更改

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>

如果对所有<math>\lambda\ ,</math>都满足关系{{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>  

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。

is homogeneous of degree 2 in <math>x</math> and <math>y</math> and of

 

For example, <math>ax^2 + bxy + cy^2</math> 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。

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

{m_1}, \lambda {m_2}, \ldots ) = \lambda S(E, V, {m_1}, {m_2}, \ldots).

{m_1}, \lambda {m_2}, \ldots ) = \lambda S(E, V, {m_1}, {m_2}, \ldots).

</math>

</math>

在[https://wiki.swarma.org/index.php/%E7%83%AD%E5%8A%9B%E5%AD%A6 热力学 Thermodynamics]中，如果一个系统的标度增加<math>\lambda</math>倍而其强度量不发生变化，则该系统所有化学组分的广度量（如熵<math>S\ ,</math>，能量<math>E\ ,</math>，体积<math>V\ ,</math>，质量<math>m_1, m_2, \ldots</math>等）也增加相同倍数。因此广度函数<math>S(E, V, m_1, m_2, \ldots)</math>在广义论证中满足齐次关系：{{NumBlk|:|<math>S(\lambda E, \lambda V, \lambda

在[https://wiki.swarma.org/index.php/%E7%83%AD%E5%8A%9B%E5%AD%A6 热力学 Thermodynamics]中，如果一个系统的标度增加<math>\lambda</math>倍而其强度量不发生变化，则该系统所有化学组分的广度量（如熵<math>S\ ,</math>能量<math>E\ ,</math>体积<math>V\ ,</math>质量<math>m_1, m_2, \ldots</math>等）也增加相同倍数。因此广度函数<math>S(E, V, m_1, m_2, \ldots)</math>在广义论证中满足齐次关系：{{NumBlk|:|<math>S(\lambda E, \lambda V, \lambda

{m_1}, \lambda {m_2}, \ldots ) = \lambda S(E, V, {m_1}, {m_2}, \ldots).</math>|{{EquationRef|4}}}}

{m_1}, \lambda {m_2}, \ldots ) = \lambda S(E, V, {m_1}, {m_2}, \ldots).</math>|{{EquationRef|4}}}}

\frac{1}{T} (E + pV - \mu_1m_1 - \mu_2m_2 -

\frac{1}{T} (E + pV - \mu_1m_1 - \mu_2m_2 -

\cdots) =S, </math>

\cdots) =S, </math>

以<math>T</math>，<math>p</math>，<math>\mu_i</math>分别表示温度，压力和不同组分<math>i\ ,</math>的化学势，根据热力学关系 <math>\partial

以<math>T</math>，<math>p</math>，<math>\mu_i</math> 分别表示温度，压力和不同组分<math>i\ </math>的化学势，根据热力学关系 <math>\partial

S/\partial E = 1/T\ ,</math> <math>\partial S/\partial V = p/T\ ,</math> 和<math>\partial S/\partial m_i = - \mu_i/T\ ;</math>；再由欧拉定理可得：{{NumBlk|:|<math>\frac{1}{T} (E + pV - \mu_1m_1 - \mu_2m_2 -

S/\partial E = 1/T\ ,</math> <math>\partial S/\partial V = p/T\ ,</math> 和<math>\partial S/\partial m_i = - \mu_i/T\ ;</math>再由欧拉定理可得：{{NumBlk|:|<math>\frac{1}{T} (E + pV - \mu_1m_1 - \mu_2m_2 -

\cdots) =S,</math>|{{EquationRef|5}}}}

\cdots) =S,</math>|{{EquationRef|5}}}}

X = m_1 \frac{\partial X}{\partial

X = m_1 \frac{\partial X}{\partial

m_1} + m_2 \frac{\partial X}{\partial m_2} + \cdots , </math>

m_1} + m_2 \frac{\partial X}{\partial m_2} + \cdots , </math>

任何广度函数<math>X(T, p, m_1, m_2, \ldots)\ ,</math>（如体积<math>V\ ,</math>或者吉布斯自由能<math>E+pV-TS\ ,</math>）在等温等压状态下，对<math>m_i</math>都是一次齐次的，因此{{NumBlk|:|<math>X = m_1 \frac{\partial X}{\partial

任何广度函数<math>X(T, p, m_1, m_2, \ldots)\ ,</math>（如体积<math>V\ </math>或者吉布斯自由能<math>E+pV-TS\ ,</math>）在等温等压状态下，对<math>m_i</math>都是一次齐次的，因此{{NumBlk|:|<math>X = m_1 \frac{\partial X}{\partial

m_1} + m_2 \frac{\partial X}{\partial m_2} + \cdots ,</math>|{{EquationRef|6}}}}

m_1} + m_2 \frac{\partial X}{\partial m_2} + \cdots ,</math>|{{EquationRef|6}}}}

Near the Curie point (critical point) of a ferromagnet, which occurs

Near the Curie point (critical point) of a ferromagnet, which occurs

at <math>T = T_c\ ,</math> the magnetic field <math>H\ ,</math>

at <math>T = T_c\ ， </math> the magnetic field <math>H\ ,</math>

magnetization <math>M\ ,</math> and <math>t = T/T_c-1\ ,</math> are

magnetization <math>M\ ,</math> and <math>t = T/T_c-1\ ,</math> are

related by  

related by  

H = M\mid M\mid ^{\delta-1} j(t/\mid M\mid

H = M\mid M\mid ^{\delta-1} j(t/\mid M\mid

^{1/\beta}) </math>

^{1/\beta}) </math>

当<math>t = T/T_c-1\ ,</math>时，铁磁物质临近居里点（临界点），磁场强度<math>H\ ,</math>，磁化强度<math>M\ ,</math>和<math>t = T/T_c-1\ ,</math>满足{{NumBlk|:|<math>H = M\mid M\mid ^{\delta-1} j(t/\mid M\mid

当<math>t = T/T_c-1\ </math>时，铁磁物质临近居里点（临界点），磁场强度<math>H\,</math>磁化强度<math>M\,</math>和<math>t = T/T_c-1\ </math>满足{{NumBlk|:|<math>H = M\mid M\mid ^{\delta-1} j(t/\mid M\mid

^{1/\beta})</math>|{{EquationRef|7}}}}

^{1/\beta})</math>|{{EquationRef|7}}}}

^{1/\beta}</math> of degree <math>\beta \delta\ .</math>  The scaling function <math>j(x)</math> vanishes proportionally to <math>x+b</math> as <math>x</math> approaches <math>-b\ ,</math> with <math>b</math> a positive constant; it diverges proportionally to <math>x^{\beta(\delta-1)}</math> as <math>x\rightarrow \infty\ ;</math> and <math>j(0) = c\ ,</math> another positive constant (Fig. 1). Although ({{EquationNote|7}}) is confined to the immediate neighborhood of the critical point <math>(t, M, H</math> all near 0), the scaling variable <math>x = t/\mid M\mid ^{1/\beta}</math> nevertheless traverses the infinite range <math>-b < x < \infty\ .</math>

^{1/\beta}</math> of degree <math>\beta \delta\ .</math>  The scaling function <math>j(x)</math> vanishes proportionally to <math>x+b</math> as <math>x</math> approaches <math>-b\ ,</math> with <math>b</math> a positive constant; it diverges proportionally to <math>x^{\beta(\delta-1)}</math> as <math>x\rightarrow \infty\ ;</math> and <math>j(0) = c\ ,</math> another positive constant (Fig. 1). Although ({{EquationNote|7}}) is confined to the immediate neighborhood of the critical point <math>(t, M, H</math> all near 0), the scaling variable <math>x = t/\mid M\mid ^{1/\beta}</math> nevertheless traverses the infinite range <math>-b < x < \infty\ .</math>

其中<math>j(x)</math>是“标度”函数，<math>\beta</math>和<math>\delta</math>是临界点指数。因此由({{EquationNote|2}})和({{EquationNote|7}})，当铁磁物质趋近于临界点时（<math>(H\rightarrow 0</math>且<math>t\rightarrow 0)\ ,</math>），<math>\mid H\mid</math>是<math>t</math>和<math>\mid M\mid

其中<math>j(x)</math>是“标度”函数，<math>\beta</math>和<math>\delta</math>是临界点指数。因此由({{EquationNote|2}})和({{EquationNote|7}})，当铁磁物质趋近于临界点时<math>(H\rightarrow 0</math>且<math>t\rightarrow 0)\ ,</math>，<math>\mid H\mid</math>是<math>t</math>和<math>\mid M\mid

^{1/\beta}</math>的<math>\beta \delta\ .</math>次齐次函数。当<math>x</math>趋近于<math>-b\ ,</math>（正常数）时，标度函数<math>j(x)</math>趋近于零；当<math>x\rightarrow \infty\ ;</math>时，它发散至<math>x^{\beta(\delta-1)}</math>，且<math>j(0) = c\ </math>（正常数）（如图一）。尽管({{EquationNote|7}})局限在临界点<math>(t, M, H</math>都接近零）附近的极小范围内，但标度变量<math>x = t/\mid M\mid ^{1/\beta}</math>却遍历<math>-b < x < \infty\ .</math>的无穷范围。

^{1/\beta}</math>的<math>\beta \delta\ </math>次齐次函数。当<math>x</math>趋近于<math>-b\</math>（正常数）时，标度函数<math>j(x)</math>趋近于零；当<math>x\rightarrow \infty\ ;</math>时，它发散至<math>x^{\beta(\delta-1)}</math>，且<math>j(0) = c\ </math>（正常数）（如图一）。尽管({{EquationNote|7}})局限在临界点<math>(t, M, H</math>都接近零）附近的极小范围内，但标度变量<math>x = t/\mid M\mid ^{1/\beta}</math>却遍历<math>-b < x < \infty\</math>的无穷范围。

[[Image:scaling_laws_widom_nocaption_Fig1.png|thumb|300px|right|Scaling function  

[[Image:scaling_laws_widom_nocaption_Fig1.png|thumb|300px|right|Scaling function