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第192行：
第192行： − 系统的自由度集合是独立的，如果与该自由度集合相关的能量可以写成下列形式:+ + − − <math>E = \sum_{i=1}^N E_i(X_i),</math> − − [数学] ，[数学] − 第206行：
第202行： − 这里是唯一变量的函数。+ 第214行：
第210行： − 例如: 如果和是两个自由度，关联的能量是:+ + + 第220行：
第218行： − + + + 第226行：
第226行： − + 第234行：
第234行： − 对于从1到1，第四自由度的值是按照美国波兹曼分布协会的标准分配的。它的概率密度函数如下:+ + − − <math>p_i(X_i) = \frac{e^{-\frac{E_i}{k_B T}}}{\int dX_i \, e^{-\frac{E_i}{k_B T}}}</math>, − − [数学][数学] 第248行：
第245行： − 在本节中，整篇文章中括号 < math > langle rangle </math > 表示它们所包含的量的平均值。+ 第256行：
第253行： − 系统的内能是与每个自由度有关的平均能量之和:+ − − <math>\langle E \rangle = \sum_{i=1}^N \langle E_i \rangle.</math> − − 数学，数学，数学 − −
→Independent degrees of freedom
The set of degrees of freedom of a system is independent if the energy associated with the set can be written in the following form:
The set of degrees of freedom of a system is independent if the energy associated with the set can be written in the following form:
如果系统的自由度{{math|''X''<sub>1</sub>, ... , ''X''<sub>''N''</sub>}}的集合可以用以下形式表示，则它们是独立的：
:<math>E = \sum_{i=1}^N E_i(X_i),</math>
:<math>E = \sum_{i=1}^N E_i(X_i),</math>
where is a function of the sole variable .
where is a function of the sole variable .
其中{{mvar|E<sub>i</sub>}}是唯一变量{{mvar|X<sub>i</sub>}}的函数。
example: if and are two degrees of freedom, and is the associated energy:
example: if and are two degrees of freedom, and is the associated energy:
例如：如果{{math|''X''<sub>1</sub>}} 和 {{math|''X''<sub>2</sub>}}是两个自由度，并且E是关联的能量。
:* If <math>E = X_1^4 + X_2^4</math>, then the two degrees of freedom are independent.
:* If <math>E = X_1^4 + X_2^4</math>, then the two degrees of freedom are independent.
* If <math>E = X_1^4 + X_2^4</math>, then the two degrees of freedom are independent.
* If <math>E = X_1^4 + X_2^4</math>, then the two degrees of freedom are independent.
* 如果 < math > e = x _ 1 ^ 4 + x _ 2 ^ 4 </math > ，那么这两个自由度是独立的。
* 如果<math>E = X_1^4 + X_2^4</math>，则两个自由度是独立的。
:* If <math>E = X_1^4 + X_1 X_2 + X_2^4</math>, then the two degrees of freedom are ''not'' independent. The term involving the product of {{math|''X''<sub>1</sub>}} and {{math|''X''<sub>2</sub>}} is a coupling term that describes an interaction between the two degrees of freedom.
:* If <math>E = X_1^4 + X_1 X_2 + X_2^4</math>, then the two degrees of freedom are ''not'' independent. The term involving the product of {{math|''X''<sub>1</sub>}} and {{math|''X''<sub>2</sub>}} is a coupling term that describes an interaction between the two degrees of freedom.
* If <math>E = X_1^4 + X_1 X_2 + X_2^4</math>, then the two degrees of freedom are not independent. The term involving the product of and is a coupling term that describes an interaction between the two degrees of freedom.
* If <math>E = X_1^4 + X_1 X_2 + X_2^4</math>, then the two degrees of freedom are not independent. The term involving the product of and is a coupling term that describes an interaction between the two degrees of freedom.
* 如果 < math > e = x 1 ^ 4 + x 1 x 2 + x 2 ^ 4 </math > ，那么这两个自由度不是独立的。包含和乘积的术语是描述两个自由度之间相互作用的耦合术语。
* 如果<math>E = X_1^4 + X_1 X_2 + X_2^4</math>，则两个自由度不是独立的。其中{{math|''X''<sub>1</sub>}}和{{math|''X''<sub>2</sub>}}的乘积是描述两个自由度之间相互作用的耦合项。
For from 1 to , the value of the th degree of freedom is distributed according to the Boltzmann distribution. Its probability density function is the following:
For from 1 to , the value of the th degree of freedom is distributed according to the Boltzmann distribution. Its probability density function is the following:
这里的i值可以取1到N，第i个自由度Xi的值依据'''<font color="#ff8000"> 波尔兹曼分布Boltzmann distribution </font>'''。其'''<font color="#ff8000"> 概率密度函数Probability density function</font>'''如下：
: <math>p_i(X_i) = \frac{e^{-\frac{E_i}{k_B T}}}{\int dX_i \, e^{-\frac{E_i}{k_B T}}}</math>,
: <math>p_i(X_i) = \frac{e^{-\frac{E_i}{k_B T}}}{\int dX_i \, e^{-\frac{E_i}{k_B T}}}</math>,
In this section, and throughout the article the brackets <math>\langle \rangle</math> denote the mean of the quantity they enclose.
In this section, and throughout the article the brackets <math>\langle \rangle</math> denote the mean of the quantity they enclose.
在本节以及整篇文章中，方括号<math>\langle \rangle</math>表示其所包含数量的平均值。
The internal energy of the system is the sum of the average energies associated with each of the degrees of freedom:
The internal energy of the system is the sum of the average energies associated with each of the degrees of freedom:
系统的内能是每个自由度相关的平均能量之和：
:<math>\langle E \rangle = \sum_{i=1}^N \langle E_i \rangle.</math>
:<math>\langle E \rangle = \sum_{i=1}^N \langle E_i \rangle.</math>
==Quadratic degrees of freedom==
==Quadratic degrees of freedom==