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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，与<math>r^{-\frac{1}{2}(d-1)}e^{-r/\xi}\ ,</math>成正比（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>的增加而衰减，与<math>r^{-\frac{1}{2}(d-1)}e^{-r/\xi}\ </math>成正比（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>

{{NumBlk|:|<math>\rho kT

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

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

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

with <math>k</math> Boltzmann's constant and where the integral is over all space with <math>\rm{d} \tau</math> the element of volume. The same relation holds in the ferromagnets with <math>\chi</math> then the magnetic susceptibility and with the deviation of <math>\rho</math> from the critical density <math>\rho_c</math> then the magnetization <math>M\ .</math>  At the critical point <math>\chi</math> is infinite and correspondingly the integral diverges because the decay length <math>\xi</math> is then also infinite.  The density <math>\rho</math> is there just the finite positive constant <math>\rho_c</math> and <math>T</math> the finite <math>T_c\ .</math>  Then from the scaling law ({{EquationNote|1=10}}), because of the homogeneity of <math>h(r,t)</math> and because the main contribution to the diverging integral comes from large <math>r\ ,</math> where ({{EquationNote|1=10}}) holds, it follows that <math>\chi</math> diverges proportionally to <math>\xi^{2-\eta} \int

with <math>k</math> Boltzmann's constant and where the integral is over all space with <math>\rm{d} \tau</math> the element of volume. The same relation holds in the ferromagnets with <math>\chi</math> then the magnetic susceptibility and with the deviation of <math>\rho</math> from the critical density <math>\rho_c</math> then the magnetization <math>M\ .</math>  At the critical point <math>\chi</math> is infinite and correspondingly the integral diverges because the decay length <math>\xi</math> is then also infinite.  The density <math>\rho</math> is there just the finite positive constant <math>\rho_c</math> and <math>T</math> the finite <math>T_c\ .</math>  Then from the scaling law ({{EquationNote|1=10}}), because of the homogeneity of <math>h(r,t)</math> and because the main contribution to the diverging integral comes from large <math>r\ ,</math> where ({{EquationNote|1=10}}) holds, it follows that <math>\chi</math> diverges proportionally to <math>\xi^{2-\eta} \int

:<math>\label{eq:14}

:<math>\label{eq:14}

(2-\eta)\nu = \gamma .  </math>

(2-\eta)\nu = \gamma .  </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}}),