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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，'''<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。
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.
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，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。
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 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，，需要两个角度来描述每个'''<font color="#ff8000">单体monomer </font>'''的方向。
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.
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。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>N = 6 = 3 + 2 + 1.</math>
:<math>N = 6 = 3 + 2 + 1.</math>
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.
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'''<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.
For from 1 to , the value of the th degree of freedom is distributed according to the Boltzmann distribution. Its probability density function is the following:
For from 1 to , the value of the th degree of freedom is distributed according to the Boltzmann distribution. Its probability density function is the following:
这里的i值可以取1到N，第i个自由度Xi的值依据'''<font color="#ff8000"> 波尔兹曼分布Boltzmann distribution </font>'''。其'''<font color="#ff8000"> 概率密度函数Probability density function</font>'''如下：
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>,
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。
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 ===
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：
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>
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。
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 总结 ==
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 (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.
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。
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>'''（对应于转动自由度）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==