In physics, a degree of freedom is an independent physical parameter in the formal description of the state of a physical system. The set of all dimensions of a system is known as a phase space, and degrees of freedom are sometimes referred to as its dimensions.
Definition
A degree of freedom of a physical system refers to a (typically real) parameter that is necessary to characterize the state of a physical system.
Consider a point particle that is free to move in three dimensions. The location of any particle in threedimensional space can be specified by three position coordinates: x, y, and z. The direction and speed at which a particle moves can be described in terms of three velocity components, e.g. v_{x}, v_{y}, and v_{z}. 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 will have six degrees of freedom. If the motion of the particle is constrained to a lower number of dimensions – if, for example, the particle must move along a wire or on a fixed surface – then the system will have less than six degrees of freedom. On the other hand, a system with an extended object that may rotate or vibrate can have more than six degrees of freedom. A force on the particle that depends only upon time and the particle's position and velocity fits this description.
In mechanics, a point particle's state at any given time can be described with position and velocity coordinates in the Lagrangian formalism, or with position and momentum coordinates in the Hamiltonian formalism.
Similarly, 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.
A degree of freedom may be any useful property that is not dependent on other variables. For example, in the 3D ideal chain model, two angles are necessary to describe each monomer's orientation.
In statistical mechanics and thermodynamics, it is often useful to specify quadratic degrees of freedom. These are degrees of freedom that contribute in a quadratic way to the energy of the system. They are also variables that contribute quadratically to the Hamiltonian.
Degrees of freedom of gas molecules
In threedimensional space, three degrees of freedom are associated with the movement of a particle. A diatomic gas molecule thus 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:

3N = 6 = 3 + 2 + 1.
For a general (nonlinear) molecule with N > 2 atoms, all 3 rotational degrees of freedom are considered, resulting in the decomposition:

3N = 3 + 3 + (3N  6)
which means that an Natom 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]}
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:

For a single particle we need 2 coordinates in a 2D plane to specify its position and 3 coordinates in 3D plane. Thus its degree of freedom in a 3D plane is 3.

For a body consisting of 2 particles (ex. a diatomic molecule) in a 3D plane 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 (x_{1}, y_{1}, z_{1}) and the other has coordinate (x_{2}, y_{2}, z_{2}) with z_{2} unknown. Application of the formula for distance between two coordinates

d=\sqrt{(x_2x_1)^2+(y_2y_1)^2+(z_2z_1)^2}
results in one equation with one unknown, in which we can solve for z_{2}. One of x_{1}, x_{2}, y_{1}, y_{2}, z_{1}, or z_{2} 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_{B}T). In the following table such degrees of freedom are disregarded because of their low effect on total energy. However, at very high temperatures they cannot be neglected.
Independent degrees of freedom
The set of degrees of freedom X_{1}, ... , X_{N} of a system is independent if the energy associated with the set can be written in the following form:

E = \sum_{i=1}^N E_i(X_i),
where E_{i} is a function of the sole variable X_{i}.
example: if X_{1} and X_{2} are two degrees of freedom, and E is the associated energy:


If E = X_1^4 + X_2^4, then the two degrees of freedom are independent.

If 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 X_{1} and X_{2} 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:

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 \langle \rangle denote the mean of the quantity they enclose.
The internal energy of the system is the sum of the average energies associated to each of the degrees of freedom:

\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 to this degree of freedom can be written as

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 X_{1} and X_{2} are two degrees of freedom, and E is the associated energy:


If E = X_1^4 + X_1^3 X_2 + X_2^4, then the two degrees of freedom are not independent and nonquadratic.

If E = X_1^4 + X_2^4, then the two degrees of freedom are independent and nonquadratic.

If E = X_1^2 + X_1 X_2 + 2X_2^2, then the two degrees of freedom are not independent but are quadratic.

If 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
X_{1}, ... , X_{N} are quadratic and independent degrees of freedom if the energy associated to a microstate of the system they represent can be written as:

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:

U = \langle E \rangle = N\,\frac{k_B T}{2}
Here, the mean energy associated with a degree of freedom is:

\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}}}

\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 have only two eigenvalues, and a continuous rotational freedom of classical bodies becomes reduced to so named spin for these and other microscopic particles. This effect of discreteness (sometimes referred to as quantization, although the latter is a much broader concept) becomes dominant when action has an order of magnitude of the Planck constant, and individual degrees of freedom cannot be distinguished then.
References

^ Reif, F. (2009). Fundamentals of Statistical and Thermal Physics. Long Grove, IL: Waveland Press, Inc. p. 51.

^ Thomas Waldmann, Jens Klein, Harry E. Hoster, R. Jürgen Behm (2012), "Stabilization of Large Adsorbates by Rotational Entropy: A TimeResolved VariableTemperature STM Study" (in German), ChemPhysChem: pp. n/a–n/a, doi:10.1002/cphc.201200531
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