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第1行： + − + − − 第11行：
第10行： − 在物理和化学领域中，'''<font color="#ff8000"> 自由度Degree of freedom</font>'''是对物理系统状态形式描述的独立物理参数。系统所有状态的集合称为系统的'''<font color="#ff8000"> 相空间Phase space</font>'''，系统的自由度是相空间的维数。+ − − 第19行：
第16行： − 一个粒子在三维空间中的位置需要三个位置坐标进行标识。类似地，我们也可以根据三个速度分量来描述粒子运动的方向和速度，每个速度分量都参考该空间的三个维度。如果系统的时间演变是确定的，那么其中某个瞬间以及其过去和未来瞬间的位置和速度，均可以被该瞬间的状态确定为时间函数，相应地，这种系统具有六个自由度。如果将粒子的运动轨迹限制在更小的维度里面，例如，粒子必须沿着电线或在固定的表面上移动，那么系统的自由度则小于6。另一方面，如果一个系统带有可旋转或振动的扩展对象，那么它的自由度将大于6。+ − + − + 第27行：
第24行： − + − − 第35行：
第30行： − 在统计力学中，自由度是描述系统微观状态的单个标量数。一个系统所有微观状态的规格参数都是基于该系统相空间中的一个点。+ − − 第43行：
第36行： − + − − 第51行：
第42行： − 指定二次自由度通常很重要。因为它们有助于建立系统能量的二次函数。+ − − 第63行：
第52行： − 在三维空间中，粒子的运动与它的三个自由度有关。双原子气体分子具有6个自由度（存在争议）。因此可以根据其分子的平移，旋转和振动来分解该运动集合。整个分子的质心运动具有3个自由度，除此之外，还具有两个旋转自由度和一个[存在争议]振动自由度。其中旋转运动是围绕垂直于两个原子之间的连线轴发生的。值得注意的是这里围绕原子和原子键的旋转并不是物理旋转[存在争议]。因此对于双原子分子，可以将其分解为：+ − − 第73行：
第60行： − 对于一般的非线性分子，其所有的3个旋转自由度均需考虑，因此它的分解形式为：+ − − 第83行：
第68行： − + − − 第91行：
第74行： − 如上所述，还可以使用指定维度空间所需的最少坐标数来确定自由度。比如：+ − − 第100行：
第81行： − − 第107行：
第86行： − 在三维空间中，由2个粒子（例如双原子分子）组成且彼此之间具有恒定距离（假设d）的物体，我们可以（在下面）显示其自由度为5。+ − − 第116行：
第93行： − − 第126行：
第101行： − − − 第135行：
第107行： − − 第142行：
第112行： − + − − 第151行：
第119行： − 第192行：
第159行： − 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：+ − 第203行：
第169行： − − 第211行：
第175行： − − 第219行：
第181行： − − 第227行：
第187行： − − 第235行：
第193行： − − − 第246行：
第201行： − − 第264行：
第217行： − − 第274行：
第225行： − − 第281行：
第230行： − + − − 第290行：
第237行： − − 第298行：
第243行： − − 第306行：
第249行： − − 第314行：
第255行： − − 第321行：
第260行： − 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。+ − − 第332行：
第269行： − 第342行：
第278行： − 在'''<font color="#ff8000"> 统计力学Statistical mechanics</font>'''的经典极限中，在'''<font color="#ff8000"> 热力学平衡Thermodynamic equilibrium</font>'''状态下，N个二次且独立自由度的系统内部能量为：+ − − 第353行：
第287行： − − − 第365行：
第296行： − 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。+ 第373行：
第304行： − 尽管在数学上很方便，但将系统状态描述为相空间中的一个点，从根本上讲是不准确的。在'''<font color="#ff8000"> 量子力学Quantum mechanics</font>'''中，体系运动状态的自由度被波函数的概念所取代，并且对应于其他自由度的'''<font color="#ff8000"> 算子Operator</font>'''具有离散的光谱。例如，电子或光子的本征'''<font color="#ff8000"> 角动量算符Angular momentum operator </font>'''（对应于旋转自由度）只有两个特征值。当运动具有'''<font color="#ff8000"> 普朗克常数Planck constant</font>'''的量级时，这种离散变得非常明显，并且可以区分出各个自由度。+ − − − − − − − − − − − − − − − − − − − −
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已由 Sizuka 初次审校
{{about|physics and chemistry|other fields|Degrees of freedom}}
{{about|physics and chemistry|other fields|Degrees of freedom}}
{{Refimprove|date=November 2009}}
{{Refimprove|date=November 2009}
In [[physics]] and [[chemistry]], a '''degree of freedom''' is an independent physical [[parameter]] in the formal description of the state of a [[physical system]]. The set of all states of a system is known as the system's [[phase space]], and the degrees of freedom of the system are the [[dimension]]s of the phase space.
In [[physics]] and [[chemistry]], a '''degree of freedom''' is an independent physical [[parameter]] in the formal description of the state of a [[physical system]]. The set of all states of a system is known as the system's [[phase space]], and the degrees of freedom of the system are the [[dimension]]s of the phase space.
In physics and chemistry, a degree of freedom is an independent physical parameter in the formal description of the state of a physical system. The set of all states of a system is known as the system's phase space, and the degrees of freedom of the system are the dimensions of the phase space.
In physics and chemistry, a degree of freedom is an independent physical parameter in the formal description of the state of a physical system. The set of all states of a system is known as the system's phase space, and the degrees of freedom of the system are the dimensions of the phase space.
在物理和化学中，'''<font color="#ff8000"> 自由度Degree of freedom</font>'''指的是形式化描述某一物理系统的状态时，一个独立的物理量。某一系统所有状态的集合称为该系统的'''<font color="#ff8000"> 相空间Phase space</font>'''，该系统的自由度便是相空间的维数。
The location of a particle in three-dimensional space requires three position coordinates. Similarly, the direction and speed at which a particle moves can be described in terms of three velocity components, each in reference to the three dimensions of space. If the time evolution of the system is deterministic, where the state at one instant uniquely determines its past and future position and velocity as a function of time, such a system has six degrees of freedom.{{citation needed|date=November 2015}} If the motion of the particle is constrained to a lower number of dimensions – for example, the particle must move along a wire or on a fixed surface – then the system has fewer than six degrees of freedom. On the other hand, a system with an extended object that can rotate or vibrate can have more than six degrees of freedom.
The location of a particle in three-dimensional space requires three position coordinates. Similarly, the direction and speed at which a particle moves can be described in terms of three velocity components, each in reference to the three dimensions of space. If the time evolution of the system is deterministic, where the state at one instant uniquely determines its past and future position and velocity as a function of time, such a system has six degrees of freedom.{{citation needed|date=November 2015}} If the motion of the particle is constrained to a lower number of dimensions – for example, the particle must move along a wire or on a fixed surface – then the system has fewer than six degrees of freedom. On the other hand, a system with an extended object that can rotate or vibrate can have more than six degrees of freedom.
The location of a particle in three-dimensional space requires three position coordinates. Similarly, the direction and speed at which a particle moves can be described in terms of three velocity components, each in reference to the three dimensions of space. If the time evolution of the system is deterministic, where the state at one instant uniquely determines its past and future position and velocity as a function of time, such a system has six degrees of freedom. If the motion of the particle is constrained to a lower number of dimensions – for example, the particle must move along a wire or on a fixed surface – then the system has fewer than six degrees of freedom. On the other hand, a system with an extended object that can rotate or vibrate can have more than six degrees of freedom.
The location of a particle in three-dimensional space requires three position coordinates. Similarly, the direction and speed at which a particle moves can be described in terms of three velocity components, each in reference to the three dimensions of space. If the time evolution of the system is deterministic, where the state at one instant uniquely determines its past and future position and velocity as a function of time, such a system has six degrees of freedom. If the motion of the particle is constrained to a lower number of dimensions – for example, the particle must move along a wire or on a fixed surface – then the system has fewer than six degrees of freedom. On the other hand, a system with an extended object that can rotate or vibrate can have more than six degrees of freedom.
'''<font color="#ff8000"> 粒子 particle </font>'''在'''<font color="#ff8000">三维 three-dimensional </font>'''空间中的位置需要三个位置坐标来标识。
同样地，我们也能够根据三个速度分量来描述粒子运动的方向和速度，每个速度分量都参照该空间的三个维度。
如果系统的时间演变是确定的，其中某一时刻的状态唯一地确定其过去和未来的位置和速度随时间的变化，则该系统具有六个自由度。如果将粒子的运动限制在更低的维度，例如，粒子必须沿着一条线或在固定的表面上移动，那么系统的自由度则小于6。另一方面，如果一个系统带有可旋转或振动的扩展对象，那么它的自由度将大于6。
In [[classical mechanics]], the state of a [[point particle]] at any given time is often described with position and velocity coordinates in the [[Lagrangian mechanics|Lagrangian]] formalism, or with position and momentum coordinates in the [[Hamiltonian (quantum mechanics)|Hamiltonian]] formalism.
In [[classical mechanics]], the state of a [[point particle]] at any given time is often described with position and velocity coordinates in the [[Lagrangian mechanics|Lagrangian]] formalism, or with position and momentum coordinates in the [[Hamiltonian (quantum mechanics)|Hamiltonian]] formalism.
In classical mechanics, the state of a point particle at any given time is often described with position and velocity coordinates in the Lagrangian formalism, or with position and momentum coordinates in the Hamiltonian formalism.
In classical mechanics, the state of a point particle at any given time is often described with position and velocity coordinates in the Lagrangian formalism, or with position and momentum coordinates in the Hamiltonian formalism.
在经典力学中，任何给定时间下'''<font color="#ff8000"> 质点Point particle</font>'''的状态，不同的力学形式会有不一样的描述，在拉格朗日力学中描述为位置和速度坐标，而在哈密顿力学中则描述为位置和动量坐标。
在经典力学中，任何给定时间下'''<font color="#ff8000"> 质点Point particle</font>'''的状态，通常是用'''<font color="#ff8000"> 拉格朗日形式Lagrangian formalism </font>'''中的位置和速度坐标来描述，或者用'''<font color="#ff8000"> 哈密顿形式Hamiltonian formalism </font>'''中的位置和动量坐标来描述。
In [[statistical mechanics]], a degree of freedom is a single [[scalar (physics)|scalar]] number describing the [[microstate (statistical mechanics)|microstate]] of a system.<ref name=":0">{{cite book |last=Reif |first=F. |title=Fundamentals of Statistical and Thermal Physics |year=2009 |publisher=Waveland Press, Inc. |location=Long Grove, IL |isbn=1-57766-612-7 |page=51}}</ref> The specification of all microstates of a system is a point in the system's [[phase space]].
In [[statistical mechanics]], a degree of freedom is a single [[scalar (physics)|scalar]] number describing the [[microstate (statistical mechanics)|microstate]] of a system.<ref name=":0">{{cite book |last=Reif |first=F. |title=Fundamentals of Statistical and Thermal Physics |year=2009 |publisher=Waveland Press, Inc. |location=Long Grove, IL |isbn=1-57766-612-7 |page=51}}</ref> The specification of all microstates of a system is a point in the system's [[phase space]].
In statistical mechanics, a degree of freedom is a single scalar number describing the microstate of a system. The specification of all microstates of a system is a point in the system's phase space.
In statistical mechanics, a degree of freedom is a single scalar number describing the microstate of a system. The specification of all microstates of a system is a point in the system's phase space.
在统计力学中，自由度指的是在描述某个系统的微观状态时，单个的标量数。<ref name=":0">{{cite book |last=Reif |first=F. |title=Fundamentals of Statistical and Thermal Physics |year=2009 |publisher=Waveland Press, Inc. |location=Long Grove, IL |isbn=1-57766-612-7 |page=51}}</ref>某个系统所有微观状态的规格参数都是该系统相空间中的一个点。
In the 3D [[ideal chain]] model in chemistry, two [[angle]]s are necessary to describe the orientation of each monomer.
In the 3D [[ideal chain]] model in chemistry, two [[angle]]s are necessary to describe the orientation of each monomer.
In the 3D ideal chain model in chemistry, two angles are necessary to describe the orientation of each monomer.
In the 3D ideal chain model in chemistry, two angles are necessary to describe the orientation of each monomer.
在化学的三维'''<font color="#ff8000"> 理想链Ideal chain</font>'''模型中，描述每个单元结构方向的必要参数是它们的两个角度。
在化学的三维'''<font color="#ff8000"> 理想链Ideal chain</font>'''模型中，，需要两个角度来描述每个'''<font color="#ff8000">单体monomer </font>'''的方向。
It is often useful to specify quadratic degrees of freedom. These are degrees of freedom that contribute in a quadratic function to the energy of the system.
It is often useful to specify quadratic degrees of freedom. These are degrees of freedom that contribute in a quadratic function to the energy of the system.
It is often useful to specify quadratic degrees of freedom. These are degrees of freedom that contribute in a quadratic function to the energy of the system.
It is often useful to specify quadratic degrees of freedom. These are degrees of freedom that contribute in a quadratic function to the energy of the system.
指定二自由度通常很有用。因为它们有助于建立系统能量的二次函数。
== Gas molecules 气体分子 ==
== Gas molecules 气体分子 ==
In three-dimensional space, three degrees of freedom are associated with the movement of a particle. A diatomic gas molecule has 6 degrees of freedom. This set may be decomposed in terms of translations, rotations, and vibrations of the molecule. The center of mass motion of the entire molecule accounts for 3 degrees of freedom. In addition, the molecule has two rotational degrees of motion and one vibrational mode. The rotations occur around the two axes perpendicular to the line between the two atoms. The rotation around the atom–atom bond is not a physical rotation. This yields, for a diatomic molecule, a decomposition of:
In three-dimensional space, three degrees of freedom are associated with the movement of a particle. A diatomic gas molecule has 6 degrees of freedom. This set may be decomposed in terms of translations, rotations, and vibrations of the molecule. The center of mass motion of the entire molecule accounts for 3 degrees of freedom. In addition, the molecule has two rotational degrees of motion and one vibrational mode. The rotations occur around the two axes perpendicular to the line between the two atoms. The rotation around the atom–atom bond is not a physical rotation. This yields, for a diatomic molecule, a decomposition of:
在三维空间中，粒子的运动与它的三个自由度有关。双原子气体分子具有6个自由度。我们可以根据其分子的平动、转动和振动来分解这6个自由度。整个分子的质心运动具有三个自由度。'''<font color="#32CD32">除此之外，分子还有两个转动自由度和一个振动自由度</font>'''。--[[用户:Sizuka|Sizuka]]（[[用户讨论:Sizuka|不太确定] rotational degrees of motion和vibrational mode的含义]）。其中，这两个转动自由度，是在绕垂直于两个原子间直线的两个轴而发生的。但是，绕原子-原子键的转动并不是物理旋转。因此，在此种情况下，可以将双原子分子的自由度分解为：
:<math>N = 6 = 3 + 2 + 1.</math>
:<math>N = 6 = 3 + 2 + 1.</math>
For a general, non-linear molecule, all 3 rotational degrees of freedom are considered, resulting in the decomposition:
For a general, non-linear molecule, all 3 rotational degrees of freedom are considered, resulting in the decomposition:
For a general, non-linear molecule, all 3 rotational degrees of freedom are considered, resulting in the decomposition:
For a general, non-linear molecule, all 3 rotational degrees of freedom are considered, resulting in the decomposition:
对于一般的非线性分子，则需要考虑该分子的三个转动自由度，因此它的分解形式为：
:<math>3N = 3 + 3 + (3N - 6)</math>
:<math>3N = 3 + 3 + (3N - 6)</math>
which means that an {{mvar|N}}-atom molecule has {{math|3''N'' − 6}} vibrational degrees of freedom for {{math|''N'' > 2}}. In special cases, such as adsorbed large molecules, the rotational degrees of freedom can be limited to only one.<ref>{{cite journal|doi=10.1002/cphc.201200531|pmid=23047526|title=Stabilization of Large Adsorbates by Rotational Entropy: A Time-Resolved Variable-Temperature STM Study|journal=ChemPhysChem|volume=14|issue=1|pages=162–9|year=2013|last1=Waldmann|first1=Thomas|last2=Klein|first2=Jens|last3=Hoster|first3=Harry E.|last4=Behm|first4=R. Jürgen}}</ref>
which means that an {{mvar|N}}-atom molecule has {{math|3''N'' − 6}} vibrational degrees of freedom for {{math|''N'' > 2}}. In special cases, such as adsorbed large molecules, the rotational degrees of freedom can be limited to only one.<ref>{{cite journal|doi=10.1002/cphc.201200531|pmid=23047526|title=Stabilization of Large Adsorbates by Rotational Entropy: A Time-Resolved Variable-Temperature STM Study|journal=ChemPhysChem|volume=14|issue=1|pages=162–9|year=2013|last1=Waldmann|first1=Thomas|last2=Klein|first2=Jens|last3=Hoster|first3=Harry E.|last4=Behm|first4=R. Jürgen}}</ref>
which means that an -atom molecule has vibrational degrees of freedom for . In special cases, such as adsorbed large molecules, the rotational degrees of freedom can be limited to only one.
which means that an -atom molecule has vibrational degrees of freedom for . In special cases, such as adsorbed large molecules, the rotational degrees of freedom can be limited to only one.
这意味着当N>2时，N原子分子具有3N-6个振动自由度。不过在特殊情况下，例如一个吸附的大分子，旋转自由度只能限制为一个。
这意味着当N>2时，N原子分子具有3N-6个振动自由度。不过在特殊情况下，例如一个吸附的大分子，转动自由度只能限制为一个。<ref>{{cite journal|doi=10.1002/cphc.201200531|pmid=23047526|title=Stabilization of Large Adsorbates by Rotational Entropy: A Time-Resolved Variable-Temperature STM Study|journal=ChemPhysChem|volume=14|issue=1|pages=162–9|year=2013|last1=Waldmann|first1=Thomas|last2=Klein|first2=Jens|last3=Hoster|first3=Harry E.|last4=Behm|first4=R. Jürgen}}</ref>
As defined above one can also count degrees of freedom using the minimum number of coordinates required to specify a position. This is done as follows:
As defined above one can also count degrees of freedom using the minimum number of coordinates required to specify a position. This is done as follows:
As defined above one can also count degrees of freedom using the minimum number of coordinates required to specify a position. This is done as follows:
As defined above one can also count degrees of freedom using the minimum number of coordinates required to specify a position. This is done as follows:
如上所述，还可以使用指定维度空间所需的最少坐标数来计算自由度。比如：
# For a single particle we need 2 coordinates in a 2-D plane to specify its position and 3 coordinates in 3-D space. Thus its degree of freedom in a 3-D space is 3.
# For a single particle we need 2 coordinates in a 2-D plane to specify its position and 3 coordinates in 3-D space. Thus its degree of freedom in a 3-D space is 3.
对于单个粒子，我们需要在二维平面中指定2个坐标，在三维空间中指定3个坐标。因此，它在三维空间中的自由度为3。
对于单个粒子，我们需要在二维平面中指定2个坐标，在三维空间中指定3个坐标。因此，它在三维空间中的自由度为3。
# For a body consisting of 2 particles (ex. a diatomic molecule) in a 3-D space with constant distance between them (let's say d) we can show (below) its degrees of freedom to be 5.
# For a body consisting of 2 particles (ex. a diatomic molecule) in a 3-D space with constant distance between them (let's say d) we can show (below) its degrees of freedom to be 5.
For a body consisting of 2 particles (ex. a diatomic molecule) in a 3-D space with constant distance between them (let's say d) we can show (below) its degrees of freedom to be 5.
For a body consisting of 2 particles (ex. a diatomic molecule) in a 3-D space with constant distance between them (let's say d) we can show (below) its degrees of freedom to be 5.
在三维空间中，由2个粒子（例如双原子分子）组成且彼此之间具有恒定距离（假设d）的物体，（如下所示），可以表明其自由度为5。
Let's say one particle in this body has coordinate {{math|(''x''<sub>1</sub>, ''y''<sub>1</sub>, ''z''<sub>1</sub>)}} and the other has coordinate {{math|(''x''<sub>2</sub>, ''y''<sub>2</sub>, ''z''<sub>2</sub>)}} with {{math| ''z''<sub>2</sub>}} unknown. Application of the formula for distance between two coordinates
Let's say one particle in this body has coordinate {{math|(''x''<sub>1</sub>, ''y''<sub>1</sub>, ''z''<sub>1</sub>)}} and the other has coordinate {{math|(''x''<sub>2</sub>, ''y''<sub>2</sub>, ''z''<sub>2</sub>)}} with {{math| ''z''<sub>2</sub>}} unknown. Application of the formula for distance between two coordinates
假设这个物体中的一个粒子的坐标为{{math|(''x''<sub>1</sub>, ''y''<sub>1</sub>, ''z''<sub>1</sub>)}}，另一个粒子的坐标为{{math|(''x''<sub>2</sub>, ''y''<sub>2</sub>, ''z''<sub>2</sub>)}}，其中{{math| ''z''<sub>2</sub>}}未知。那么两个坐标之间距离的公式可以描述为：
假设这个物体中的一个粒子的坐标为{{math|(''x''<sub>1</sub>, ''y''<sub>1</sub>, ''z''<sub>1</sub>)}}，另一个粒子的坐标为{{math|(''x''<sub>2</sub>, ''y''<sub>2</sub>, ''z''<sub>2</sub>)}}，其中{{math| ''z''<sub>2</sub>}}未知。那么两个坐标之间距离的公式可以描述为：
:<math>d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2+(z_2-z_1)^2}</math>
:<math>d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2+(z_2-z_1)^2}</math>
results in one equation with one unknown, in which we can solve for {{math|''z''<sub>2</sub>}}. One of {{math|''x''<sub>1</sub>}}, {{math|''x''<sub>2</sub>}}, {{math|''y''<sub>1</sub>}}, {{math|''y''<sub>2</sub>}}, {{math|''z''<sub>1</sub>}}, or {{math|''z''<sub>2</sub>}} can be unknown.
results in one equation with one unknown, in which we can solve for {{math|''z''<sub>2</sub>}}. One of {{math|''x''<sub>1</sub>}}, {{math|''x''<sub>2</sub>}}, {{math|''y''<sub>1</sub>}}, {{math|''y''<sub>2</sub>}}, {{math|''z''<sub>1</sub>}}, or {{math|''z''<sub>2</sub>}} can be unknown.
其等式含有一个未知数{{math|''z''<sub>2</sub>}}，不过我们可以对其求解。因此实际上是允许{{math|''x''<sub>1</sub>}}, {{math|''x''<sub>2</sub>}}, {{math|''y''<sub>1</sub>}}, {{math|''y''<sub>2</sub>}}, {{math|''z''<sub>1</sub>}}, 或者 {{math|''z''<sub>2</sub>}}其中之一是未知的。
其等式含有一个未知数{{math|''z''<sub>2</sub>}}，不过我们可以对其求解。因此实际上是允许{{math|''x''<sub>1</sub>}}, {{math|''x''<sub>2</sub>}}, {{math|''y''<sub>1</sub>}}, {{math|''y''<sub>2</sub>}}, {{math|''z''<sub>1</sub>}}, 或者 {{math|''z''<sub>2</sub>}}其中之一是未知的。
Contrary to the classical [[equipartition theorem]], at room temperature, the vibrational motion of molecules typically makes negligible contributions to the [[heat capacity]]. This is because these degrees of freedom are ''frozen'' because the spacing between the energy [[eigenvalue]]s exceeds the energy corresponding to ambient [[temperature]]s ({{math|''k''<sub>B</sub>''T''}}). In the following table such degrees of freedom are disregarded because of their low effect on total energy. Then only the translational and rotational degrees of freedom contribute, in equal amounts, to the [[heat capacity ratio]]. This is why {{mvar|γ}}={{math|{{sfrac|5|3}}}} for [[monatomic]] gases and {{mvar|γ}}={{math|{{sfrac|7|5}}}} for [[diatomic]] gases at room temperature.
Contrary to the classical [[equipartition theorem]], at room temperature, the vibrational motion of molecules typically makes negligible contributions to the [[heat capacity]]. This is because these degrees of freedom are ''frozen'' because the spacing between the energy [[eigenvalue]]s exceeds the energy corresponding to ambient [[temperature]]s ({{math|''k''<sub>B</sub>''T''}}). In the following table such degrees of freedom are disregarded because of their low effect on total energy. Then only the translational and rotational degrees of freedom contribute, in equal amounts, to the [[heat capacity ratio]]. This is why {{mvar|γ}}={{math|{{sfrac|5|3}}}} for [[monatomic]] gases and {{mvar|γ}}={{math|{{sfrac|7|5}}}} for [[diatomic]] gases at room temperature.
与经典的'''<font color="#ff8000"> 能量均分定理Equipartition theorem</font>'''相反，在室温下，分子的振动对'''<font color="#ff8000"> 热容量Heat capacity</font>'''的贡献通常可忽略不计。这是因为这些自由度被冻结了，因为能量本征值之间的间隔超过了与环境温度（kBT）相对应的能量。在下表中，这些自由度均被忽略，因为它们对总能量的影响非常小。只有平移和旋转自由度对'''<font color="#ff8000"> 热容比Heat capacity ratio</font>'''有些许贡献（等量）。这就是为什么在室温下，单原子气体 γ=5/3和双原子气体 γ=7/5的原因。
与经典的'''<font color="#ff8000"> 能量均分定理Equipartition theorem</font>'''相反，在室温下，分子的振动对'''<font color="#ff8000"> 热容量Heat capacity</font>'''的贡献通常可忽略不计。这是因为这些自由度被冻结了，因为能量本征值之间的间隔超过了与环境温度（kBT）相对应的能量。在下表中，这些自由度均被忽略，因为它们对总能量的影响非常小。只有平移和旋转自由度对'''<font color="#ff8000"> 热容比Heat capacity ratio</font>'''有些许贡献（等量）。这就是为什么在室温下，单原子气体 γ=5/3和双原子气体 γ=7/5的原因。
However, at very high temperatures, on the order of the vibrational temperature (Θ<sub>vib</sub>), vibrational motion cannot be neglected.
However, at very high temperatures, on the order of the vibrational temperature (Θ<sub>vib</sub>), vibrational motion cannot be neglected.
However, at very high temperatures, on the order of the vibrational temperature (Θ<sub>vib</sub>), vibrational motion cannot be neglected.
However, at very high temperatures, on the order of the vibrational temperature (Θ<sub>vib</sub>), vibrational motion cannot be neglected.
不过，在非常高的温度下，差不多在振动温度(Θ<sub>vib</sub>)的量级上，振动运动就不能忽略了。
不过，在非常高的温度下，差不多在振动温度(Θ<sub>vib</sub>)的量级上，振动运动就不能被忽略了。
Vibrational temperatures are between 10<sup>3</sup> K and 10<sup>4</sup> K.<ref name=":0" />
Vibrational temperatures are between 10<sup>3</sup> K and 10<sup>4</sup> K.<ref name=":0" />
振动温度在10<sup>3</sup> K和10<sup>4</sup> K之间。
振动温度在10<sup>3</sup> K和10<sup>4</sup> K之间。
{| class="wikitable"
{| class="wikitable"
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:
某个系统的自由度{{math|''X''<sub>1</sub>, ... , ''X''<sub>''N''</sub>}}集合，如果集合的能量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 {{mvar|E<sub>i</sub>}} is a function of the sole variable {{mvar|X<sub>i</sub>}}.
where {{mvar|E<sub>i</sub>}} is a function of the sole variable {{mvar|X<sub>i</sub>}}.
其中{{mvar|E<sub>i</sub>}}是唯一变量{{mvar|X<sub>i</sub>}}的函数。
其中{{mvar|E<sub>i</sub>}}是唯一变量{{mvar|X<sub>i</sub>}}的函数。
example: if {{math|''X''<sub>1</sub>}} and {{math|''X''<sub>2</sub>}} are two degrees of freedom, and {{mvar|E}} is the associated energy:
example: if {{math|''X''<sub>1</sub>}} and {{math|''X''<sub>2</sub>}} are two degrees of freedom, and {{mvar|E}} 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。
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.
* 如果<math>E = X_1^4 + X_2^4</math>，则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.
* 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。
* 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 {{mvar|i}} from 1 to {{mvar|N}}, the value of the {{mvar|i}}th degree of freedom {{mvar|X<sub>i</sub>}} is distributed according to the [[Boltzmann distribution]]. Its [[probability density function]] is the following:
For {{mvar|i}} from 1 to {{mvar|N}}, the value of the {{mvar|i}}th degree of freedom {{mvar|X<sub>i</sub>}} 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：
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。
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:
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
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 {{mvar|Y}} is a [[linear combination]] of other quadratic degrees of freedom.
where {{mvar|Y}} 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。
其中{{mvar|Y}}是其他二次自由度的线性组合。
example: if {{math|''X''<sub>1</sub>}} and {{math|''X''<sub>2</sub>}} are two degrees of freedom, and {{mvar|E}} is the associated energy:
example: if {{math|''X''<sub>1</sub>}} and {{math|''X''<sub>2</sub>}} are two degrees of freedom, and {{mvar|E}} is the associated energy:
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：
例如：如果{{math|''X''<sub>1</sub>}} 和 {{math|''X''<sub>2</sub>}}是两个自由度，而E是关联的能量：
* 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.
* 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。
* 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.
* 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。
* 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.
* 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。
* 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.
* 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。
* 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 (mechanics)|dynamics]] of a system of quadratic degrees of freedom are controlled by a set of homogeneous [[linear differential equation]]s with [[constant coefficients]].
For example, in [[Newtonian mechanics]], the [[Dynamics (mechanics)|dynamics]] of a system of quadratic degrees of freedom are controlled by a set of homogeneous [[linear differential equation]]s 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.
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，一个二自由度的动力学系统是由一组具有恒定系数的'''<font color="#ff8000"> 齐次线性微分方程 homogeneous linear differential equations </font>'''控制的。
=== Quadratic and independent degree of freedom 二次独立的自由度 ===
=== Quadratic and independent degree of freedom 二次独立的自由度 ===
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：
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>
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:
在统计力学的经典极限中，在'''<font color="#ff8000"> 热力学平衡Thermodynamic equilibrium</font>'''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：
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.
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带有的平均能量之和，由此得到了结果。
== Generalizations 概括 ==
== Generalizations 概括 ==
The description of a system's state as a point in its phase space, although mathematically convenient, is thought to be fundamentally inaccurate. In quantum mechanics, the motion degrees of freedom are superseded with the concept of wave function, and operators which correspond to other degrees of freedom have discrete spectra. For example, intrinsic angular momentum operator (which corresponds to the rotational freedom) for an electron or photon has only two eigenvalues. This discreteness becomes apparent when action has an order of magnitude of the Planck constant, and individual degrees of freedom can be distinguished.
The description of a system's state as a point in its phase space, although mathematically convenient, is thought to be fundamentally inaccurate. In quantum mechanics, the motion degrees of freedom are superseded with the concept of wave function, and operators which correspond to other degrees of freedom have discrete spectra. For example, intrinsic angular momentum operator (which corresponds to the rotational freedom) for an electron or photon has only two eigenvalues. This discreteness becomes apparent when action has an order of magnitude of the Planck constant, and individual degrees of freedom can be distinguished.
尽管在数学上很方便，但将系统状态描述为相空间中的一个点，从根本上讲是不准确的。在量子力学中，体系运动状态的自由度被波函数的概念所取代，并且对应于其他自由度的'''<font color="#ff8000"> 算子Operator</font>'''具有离散的光谱。例如，电子或光子的本征'''<font color="#ff8000"> 角动量算符Angular momentum operator </font>'''（对应于旋转自由度）只有两个特征值。当运动具有'''<font color="#ff8000"> 普朗克常数Planck constant</font>'''的量级时，这种离散变得非常明显，并且可以区分出各个自由度。
==References==
==References==
{{reflist}}
{{reflist}}
[[Category:Concepts in physics]]
[[Category:Concepts in physics]]