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第1行： − 此词条暂由彩云小译翻译，翻译字数共930，未经人工整理和审校，带来阅读不便，请见谅。+ 第33行：
第33行： − + − 在统计力学中，一个自由度是描述系统微观状态的单个标量数。+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + 第45行：
第175行： − + 第51行：
第181行： − 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.+ 第57行：
第187行： − + 第63行：
第193行： − ==Gas molecules==+ 第69行：
第199行： − [[Image:Degrees of freedom (diatomic molecule).png|thumb|right|Different ways of visualizing the 6 degrees of freedom of a diatomic molecule. (CM: [[center of mass]] of the system, T: [[translational motion]], R: [[rotation]]al motion, V: [[molecular vibration|vibrational motion]].)]]+ 第75行：
第205行： − + 第81行：
第211行： − 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{{Dubious|reason=diatomic molecules have at most 6, usually 5|date=April 2017}}. This set may be decomposed in terms of translations, rotations, and [[molecular vibration|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 [[rotation]]al degrees of motion and one{{Dubious|reason=diatomic molecules have at most 1, usually zero vibrational modes|date=April 2017}} [[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{{Dubious|reason=Diatomic molecules rotate and precess|date=April 2017}}. This yields, for a diatomic molecule, a decomposition of:+ 第87行：
第217行： − :<math>N = 6 = 3 + 2 + 1.</math>+ 第93行：
第223行： − + 第99行：
第229行： − For a general, non-linear molecule, all 3 rotational degrees of freedom are considered, resulting in the decomposition:+ 第105行：
第235行： − :<math>3N = 3 + 3 + (3N - 6)</math>+ 第111行：
第241行： − + 第117行：
第247行： − 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>+ 第123行：
第253行： − + 第129行：
第259行： − 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:+ 第135行：
第265行： − # 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.+ 第141行：
第271行： − # 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.+ 第147行：
第277行： − 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+ 第153行：
第283行： − :<math>d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2+(z_2-z_1)^2}</math>+ 第159行：
第289行： − results in one equation with one unknown, in which we can solve for {{math|''z''<sub>2</sub>}}.+ 第165行：
第295行： − 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.+ 第171行：
第301行： − + 第177行：
第307行： − 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.+ 第183行：
第313行： − + 第189行：
第319行： − However, at very high temperatures, on the order of the vibrational temperature (Θ<sub>vib</sub>), vibrational motion cannot be neglected.+ 第195行：
第325行： − + 第201行：
第331行： − Vibrational temperatures are between 10<sup>3</sup> K and 10<sup>4</sup> K.<ref name=":0" />+ 第207行：
第337行： − + 第213行：
第343行： − {| class="wikitable"+ 第219行：
第349行： − + 第225行：
第355行： − !+ 第231行：
第361行： − ! [[Monatomic]] − ! [[Linear molecule]]s − ! [[molecular geometry|Non-linear molecules]]+ + + 第241行：
第371行： − |-+ 第247行：
第377行： − | Translation ({{mvar|x}}, {{mvar|y}}, and {{mvar|z}}) − + + 第255行：
第385行： − | align="center" | 3 − + + 第263行：
第393行： − |-+ 第269行：
第399行： − | Rotation ({{mvar|x}}, {{mvar|y}}, and {{mvar|z}})+ 第275行：
第405行： − | align="center" | 0 − + + 第283行：
第413行： − | align="center" | 3+ 第289行：
第419行： − |- − | '''Total''' (disregarding Vibration at room temperatures)+ + 第297行：
第427行： − | align="center" | 3 − | align="center" | 5+ + 第305行：
第435行： − | align="center" | 6+ 第311行：
第441行： − |- − | Vibration − | align="center" | 0+ + + 第321行：
第451行： − | align="center" | {{math|3''N'' − 5}}+ 第327行：
第457行： − | align="center" | {{math|3''N'' − 6}} − + + 第335行：
第465行： − | '''Total''' (including Vibration) − | align="center" | '''3'''+ + 第343行：
第473行： − | align="center" | '''3''N'''''+ 第349行：
第479行： − | align="center" | '''3''N'''''+ 第355行：
第485行： − |}+ 第361行：
第491行： − + 第367行：
第497行： − ==Independent degrees of freedom== − The set of degrees of freedom {{math|''X''<sub>1</sub>, ... , ''X''<sub>''N''</sub>}} of a system is independent if the energy associated with the set can be written in the following form:+ + 第375行：
第505行： − :<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>}}.+ 第385行：
第515行： − + 第391行：
第521行： − 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: − :* 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_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.+ + + 第401行：
第531行： − + 第407行：
第537行： − 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: − + + 第415行：
第545行： − + 第421行：
第551行： − In this section, and throughout the article the brackets <math>\langle \rangle</math> denote the [[mean]] of the quantity they enclose.+ − + − The [[internal energy]] of the system is the sum of the average energies associated with each of the degrees of freedom:+ 第435行：
第565行： − :<math>\langle E \rangle = \sum_{i=1}^N \langle E_i \rangle.</math> + − ==Quadratic degrees of freedom==+ − + + + + + + + + + + − 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 − :<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. − 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: − :* 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. − 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]]. − ===Quadratic and independent degree of freedom===+ 第477行：
第607行： − {{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:+
Moved page from wikipedia:en:Degrees of freedom (physics and chemistry) (history)
此词条暂由彩云小译翻译，翻译字数共1459，未经人工整理和审校，带来阅读不便，请见谅。
{{about|physics and chemistry|other fields|Degrees of freedom}}
{{about|physics and chemistry|other fields|Degrees of freedom}}
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.
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，一个自由度是描述系统微观状态的单个标量数。一个系统的所有微观状态的描述是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.
在化学三维理想链模型中，每个单体的取向需要用两个角度来描述。
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.
指定二次自由度常常是有用的。这些自由度以二次函数的形式，对系统的能量做出贡献。
==Gas molecules==
[[Image:Degrees of freedom (diatomic molecule).png|thumb|right|Different ways of visualizing the 6 degrees of freedom of a diatomic molecule. (CM: [[center of mass]] of the system, T: [[translational motion]], R: [[rotation]]al motion, V: [[molecular vibration|vibrational motion]].)]]
Different ways of visualizing the 6 degrees of freedom of a diatomic molecule. (CM: [[center of mass of the system, T: translational motion, R: rotational motion, V: vibrational motion.)]]
用不同的方式可视化双原子分子的6个自由度。(CM: [系统的质心，t: 平动运动，r: 旋转轴，v: 振动运动]]
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{{Dubious|reason=diatomic molecules have at most 6, usually 5|date=April 2017}}. This set may be decomposed in terms of translations, rotations, and [[molecular vibration|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 [[rotation]]al degrees of motion and one{{Dubious|reason=diatomic molecules have at most 1, usually zero vibrational modes|date=April 2017}} [[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{{Dubious|reason=Diatomic molecules rotate and precess|date=April 2017}}. 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，3个自由度与一个粒子的运动有关。一个双原子气体分子有6个自由度。这个集合可以根据分子的平动、转动和振动来分解。整个分子的质心运动占3个自由度。此外，分子有两个转动度和一个振动模式。旋转发生在两个轴与两个原子之间的直线垂直的周围。原子-原子键的旋转不是物理旋转。这样，在一个双原子分子的时间里，产生了一个分解过程:
:<math>N = 6 = 3 + 2 + 1.</math>
<math>N = 6 = 3 + 2 + 1.</math>
6 = 3 + 2 + 1
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，考虑了所有3个转动自由度，导致分解:
:<math>3N = 3 + 3 + (3N - 6)</math>
<math>3N = 3 + 3 + (3N - 6)</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 -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.
这意味着一个原子分子的振动自由度。在特殊情况下，如吸附大分子，转动自由度可以限制为只有一个。
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:
根据上面的定义，还可以使用指定位置所需的最小坐标数来计算自由度。具体做法如下:
# 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，我们需要二维平面中的两个坐标来指定它的位置，以及三维空间中的三个坐标。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.
对于由两个粒子组成的物体(例如。在一个三维空间中，两个双原子分子之间的距离不变(比如说 d) ，我们可以显示它的自由度为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 and the other has coordinate with unknown. Application of the formula for distance between two coordinates
假设这个物体中的一个粒子具有坐标，而另一个粒子具有未知坐标。两坐标间距公式的应用
:<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>
[ 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>}}.
results in one equation with one unknown, in which we can solve for .
结果是一个方程有一个未知数，在这个方程中我们可以解出。
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.
One of , , , , , or can be unknown.
其中之一，，，，，或者可能是未知的。
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 eigenvalues exceeds the energy corresponding to ambient temperatures (). 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 =}} for monatomic gases and =}} for diatomic gases at room temperature.
与经典的能量均分定理相反，在室温下，分子的振动运动对热容的贡献可以忽略不计。这是因为这些自由度是冻结的，因为能量本征值之间的间距超过了对应于环境温度的能量()。在下表中，这种自由度不予考虑，因为它们对总能量的影响很小。那么，只有平动自由度和转动自由度，以相等的数量贡献于热容比。这就是为什么对于单原子气体和对于室温下的双原子气体。
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.
然而，在非常高的温度下，按振动温度(θ < sub > vib </sub >)的顺序，振动运动是不可忽略的。
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.
振动温度在10 < sup > 3 </sup > k 和10 < sup > 4 </sup > k 之间。
{| class="wikitable"
{| class="wikitable"
{| class="wikitable"
{ | class = “ wikitable”
{ | class = “ wikitable”
|-
|-
|-
|-
|-
!
!
!
!
!
! [[Monatomic]]
! Monatomic
! Monatomic
!单原子的
!单原子的
! [[Linear molecule]]s
! Linear molecules
! Linear molecules
!线性分子
!线性分子
! [[molecular geometry|Non-linear molecules]]
! Non-linear molecules
! Non-linear molecules
!非线性分子
!非线性分子
|-
|-
|-
|-
|-
| Translation ({{mvar|x}}, {{mvar|y}}, and {{mvar|z}})
| Translation (, , and )
| Translation (, , and )
| 翻译(，，和)
| 翻译(，，和)
| align="center" | 3
| align="center" | 3
| align="center" | 3
| align = “ center” | 3
| align = “ center” | 3
| align="center" | 3
| align="center" | 3
| align="center" | 3
| align = “ center” | 3
| align = “ center” | 3
| align="center" | 3
| align="center" | 3
| align="center" | 3
| align = “ center” | 3
| align = “ center” | 3
|-
|-
|-
|-
|-
| Rotation ({{mvar|x}}, {{mvar|y}}, and {{mvar|z}})
| Rotation (, , and )
| Rotation (, , and )
| 旋转(、和)
| 旋转(、和)
| align="center" | 0
| align="center" | 0
| align="center" | 0
| align = “ center” | 0
| align = “ center” | 0
| align="center" | 2
| align="center" | 2
| align="center" | 2
| align = “ center” | 2
| align = “ center” | 2
| align="center" | 3
| align="center" | 3
| align="center" | 3
| align = “ center” | 3
| align = “ center” | 3
|-
|-
|-
|-
|-
| '''Total''' (disregarding Vibration at room temperatures)
| Total (disregarding Vibration at room temperatures)
| Total (disregarding Vibration at room temperatures)
| 总计(不包括室温下的振动)
| 总计(不包括室温下的振动)
| align="center" | 3
| align="center" | 3
| align="center" | 3
| align = “ center” | 3
| align = “ center” | 3
| align="center" | 5
| align="center" | 5
| align="center" | 5
| align = “ center” | 5
| align = “ center” | 5
| align="center" | 6
| align="center" | 6
| align="center" | 6
| align = “ center” | 6
| align = “ center” | 6
|-
|-
|-
|-
|-
| Vibration
| Vibration
| Vibration
| 震动
| 震动
| align="center" | 0
| align="center" | 0
| align="center" | 0
| align = “ center” | 0
| align = “ center” | 0
| align="center" | {{math|3''N'' − 5}}
| align="center" |
| align="center" |
| align = “ center” |
| align = “ center” |
| align="center" | {{math|3''N'' − 6}}
| align="center" |
| align="center" |
| align = “ center” |
| align = “ center” |
|-
|-
|-
|-
|-
| '''Total''' (including Vibration)
| Total (including Vibration)
| Total (including Vibration)
| 总数(包括振动)
| 总数(包括振动)
| align="center" | '''3'''
| align="center" | 3
| align="center" | 3
| align = “ center” | 3
| align = “ center” | 3
| align="center" | '''3''N'''''
| align="center" | 3N
| align="center" | 3N
| align = “ center” | 3N
| align = “ center” | 3N
|-
| align="center" | '''3''N'''''
| align="center" | 3N
| align="center" | 3N
| align = “ center” | 3N
| align = “ center” | 3N
|}
|}
|}
|}
|}
==Independent degrees of freedom==
The set of degrees of freedom {{math|''X''<sub>1</sub>, ... , ''X''<sub>''N''</sub>}} 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:
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:
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:
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>
<math>E = \sum_{i=1}^N E_i(X_i),</math>
<math>E = \sum_{i=1}^N E_i(X_i),</math>
[数学] ，[数学]
[数学] ，[数学]
| align="center" | 3
where {{mvar|E<sub>i</sub>}} is a function of the sole variable {{mvar|X<sub>i</sub>}}.
where is a function of the sole variable .
where is a function of the sole variable .
这里是唯一变量的函数。
这里是唯一变量的函数。
| align="center" | 3
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: 如果和是两个自由度，关联的能量是:
例如: 如果和是两个自由度，关联的能量是:
:* 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 > ，那么这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 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 > ，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。
| align="center" | 2
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 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:
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:
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>,
<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>,
[数学][数学]
[数学][数学]
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.
In this section, and throughout the article the brackets <math>\langle \rangle</math> denote the mean of the quantity they enclose.
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。
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。
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:
The internal energy of the system is the sum of the average energies associated with each of the degrees of freedom:
系统的内能是与每个自由度有关的平均能量之和:
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>
<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>
数学，数学，数学
数学，数学，数学
==Quadratic degrees of freedom==
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
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>,
<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>,
数学 e = alpha _ i，，x _ i ^ 2 + beta _ i，，x _ i y,
数学 e = alpha _ i，，x _ i ^ 2 + beta _ i，，x _ i y,
|-
where {{mvar|Y}} is a [[linear combination]] of other quadratic degrees of freedom.
where is a linear combination of other quadratic degrees of freedom.
where is a linear combination of other quadratic degrees of freedom.
其他二次自由度的线性组合。
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 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: 如果和是两个自由度，关联的能量是:
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_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 > ，那么这两个自由度不是独立的和非二次的。
:* 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 > ，那么这两个自由度是独立的和非二次的。
:* 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 + 2 x 2 ^ 2 </math > ，那么这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 + 2 x _ 2 ^ 2 </math > ，那么这两个自由度是独立的和二次的。
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 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。
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。
===Quadratic and independent degree of freedom===
{{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:
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:
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</math>
<math>E = \sum_{i=1}^N \alpha_i X_i^2</math>
< math > e = sum { i = 1} ^ n alpha _ i x _ i ^ 2
< math > e = sum { i = 1} ^ n alpha _ i x _ i ^ 2
===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 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下，在热力学平衡，一个二次独立自由度系统的内能是:
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>
<math>U = \langle E \rangle = N\,\frac{k_B T}{2}</math>
<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]
[数学] u = langle e rangle = n，frac { k _ b t }{2}[数学]
: <math>p_i(X_i) = \frac{e^{-\frac{E_i}{k_B T}}}{\int dX_i \, e^{-\frac{E_i}{k_B T}}}</math>,
Here, the [[mean]] energy associated with a degree of freedom is:
Here, the mean energy associated with a degree of freedom is:
Here, the mean energy associated with a degree of freedom is:
在这里，与一定自由度相关的平均能是:
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>
<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>
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 >
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>
<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>
2} frac { k _ b }{ 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}{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 >
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，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。
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。
==Generalizations==
The description of a system's state as a [[point (geometry)|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 [[operator (physics)|operators]] which correspond to other degrees of freedom have [[point spectrum|discrete spectra]]. For example, [[angular momentum operator#Spin, orbital, and total angular momentum|intrinsic angular momentum]] operator (which corresponds to the rotational freedom) for an [[electron]] or [[photon]] has only two [[eigenvalue]]s. This discreteness becomes apparent when [[action (physics)|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.
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.
将一个系统的状态描述为其相空间中的一个点，虽然在数学上很方便，但被认为是根本不准确的。在量子力学中，运动自由度被波函数的概念所取代，相应于其他自由度的运算符具有离散谱。例如，一个电子或光子的内禀角动量算符(相当于转动自由度)只有两个本征值。当作用力的数量级为普朗克常数时，这种离散性就变得明显，个体自由度也可以区分开来。
将一个系统的状态描述为其相空间中的一个点，虽然在数学上很方便，但被认为是根本不准确的。在量子力学中，运动自由度被波函数的概念所取代，相应于其他自由度的运算符具有离散谱。例如，一个电子或光子的内禀角动量算符(相当于转动自由度)只有两个本征值。当作用力的数量级为普朗克常数时，这种离散性变得显而易见，个体自由度也可以区分。
==References==
{{reflist}}
[[Category:Concepts in physics]]
Category:Concepts in physics
Category:Concepts in physics
分类: 物理概念
分类: 物理概念
[[Category:Dimension]]
Category:Dimension
Category:Dimension