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要项。而本身在<math>d<4</math>时，包含依赖于<math>d</math>的指数的主要项，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

在理想玻色气体的解析溶解模型中，可以清楚地看到临界点性质从<math>d<4</math>到<math>d=4</math>再到<math>d>4</math>的变化过程。在<math>d \le 2\ </math>的情形下，不存在相变或者临界点。当<math>d>2</math>时，对于所有<math>\rho \Lambda ^d

在理想玻色气体的解析溶解模型中，可以清楚地看到临界点性质从<math>d<4</math>到<math>d=4</math>再到<math>d>4</math>的变化过程。在<math>d \le 2\ </math>的情形下，不存在相变或者临界点。当<math>d>2</math>时，对于所有<math>\rho \Lambda ^d

\ge \zeta (d/2)\ </math>，化学势<math>\mu</math>（此处不要与表面张力指数<math>\mu</math>混淆）都会变为零。其中<math>\rho</math>是密度，<math>\Lambda</math>是热德布罗意波长，即<math>h/\sqrt {2\pi mkT}</math>（其中<math>h</math>是普朗克常数，<math>m</math>是原子质量），<math>\zeta (s)</math>是黎曼<math>\zeta</math>函数。当由下<math>\rho \Lambda^d \rightarrow

\ge \zeta (d/2)\ </math>，化学势<math>\mu</math>（此处不要与表面张力指数<math>\mu</math>混淆）都会变为零。其中<math>\rho</math>是密度，<math>\Lambda</math>是热德布罗意波长，即<math>h/\sqrt {2\pi mkT}</math>（其中<math>h</math>是普朗克常数，<math>m</math>是原子质量），<math>\zeta (s)</math>是黎曼<math>\zeta</math>函数。当由下<math>\rho \Lambda^d \rightarrow

\zeta(d/2)</math>时，<math>\mu</math>从负值范围变为零。当<math>\mu \rightarrow 0-\ ,</math>时，<math>\zeta(d/2)-\rho \Lambda^d</math>之差（在正比例因子内）变为零，且满足以下关系：{{NumBlk|1=:|2=<math>\zeta(d/2)-\rho

\zeta(d/2)</math>时，<math>\mu</math>从负值范围变为零。当<math>\mu \rightarrow 0-\ </math>时，<math>\zeta(d/2)-\rho \Lambda^d</math>之差（在正比例因子内）变为零，且满足以下关系：{{NumBlk|1=:|2=<math>\zeta(d/2)-\rho

\Lambda^d \sim \left\{ \begin{array} {lc }(-\mu)^{d/2-1}, & 2<d<4 \\

\Lambda^d \sim \left\{ \begin{array} {lc }(-\mu)^{d/2-1}, & 2<d<4 \\

\\ \mu \ln(-\mu/kT), & d=4 \\ \\ -\mu , & d>4 . \end{array}\right. </math>|3={{EquationRef|19}}}}

\\ \mu \ln(-\mu/kT), & d=4 \\ \\ -\mu , & d>4 . \end{array}\right. </math>|3={{EquationRef|19}}}}

When <math>2<d<4</math> the mean-field <math>-\mu</math> is still present but is dominated by <math>(-\mu)^{d/2-1}\ ;</math> when

When <math>2<d<4</math> the mean-field <math>-\mu</math> is still present but is dominated by <math>(-\mu)^{d/2-1}\ ;</math> when

<math>d>4</math> the singular <math>(-\mu)^{d/2-1}</math> is still present but is dominated by the mean-field <math>-\mu\ .</math>

<math>d>4</math> the singular <math>(-\mu)^{d/2-1}</math> is still present but is dominated by the mean-field <math>-\mu\ .</math>

This behavior is reflected again in the R'''enormalization-group theory''' [19-21].  In the simplest cases there are two competing fixed points for  the renormalization-group flows, one of them associated with <math>d</math>-dependent  

This behavior is reflected again in the R'''enormalization-group theory''' [19-21].  In the simplest cases there are two competing fixed points for  the renormalization-group flows, one of them associated with <math>d</math>-dependent