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→Critical exponents 临界指数
another scaling relation [16,17].
another scaling relation [16,17].
== Exponents and space dimension ==
== Exponents and space dimension 临界点指数与空间维度 ==
The critical-point exponents depend on the dimensionality <math>d\ .</math> The theory was found to be illuminated by treating
The critical-point exponents depend on the dimensionality <math>d\ .</math> The theory was found to be illuminated by treating
:<math>\label{eq:16}
:<math>\label{eq:16}
\mu = (d-1)\nu, </math>
\mu = (d-1)\nu, </math>
{{NumBlk|:|<math>\mu = (d-1)\nu,</math>|{{EquationRef|16}}}}
临界点指数取决于维数<math>d\ </math>。人们发现，将<math>d</math>视为具有任意大小的连续变量可以解释说明这一观点。在一类被称为超标度的临界点指数关系中，可以清楚地看到<math>d</math>。关联长度<math>\xi</math>为密度或磁化波动的相干长度。决定其大小的是体积<math>\xi ^d</math>中与自发波动有关的过剩自由能，且一定是<math>kT\ ,</math>阶的，在临界点处具有有限值<math>kT_c</math>。但在这样的微元体中，典型的波动只会产生共轭相。则自由能<math>kT</math>为创建区域<math>\xi^{d-1}\ ,</math>的界面<math>\sigma \xi^{d-1}\ .</math>的自由能。因此，当接近临界点时，<math>\sigma \xi^{d-1}</math>具有<math>kT_c\ </math>阶的有限极限。再由指数<math>\mu</math>和<math>\nu\ ,</math>的定义可得超标度关系：{{NumBlk|:|<math>\mu = (d-1)\nu,</math>|{{EquationRef|16}}}}
a hyperscaling relation [16]. With ({{EquationNote|1=15}}) we then have also [16]
a hyperscaling relation [16]. With ({{EquationNote|1=15}}) we then have also [16]
:<math>\label{eq:17}
:<math>\label{eq:17}
d\nu = 2-\alpha = \gamma+2\beta, </math>
d\nu = 2-\alpha = \gamma+2\beta, </math>
{{NumBlk|:|<math>d\nu = 2-\alpha = \gamma+2\beta,</math>|{{EquationRef|17}}}}
再由({{EquationNote|1=15}})有：{{NumBlk|:|<math>d\nu = 2-\alpha = \gamma+2\beta,</math>|{{EquationRef|17}}}}
which, with ({{EquationNote|1=8}}) and ({{EquationNote|1=14}}), yields also [18]
which, with ({{EquationNote|1=8}}) and ({{EquationNote|1=14}}), yields also [18]
:<math>\label{eq:18}
:<math>\label{eq:18}
2-\eta = \frac{\delta -1}{\delta +1} d. </math>
2-\eta = \frac{\delta -1}{\delta +1} d. </math>
{{NumBlk|:|<math>2-\eta = \frac{\delta -1}{\delta +1} d. </math>|{{EquationRef|18}}}}
结合({{EquationNote|1=8}})和({{EquationNote|1=14}})得到：{{NumBlk|:|<math>2-\eta = \frac{\delta -1}{\delta +1} d. </math>|{{EquationRef|18}}}}
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.