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→Introduction 引言
f}{\partial y}+z\frac{\partial f}{\partial z}+\cdots \equiv nf.</math>|{{EquationRef|3}}}}
f}{\partial y}+z\frac{\partial f}{\partial z}+\cdots \equiv nf.</math>|{{EquationRef|3}}}}
In [[thermodynamics]], if the scale of a system is merely
In [https://wiki.swarma.org/index.php/%E7%83%AD%E5%8A%9B%E5%AD%A6 thermodynamics], if the scale of a system is merely
increased by a factor <math>\lambda</math> with no change in its
increased by a factor <math>\lambda</math> with no change in its
intensive properties, then all its extensive properties including its
intensive properties, then all its extensive properties including its
and the masses <math>m_1, m_2, \ldots</math> of each of its chemical
and the masses <math>m_1, m_2, \ldots</math> of each of its chemical
constituents are increased by that factor, so the extensive function
constituents are increased by that factor, so the extensive function
<math>S(E, V, m_1, m_2, \ldots)</math> is homogeneous of degree 1 in its extensive arguments:
<math>S(E, V, m_1, m_2, \ldots)</math> is homogeneous of degree 1 in its extensive arguments:
:<math>\label{eq:4}
:<math>\label{eq:4}
{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>
{{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}}}}
With <math>T</math> the temperature, <math>p</math> the
With <math>T</math> the temperature, <math>p</math> the
\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>
{{NumBlk|:|<math>\frac{1}{T} (E + pV - \mu_1m_1 - \mu_2m_2 -
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 -
\cdots) =S,</math>|{{EquationRef|5}}}}
\cdots) =S,</math>|{{EquationRef|5}}}}
an important identity. Any extensive function <math>X(T, p, m_1, m_2, \ldots)\ ,</math> such as the volume V or the Gibbs free energy <math>E+pV-TS\ ,</math> is homogeneous of the first degree in the <math>m_i</math> at fixed <math>p</math> and <math>T\ ,</math> so
an important identity. Any extensive function <math>X(T, p, m_1, m_2, \ldots)\ ,</math> such as the volume V or the Gibbs free energy <math>E+pV-TS\ ,</math> is homogeneous of the first degree in the <math>m_i</math> at fixed <math>p</math> and <math>T\ ,</math> so