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→Introduction 引言
[[Category:Statistical Mechanics]]
[[Category:Statistical Mechanics]]
<strong><nowiki>Scaling laws </nowiki></strong> are the expression of
<strong><nowiki>Scaling laws </nowiki></strong> are the expression of physical principles in the mathematical language of homogeneous functions.
physical principles in the mathematical language of homogeneous
functions.
标度律是物理原理在齐次函数数学语言中的表达。
==Introduction 引言==
==Introduction 引言==
x, \lambda y, \lambda z, \ldots) \equiv \lambda^{n}f (x, y, z,
x, \lambda y, \lambda z, \ldots) \equiv \lambda^{n}f (x, y, z,
\ldots). </math>
\ldots). </math>
{{NumBlk|2=<math>f(\lambda
如果对所有<math>\lambda\ ,</math>都满足关系{{NumBlk|2=<math>f(\lambda
x, \lambda y, \lambda z, \ldots) \equiv \lambda^{n}f (x, y, z,
x, \lambda y, \lambda z, \ldots) \equiv \lambda^{n}f (x, y, z,
\ldots). </math>|3={{EquationRef|1}}|：}}
\ldots). </math>|3={{EquationRef|1}}|：}}
则称函数<math>f (x, y, z,\ldots)</math>是变量<math>x,y,z,\ldots</math>的<math>n</math>次齐次函数。For example, <math>ax^2 + bxy + cy^2</math>
is homogeneous of degree 2 in <math>x</math> and <math>y</math> and of
is homogeneous of degree 2 in <math>x</math> and <math>y</math> and of
the first degree in <math>a, b,</math> and <math>c\ .</math>
the first degree in <math>a, b,</math> and <math>c\ .</math>
例如，<math>ax^2 + bxy + cy^2</math>是<math>x</math>和<math>y</math>二次齐次函数，而对<math>a, b,</math><math>c\ .</math>则是一次齐次。
By setting <math>\lambda = 1/x</math> in ({{EquationNote|1}}) we have
By setting <math>\lambda = 1/x</math> in ({{EquationNote|1}}) we have
f(x, y, z, \ldots) = x^nf(1, y/x,
f(x, y, z, \ldots) = x^nf(1, y/x,
z/x, \ldots) \equiv x^n\phi(y/x, z/x, \ldots); </math>
z/x, \ldots) \equiv x^n\phi(y/x, z/x, \ldots); </math>
{{NumBlk|2=<math>f(x, y, z, \ldots) = x^nf(1, y/x,
将<math>\lambda = 1/x</math>带入({{EquationNote|1}})，则有齐次性的另一种表达式，如果<math>f (x, y, z,
\ldots)</math>满足关系：{{NumBlk|2=<math>f(x, y, z, \ldots) = x^nf(1, y/x,
z/x, \ldots) \equiv x^n\phi(y/x, z/x, \ldots);</math>|3={{EquationRef|2}}|：}}
z/x, \ldots) \equiv x^n\phi(y/x, z/x, \ldots);</math>|3={{EquationRef|2}}|：}}
则它是<math>x, y,
z, \ldots</math>的<math>n</math>次齐次函数。
i.e., the <math>n^{th}</math> power of <math>x</math> times some
i.e., the <math>n^{th}</math> power of <math>x</math> times some