因此 <math>d/x-1=1/\delta\ </math>。指数 <math>d/x</math> 与 <math>d/y</math> 可以由热容指数 <math>\alpha</math> 和临界等温线指数 <math>\delta\ </math>定义。同时再有单位自旋磁化强度<math>-(\partial

f/\partial H)_T</math>，({{EquationNote|1=20}})中<math>f(t,H)</math> 的齐次形式与({{EquationNote|1=7}})式 <math>H(t,M)</math> 的齐次形式等价，由此得到标度律 <math>\gamma=\beta(\delta-1)</math> 和<math>\alpha +

f/\partial H)_T</math>，({{EquationNote|1=20}})中<math>f(t,H)</math> 的齐次形式与({{EquationNote|1=7}})式 <math>H(t,M)</math> 的齐次形式等价，由此得到标度律 <math>\gamma=\beta(\delta-1)</math> 和<math>\alpha +

2\beta + \gamma =2</math>。

2\beta + \gamma =2</math>。