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→Scaling laws 标度律
==Introduction 引言==
==Introduction 引言==
== 生物学中的规模法则 ==
== 城市科学中 ==
== 地球科学中 ==
== 数学 ==
== Scaling laws 物理学中的标度律 ==
<strong><nowiki>Scaling laws </nowiki></strong> are the expression of physical principles in the mathematical language of homogeneous functions.
<strong><nowiki>Scaling laws </nowiki></strong> are the expression of physical principles in the mathematical language of homogeneous functions.
是重要的一类关系。
是重要的一类关系。
The foregoing are scaling relations in classical thermodynamics. In more recent times, in statistical mechanics, the expression "scaling laws" has been taken to refer to the homogeneity of form of the thermodynamic and correlation functions near critical points, and to the resulting relations among the exponents that occur in those functions. There are many excellent references for critical phenomena and the associated scaling laws, among them the superb book by Domb [1] and the historic early review by Fisher [2].
The foregoing are scaling relations in classical thermodynamics. In more recent times, in statistical mechanics, the expression "scaling laws" has been taken to refer to the homogeneity of form of the thermodynamic and correlation functions near critical points, and to the resulting relations among the exponents that occur in those functions. There are many excellent references for critical phenomena and the associated scaling laws, among them the superb book by Domb [1] and the historic early review by Fisher [2].
当 <math>2\beta+\gamma=2</math>，则有 <math>\alpha =0</math>，这通常意味着对数发散而不是幂律发散，并且在 <math>t=0+</math> 和 <math>t=0-</math> 之间存在叠加有限不连续。在二维伊辛模型中，仅有对数关系而这种不连续是不存在的；而在平均场近似中情形相反。
当 <math>2\beta+\gamma=2</math>，则有 <math>\alpha =0</math>，这通常意味着对数发散而不是幂律发散，并且在 <math>t=0+</math> 和 <math>t=0-</math> 之间存在叠加有限不连续。在二维伊辛模型中，仅有对数关系而这种不连续是不存在的；而在平均场近似中情形相反。
== Critical exponents 临界指数 ==
=== Critical exponents 临界指数 ===
What were probably the historically earliest versions of critical-point exponent relations like ({{EquationNote|8}}) and ({{EquationNote|9}}) are due to Rice [10] and to Scott [11]. It was before Domb and Sykes [12] and Fisher [13] had noted that the exponent <math>\gamma</math> was in reality greater than its mean-field value <math>\gamma =1</math> but when it was already clear from Guggenheim's corresponding-states analysis [14] that <math>\beta</math> had a value much closer to 1/3 than to its mean-field value of 1/2. Then, under the assumption <math>\gamma
What were probably the historically earliest versions of critical-point exponent relations like ({{EquationNote|8}}) and ({{EquationNote|9}}) are due to Rice [10] and to Scott [11]. It was before Domb and Sykes [12] and Fisher [13] had noted that the exponent <math>\gamma</math> was in reality greater than its mean-field value <math>\gamma =1</math> but when it was already clear from Guggenheim's corresponding-states analysis [14] that <math>\beta</math> had a value much closer to 1/3 than to its mean-field value of 1/2. Then, under the assumption <math>\gamma
another scaling relation [16,17].
another scaling relation [16,17].
== Exponents and space dimension 临界点指数与空间维度 ==
=== 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
这一行为也反映在重整化群理论中。最简单的情形是，重整化群流中有两个相互竞争的不动点，一点与依赖 <math>d</math> 的指数相关，同时满足与 <math>d</math> 无关的标度关系和超标度关系，另一点则与平均场理论的 <math>d</math> 无关指数相关。前者决定了当 <math>d<4\ </math> 时的主导临界点行为。<math>d=4</math> 时，这两个不动点重合，指数现在是平均场理论的指数，但在平均场幂律中增加了对数因子。对于 <math>d>4</math>，两固定点再次分开，此时主导临界点行为源自平均场理论的指数。综上所述，两固定点产生的影响覆盖所有 <math>d\ </math>的取值范围，但是随着 <math>d\ </math>取值的变化，主导临界点行为会在二者之间切换。
这一行为也反映在重整化群理论中。最简单的情形是，重整化群流中有两个相互竞争的不动点，一点与依赖 <math>d</math> 的指数相关，同时满足与 <math>d</math> 无关的标度关系和超标度关系，另一点则与平均场理论的 <math>d</math> 无关指数相关。前者决定了当 <math>d<4\ </math> 时的主导临界点行为。<math>d=4</math> 时，这两个不动点重合，指数现在是平均场理论的指数，但在平均场幂律中增加了对数因子。对于 <math>d>4</math>，两固定点再次分开，此时主导临界点行为源自平均场理论的指数。综上所述，两固定点产生的影响覆盖所有 <math>d\ </math>的取值范围，但是随着 <math>d\ </math>取值的变化，主导临界点行为会在二者之间切换。
==Origin of homogeneity; block spins 齐次性的成因与块自旋==
===Origin of homogeneity; block spins 齐次性的成因与块自旋===
A physical explanation for the homogeneity in ({{EquationNote|1=7}}) and ({{EquationNote|1=10}}) and for the exponent relations that are
A physical explanation for the homogeneity in ({{EquationNote|1=7}}) and ({{EquationNote|1=10}}) and for the exponent relations that are