自由度

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25px  此article介紹的是physics and chemistry。For other fields, see Degrees of freedom.

模板:Refimprove 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。

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X1 和 X2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X1和X2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X1 和 X2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. ↑ 引用错误：无效<ref>标签；未给name属性为:0的引用提供文字

Category:Concepts in physics

分类: 物理概念

Category:Dimension

类别: 维度


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