自由度

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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,自由度 Degree of freedom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相空间 Phase space,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] 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.

粒子 particle 三维 three-dimensional 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 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,任何给定时间下质点 Point particle的状态,通常是用拉格朗日形式 Lagrangian formalism 中的位置和速度坐标来描述,或者用哈密顿形式 Hamiltonian formalism 中的位置和动量坐标来描述。

In statistical mechanics, a degree of freedom is a single scalar number describing the microstate of a system.[1] 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。[1]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 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.

在化学的三维理想链 Ideal chain模型中,,需要两个角度来描述每个单体 monomer 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.

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

可视化双原子分子六个自由度的不同方式。(其中CM:系统的质心,T:平移运动,R:旋转运动,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模板:Dubious. 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模板:Dubious 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模板:Dubious. 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。除此之外,分子还有两个转动自由度和一个振动自由度。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]\displaystyle{ 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:

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]\displaystyle{ 3N = 3 + 3 + (3N - 6) }[/math]

which means that an N-atom molecule has 3N − 6 vibrational degrees of freedom for N > 2. In special cases, such as adsorbed large molecules, the rotational degrees of freedom can be limited to only one.[2]

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,转动自由度只能限制为一个。[3]

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:

  1. 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。

  1. 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 (x1, y1, z1) and the other has coordinate (x2, y2, z2) with z2 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的坐标为(x1, y1, z1),另一个粒子的坐标为(x2, y2, z2),其中z2未知。那么两个坐标之间距离的公式可以描述为:

[math]\displaystyle{ 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 z2. One of x1, x2, y1, y2, z1, or z2 can be unknown.

results in one equation with one unknown, in which we can solve for . One of , , , , , or can be unknown.

其等式含有一个未知数z2,不过我们可以对其求解。因此实际上是允许x1, x2, y1, y2, z1, 或者 z2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 eigenvalues exceeds the energy corresponding to ambient temperatures (kBT). 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 γ=模板:Sfrac for monatomic gases and γ=模板:Sfrac 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 Equipartition theorem相反,在室温下,分子的振动对热容量 Heat capacity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热容比 Heat capacity ratio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 (Θvib), vibrational motion cannot be neglected.

However, at very high temperatures, on the order of the vibrational temperature (Θvib), vibrational motion cannot be neglected.

不过,在非常高的温度下,差不多在振动温度(Θvib)的量级上,振动运动就不能被忽略了。

Vibrational temperatures are between 103 K and 104 K.[1]

Vibrational temperatures are between 103 K and 104 K.

振动温度在103 K和104 K之间。[1]

单原子 线性分子 非线性分子
平移 (x, y, and z) 3 3 3
旋转 (x, y, and z) 0 2 3
总计 (不考虑室温下的振动) 3 5 6
振动 0 3N − 5 3N − 6
总计 (包括振动) 3 3N 3N

Independent degrees of freedom 独立自由度

The set of degrees of freedom X1, ... , XN 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X1, ... , XN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]\displaystyle{ E = \sum_{i=1}^N E_i(X_i), }[/math]

where Ei is a function of the sole variable Xi.

where is a function of the sole variable .

其中Ei是唯一变量Xi的函数。

example: if X1 and X2 are two degrees of freedom, and E is the associated energy:

example: if and are two degrees of freedom, and is the associated energy:

例如:如果X1X2是两个自由度,并且E是关联的能量。

  • If [math]\displaystyle{ E = X_1^4 + X_2^4 }[/math], then the two degrees of freedom are independent.
  • If [math]\displaystyle{ E = X_1^4 + X_2^4 }[/math], then the two degrees of freedom are independent.
  • 如果[math]\displaystyle{ E = X_1^4 + X_2^4 }[/math],则两个自由度是独立的。
  • If [math]\displaystyle{ 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 X1 and X2 is a coupling term that describes an interaction between the two degrees of freedom.
  • If [math]\displaystyle{ 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]\displaystyle{ 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X1X2的乘积是描述两个自由度之间相互作用的耦合项。

For i from 1 to N, the value of the ith degree of freedom Xi 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的值依据波尔兹曼分布 Boltzmann distribution 。其概率密度函数 Probability density function如下:

[math]\displaystyle{ 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]\displaystyle{ \langle \rangle }[/math] denote the mean of the quantity they enclose.

In this section, and throughout the article the brackets [math]\displaystyle{ \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]\displaystyle{ \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]\displaystyle{ \langle E \rangle = \sum_{i=1}^N \langle E_i \rangle. }[/math]

Quadratic degrees of freedom 二次自由度

A degree of freedom Xi 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Xi是二次的

[math]\displaystyle{ E = \alpha_i\,\,X_i^2 + \beta_i \,\, X_i Y }[/math],

where Y is a linear combination of other quadratic degrees of freedom.

where is a linear combination of other quadratic degrees of freedom.

其中Y是其他二次自由度的线性组合。

example: if X1 and X2 are two degrees of freedom, and E is the associated energy:

example: if and are two degrees of freedom, and is the associated energy:

例如:如果X1X2是两个自由度,而E是关联的能量:

  • If [math]\displaystyle{ 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]\displaystyle{ 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]\displaystyle{ 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]\displaystyle{ E = X_1^4 + X_2^4 }[/math], then the two degrees of freedom are independent and non-quadratic.
  • If [math]\displaystyle{ E = X_1^4 + X_2^4 }[/math], then the two degrees of freedom are independent and non-quadratic.
  • 如果[math]\displaystyle{ 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]\displaystyle{ 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]\displaystyle{ 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]\displaystyle{ 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]\displaystyle{ E = X_1^2 + 2X_2^2 }[/math], then the two degrees of freedom are independent and quadratic.
  • If [math]\displaystyle{ E = X_1^2 + 2X_2^2 }[/math], then the two degrees of freedom are independent and quadratic.
  • 如果[math]\displaystyle{ 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 homogeneous linear differential equations 控制的。

Quadratic and independent degree of freedom 二次独立的自由度

X1, ... , XN are quadratic and independent degrees of freedom if the energy associated with a microstate of the system they represent can be written as:

X1, ... , XN 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X1, ... , XN是二次独立的自由度,它们可以写成:

[math]\displaystyle{ E = \sum_{i=1}^N \alpha_i X_i^2 }[/math]

Equipartition theorem 能量均分定理

In the classical limit of statistical mechanics, at thermodynamic equilibrium, the internal energy of a system of 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:

在统计力学的经典极限中,在热力学平衡 Thermodynamic equilibrium状态下,N个二次且独立自由度的系统内部能量为:

[math]\displaystyle{ 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]\displaystyle{ \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]\displaystyle{ \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带有的平均能量之和,由此得到了结果。

Generalizations 总结

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算子 Operator具有离散的光谱。例如,电子或光子的本征角动量算符 Angular momentum operator (对应于转动自由度)只有两个特征值。当运动具有普朗克常数 Planck constant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

  1. 1.0 1.1 1.2 1.3 Reif, F. (2009). Fundamentals of Statistical and Thermal Physics. Long Grove, IL: Waveland Press, Inc.. p. 51. ISBN 1-57766-612-7. 
  2. Waldmann, Thomas; Klein, Jens; Hoster, Harry E.; Behm, R. Jürgen (2013). "Stabilization of Large Adsorbates by Rotational Entropy: A Time-Resolved Variable-Temperature STM Study". ChemPhysChem. 14 (1): 162–9. doi:10.1002/cphc.201200531. PMID 23047526.
  3. Waldmann, Thomas; Klein, Jens; Hoster, Harry E.; Behm, R. Jürgen (2013). "Stabilization of Large Adsorbates by Rotational Entropy: A Time-Resolved Variable-Temperature STM Study". ChemPhysChem. 14 (1): 162–9. doi:10.1002/cphc.201200531. PMID 23047526.

Category:Concepts in physics

分类: 物理概念

Category:Dimension

类别: 维度


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