Degrees of freedom (physics and chemistry)

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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.

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.

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 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 the 3D ideal chain model in chemistry, two angles are necessary to describe the orientation of each monomer.

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.

Gas molecules

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.)

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:

${\displaystyle N=6=3+2+1.}$

For a general, non-linear molecule, all 3 rotational degrees of freedom are considered, resulting in the decomposition:

${\displaystyle 3N=3+3+(3N-6)}$

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.

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:

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.
2. 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.

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

${\displaystyle d={\sqrt {(x_{2}-x_{1})^{2}+(y_{2}-y_{1})^{2}+(z_{2}-z_{1})^{2}}}}$

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.

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 (k_BT). 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=5/3 for monatomic gases and γ=7/5 for diatomic gases at room temperature.

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

Vibrational temperatures are between 10³ K and 10⁴ K.

	Monatomic	Linear molecules	Non-linear molecules
Translation (x, y, and z)	3	3	3
Rotation (x, y, and z)	0	2	3
Total (disregarding Vibration at room temperatures)	3	5	6
Vibration	0	3N − 5	3N − 6
Total (including Vibration)	3	3N	3N

Independent degrees of freedom

The set of degrees of freedom X₁, ... , X_N of a system is independent if the energy associated with the set can be written in the following form:

${\displaystyle E=\sum _{i=1}^{N}E_{i}(X_{i}),}$

where E_i is a function of the sole variable X_i.

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

• If ${\displaystyle E=X_{1}^{4}+X_{2}^{4}}$, then the two degrees of freedom are independent.
• If ${\displaystyle E=X_{1}^{4}+X_{1}X_{2}+X_{2}^{4}}$, 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.

For i from 1 to N, the value of the ith degree of freedom X_i is distributed according to the Boltzmann distribution. Its probability density function is the following:

${\displaystyle p_{i}(X_{i})={\frac {e^{-{\frac {E_{i}}{k_{B}T}}}}{\int dX_{i}\,e^{-{\frac {E_{i}}{k_{B}T}}}}}}$,

In this section, and throughout the article the brackets ${\displaystyle \langle \rangle }$ 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:

${\displaystyle \langle E\rangle =\sum _{i=1}^{N}\langle E_{i}\rangle .}$

Quadratic degrees of freedom

A degree of freedom X_i is quadratic if the energy terms associated with this degree of freedom can be written as

${\displaystyle E=\alpha _{i}\,\,X_{i}^{2}+\beta _{i}\,\,X_{i}Y}$,

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

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

• If ${\displaystyle E=X_{1}^{4}+X_{1}^{3}X_{2}+X_{2}^{4}}$, then the two degrees of freedom are not independent and non-quadratic.
• If ${\displaystyle E=X_{1}^{4}+X_{2}^{4}}$, then the two degrees of freedom are independent and non-quadratic.
• If ${\displaystyle E=X_{1}^{2}+X_{1}X_{2}+2X_{2}^{2}}$, then the two degrees of freedom are not independent but are quadratic.
• If ${\displaystyle E=X_{1}^{2}+2X_{2}^{2}}$, then the two degrees of freedom are independent and quadratic.

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.

Quadratic and independent degree of freedom

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

${\displaystyle E=\sum _{i=1}^{N}\alpha _{i}X_{i}^{2}}$

Equipartition theorem

In the classical limit of statistical mechanics, at thermodynamic equilibrium, the internal energy of a system of N quadratic and independent degrees of freedom is:

${\displaystyle U=\langle E\rangle =N\,{\frac {k_{B}T}{2}}}$

Here, the mean energy associated with a degree of freedom is:

${\displaystyle \langle E_{i}\rangle =\int dX_{i}\,\,\alpha _{i}X_{i}^{2}\,\,p_{i}(X_{i})={\frac {\int dX_{i}\,\,\alpha _{i}X_{i}^{2}\,\,e^{-{\frac {\alpha _{i}X_{i}^{2}}{k_{B}T}}}}{\int dX_{i}\,\,e^{-{\frac {\alpha _{i}X_{i}^{2}}{k_{B}T}}}}}}$

${\displaystyle \langle E_{i}\rangle ={\frac {k_{B}T}{2}}{\frac {\int dx\,\,x^{2}\,\,e^{-{\frac {x^{2}}{2}}}}{\int dx\,\,e^{-{\frac {x^{2}}{2}}}}}={\frac {k_{B}T}{2}}}$

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.

Generalizations

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.