更改

:<math>\label{eq:10}

:<math>\label{eq:10}

h(r,t)=r^{-(d-2+\eta)}G(r/\xi).  </math>

h(r,t)=r^{-(d-2+\eta)}G(r/\xi).  </math>

{{NumBlk|:|<math>h(r,t)=r^{-(d-2+\eta)}G(r/\xi).  </math>|{{EquationRef|10}}}}

铁磁体和流体中的长程空间相关函数在临界点附近也表现出齐次性。简单起见，考虑磁场强度<math>H=0</math>且温度接近临界点的情况，关联函数<math>r</math>作为空间分离<math>r</math>（假设很大）的函数，如下所示：{{NumBlk|:|<math>h(r,t)=r^{-(d-2+\eta)}G(r/\xi).  </math>|{{EquationRef|10}}}}

Here <math>d</math> is the dimensionality of space, <math>\eta</math> is another critical-point exponent, and <math>\xi</math> is the correlation length (exponential

Here <math>d</math> is the dimensionality of space, <math>\eta</math> is another critical-point exponent, and <math>\xi</math> is the correlation length (exponential

:<math>\label{eq:11}

:<math>\label{eq:11}

\xi\sim \mid t\mid ^{-\nu} </math>

\xi\sim \mid t\mid ^{-\nu} </math>

{{NumBlk|:|<math>\xi\sim \mid t\mid ^{-\nu}</math>|{{EquationRef|11}}}}

其中<math>d</math>是空间维度，<math>\eta</math>是另一临界点指数，<math>\xi</math>是关联长度（相关系数的指数衰减长度），当趋近于临界点时，其发散过程满足：{{NumBlk|:|<math>\xi\sim \mid t\mid ^{-\nu}</math>|{{EquationRef|11}}}}

 

as the critical point is approached, with <math>\nu</math> still another critical-point exponent. Thus, <math>h(r,t)</math> (with <math>H=0)</math> is a homogeneous function of <math>r</math> and <math>\mid t\mid

as the critical point is approached, with <math>\nu</math> still another critical-point exponent. Thus, <math>h(r,t)</math> (with <math>H=0)</math> is a homogeneous function of <math>r</math> and <math>\mid t\mid

\begin{array} {lc }x^{\frac{1}{2}(d-3)+\eta} e^{-x}, & x\rightarrow

\begin{array} {lc }x^{\frac{1}{2}(d-3)+\eta} e^{-x}, & x\rightarrow

\infty \\ 1, & x\rightarrow 0 . \end{array} \right.  </math>

\infty \\ 1, & x\rightarrow 0 . \end{array} \right.  </math>

{{NumBlk|:|<math>G(x) \sim \left\{

^{-\nu}</math>的<math>-(d-2+\eta)\ </math>次齐次方程。标度函数<math>G(x)</math>具有以下性质(在常数比例因子范围内)：{{NumBlk|:|<math>G(x) \sim \left\{

\begin{array} {lc }x^{\frac{1}{2}(d-3)+\eta} e^{-x}, & x\rightarrow

\begin{array} {lc }x^{\frac{1}{2}(d-3)+\eta} e^{-x}, & x\rightarrow

\infty \\ 1, & x\rightarrow 0 . \end{array} \right. </math>|{{EquationRef|12}}}}

\infty \\ 1, & x\rightarrow 0 . \end{array} \right. </math>|{{EquationRef|12}}}}

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.