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第257行：
第257行： − + 第263行：
第263行： − 如果与这个自由度相关的能量项可以写成二次方程，则该自由度为二次方程+ + − − <math>E = \alpha_i\,\,X_i^2 + \beta_i \,\, X_i Y </math>, − − 数学 e = alpha _ i，，x _ i ^ 2 + beta _ i，，x _ i y, − 第277行：
第273行： − 其他二次自由度的线性组合。+ 第285行：
第281行： − 例如: 如果和是两个自由度，关联的能量是:+ + + − :* If <math>E = X_1^4 + X_1^3 X_2 + X_2^4</math>, then the two degrees of freedom are not independent and non-quadratic.+ − + + + − :* If <math>E = X_1^4 + X_2^4</math>, then the two degrees of freedom are independent and non-quadratic.+ − + + + − :* If <math>E = X_1^2 + X_1 X_2 + 2X_2^2</math>, then the two degrees of freedom are not independent but are quadratic.+ − + + + − :* If <math>E = X_1^2 + 2X_2^2</math>, then the two degrees of freedom are independent and quadratic.+ − + 第317行：
第321行： − 例如，在牛顿运动定律，二次自由度系统的动力学是由一组常系数齐次线性微分方程组控制的。+ − + 第327行：
第331行： − 是二次自由度和独立自由度，如果与它们所代表的系统的微观状态相关的能量可以写成:+ − :<math>E = \sum_{i=1}^N \alpha_i X_i^2</math> − < math > e = sum { i = 1} ^ n alpha _ i x _ i ^ 2 − + − 第343行：
第344行： − 在统计力学的经典极限下，在热力学平衡，一个二次独立自由度系统的内能是:+ − − : <math>U = \langle E \rangle = N\,\frac{k_B T}{2}</math> − − [数学] u = langle e rangle = n，frac { k _ b t }{2}[数学] 第357行：
第354行： − 在这里，与一定自由度相关的平均能是:+ + − − <math>\langle E_i \rangle = \int dX_i\,\,\alpha_i X_i^2\,\, p_i(X_i) = \frac{\int dX_i\,\,\alpha_i X_i^2\,\, e^{-\frac{\alpha_i X_i^2}{k_B T}}}{\int dX_i\,\, e^{-\frac{\alpha_i X_i^2}{k_B T}}} </math> − − 2，，p _ i (xi) = frac { int dX _ i，，alpha _ i x _ i ^ 2，，e ^ {-frac { alpha _ i x _ i ^ 2，，e ^ {-frac { i x _ i ^ 2}}{ k _ b t }}{ int dX _ i，e ^ {-frac { alpha _ i x _ i ^ 2}}{ k _ b _ i，e ^ {-frac { i x _ i ^ 2}{ k _ b _ t }} </math > − − <math>\langle E_i \rangle = \frac{k_B T}{2}\frac{\int dx\,\,x^2\,\, e^{-\frac{x^2}{2}}}{\int dx\,\, e^{-\frac{x^2}{2}}} = \frac{k_B T}{2} </math> − − {2} frac { int dx，，x ^ 2，，e ^ {-frac { x ^ 2}{2}}}{ int dx，，e ^ {-frac { x ^ 2}{2}}}{ e ^ {-frac { x ^ 2}{2}}}}} = frac { k _ b }{2} </math > 第377行：
第367行： − 由于自由度是独立的，系统的内能等于与每个自由度相关的平均能量之和，从而证明了这一结果。+
→Quadratic degrees of freedom
:<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 二次自由度 ==
A degree of freedom {{mvar|X<sub>i</sub>}} is quadratic if the energy terms associated with this degree of freedom can be written as
A degree of freedom {{mvar|X<sub>i</sub>}} is quadratic if the energy terms associated with this degree of freedom can be written as
A degree of freedom is quadratic if the energy terms associated with this degree of freedom can be written as
A degree of freedom is quadratic if the energy terms associated with this degree of freedom can be written as
如果与该自由度相关的能量项可以写成如下等式，则自由度{{mvar|X<sub>i</sub>}}是二次的
:<math>E = \alpha_i\,\,X_i^2 + \beta_i \,\, X_i Y </math>,
:<math>E = \alpha_i\,\,X_i^2 + \beta_i \,\, X_i Y </math>,
where is a linear combination of other quadratic degrees of freedom.
where is a linear combination of other quadratic degrees of freedom.
其中{{mvar|Y}}是其他二次自由度的线性组合。
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，而E是相关的能量：
* If <math>E = X_1^4 + X_1^3 X_2 + X_2^4</math>, then the two degrees of freedom are not independent and non-quadratic.
* If <math>E = X_1^4 + X_1^3 X_2 + X_2^4</math>, then the two degrees of freedom are not independent and non-quadratic.
* If <math>E = X_1^4 + X_1^3 X_2 + X_2^4</math>, then the two degrees of freedom are not independent and non-quadratic.
* 如果 < math > e = x _ 1 ^ 4 + x _ 1 ^ 3 x _ 2 + x _ 2 ^ 4 </math > ，那么这两个自由度不是独立的和非二次的。
* 如果<math>E = X_1^4 + X_1^3 X_2 + X_2^4</math>，则两个自由度既不是独立的也非二次的。
* If <math>E = X_1^4 + X_2^4</math>, then the two degrees of freedom are independent and non-quadratic.
* If <math>E = X_1^4 + X_2^4</math>, then the two degrees of freedom are independent and non-quadratic.
* If <math>E = X_1^4 + X_2^4</math>, then the two degrees of freedom are independent and non-quadratic.
* 如果 < math > e = x _ 1 ^ 4 + x _ 2 ^ 4 </math > ，那么这两个自由度是独立的和非二次的。
* 如果<math>E = X_1^4 + X_2^4</math>，则两个自由度是独立的但非二次的。
* If <math>E = X_1^2 + X_1 X_2 + 2X_2^2</math>, then the two degrees of freedom are not independent but are quadratic.
* If <math>E = X_1^2 + X_1 X_2 + 2X_2^2</math>, then the two degrees of freedom are not independent but are quadratic.
* If <math>E = X_1^2 + X_1 X_2 + 2X_2^2</math>, then the two degrees of freedom are not independent but are quadratic.
* 如果 < math > e = x 1 ^ 2 + x 1 x 2 + 2 x 2 ^ 2 </math > ，那么这两个自由度不是独立的，而是二次的。
* 如果<math>E = X_1^2 + X_1 X_2 + 2X_2^2</math>，则两个自由度不是独立的，而是二次的。
* If <math>E = X_1^2 + 2X_2^2</math>, then the two degrees of freedom are independent and quadratic.
* If <math>E = X_1^2 + 2X_2^2</math>, then the two degrees of freedom are independent and quadratic.
* If <math>E = X_1^2 + 2X_2^2</math>, then the two degrees of freedom are independent and quadratic.
* 如果 < math > e = x _ 1 ^ 2 + 2 x _ 2 ^ 2 </math > ，那么这两个自由度是独立的和二次的。
* 如果<math>E = X_1^2 + 2X_2^2</math>，则两个自由度既是独立的并且是二次的。
For example, in Newtonian mechanics, the dynamics of a system of quadratic degrees of freedom are controlled by a set of homogeneous linear differential equations with constant coefficients.
For example, in Newtonian mechanics, the dynamics of a system of quadratic degrees of freedom are controlled by a set of homogeneous linear differential equations with constant coefficients.
例如，在牛顿力学中，一个二次自由度的动力学系统是由一组具有恒定系数的齐次线性微分方程控制的。
===Quadratic and independent degree of freedom===
=== Quadratic and independent degree of freedom 二次独立的自由度 ===
{{math|''X''<sub>1</sub>, ... , ''X''<sub>''N''</sub>}} are quadratic and independent degrees of freedom if the energy associated with a microstate of the system they represent can be written as:
{{math|''X''<sub>1</sub>, ... , ''X''<sub>''N''</sub>}} are quadratic and independent degrees of freedom if the energy associated with a microstate of the system they represent can be written as:
are quadratic and independent degrees of freedom if the energy associated with a microstate of the system they represent can be written as:
are quadratic and independent degrees of freedom if the energy associated with a microstate of the system they represent can be written as:
如果能量与系统的微观状态有关，则{{math|''X''<sub>1</sub>, ... , ''X''<sub>''N''</sub>}}是二次独立的自由度，它们可以写成：
<math>E = \sum_{i=1}^N \alpha_i X_i^2</math>
<math>E = \sum_{i=1}^N \alpha_i X_i^2</math>
=== Equipartition theorem 能量均分定理 ===
===Equipartition theorem===
In the classical limit of [[statistical mechanics]], at [[thermodynamic equilibrium]], the [[internal energy]] of a system of {{mvar|N}} quadratic and independent degrees of freedom is:
In the classical limit of [[statistical mechanics]], at [[thermodynamic equilibrium]], the [[internal energy]] of a system of {{mvar|N}} quadratic and independent degrees of freedom is:
In the classical limit of statistical mechanics, at thermodynamic equilibrium, the internal energy of a system of quadratic and independent degrees of freedom is:
In the classical limit of statistical mechanics, at thermodynamic equilibrium, the internal energy of a system of quadratic and independent degrees of freedom is:
在'''<font color="#ff8000"> 统计力学Statistical mechanics</font>'''的经典极限中，在热力学平衡状态下，N个二次且独立自由度的系统内部能量为：
<math>U = \langle E \rangle = N\,\frac{k_B T}{2}</math>
<math>U = \langle E \rangle = N\,\frac{k_B T}{2}</math>
Here, the mean energy associated with a degree of freedom is:
Here, the mean energy associated with a degree of freedom is:
此时，与自由度相关的平均能量为：
:<math>\langle E_i \rangle = \int dX_i\,\,\alpha_i X_i^2\,\, p_i(X_i) = \frac{\int dX_i\,\,\alpha_i X_i^2\,\, e^{-\frac{\alpha_i X_i^2}{k_B T}}}{\int dX_i\,\, e^{-\frac{\alpha_i X_i^2}{k_B T}}} </math>
:<math>\langle E_i \rangle = \int dX_i\,\,\alpha_i X_i^2\,\, p_i(X_i) = \frac{\int dX_i\,\,\alpha_i X_i^2\,\, e^{-\frac{\alpha_i X_i^2}{k_B T}}}{\int dX_i\,\, e^{-\frac{\alpha_i X_i^2}{k_B T}}} </math>
:<math>\langle E_i \rangle = \frac{k_B T}{2}\frac{\int dx\,\,x^2\,\, e^{-\frac{x^2}{2}}}{\int dx\,\, e^{-\frac{x^2}{2}}} = \frac{k_B T}{2} </math>
:<math>\langle E_i \rangle = \frac{k_B T}{2}\frac{\int dx\,\,x^2\,\, e^{-\frac{x^2}{2}}}{\int dx\,\, e^{-\frac{x^2}{2}}} = \frac{k_B T}{2} </math>
Since the degrees of freedom are independent, the internal energy of the system is equal to the sum of the mean energy associated with each degree of freedom, which demonstrates the result.
Since the degrees of freedom are independent, the internal energy of the system is equal to the sum of the mean energy associated with each degree of freedom, which demonstrates the result.
由于自由度是独立的，因此系统的内部能量等于每个自由度相关的平均能量之和，由此得到了结果。
==Generalizations==
==Generalizations==