Degrees of freedom (physics and chemistry)
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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.
A degree of freedom of a physical system is a formal description of a parameter that contributes[clarification needed] to the state of a physical system.
The location of any particle in three-dimensional space can be specified by three coordinates, x, y, and z. The direction and speed at which a particle moves can be described in terms of three velocity components --- often vx, vy, and vz. A point particle moving freely in three dimensions therefore has six degrees of freedom. If the motion of the particle is constrained - if, for example, the particle must move along a wire or on a fixed surface - then it will have less than six degrees of freedom. On the other hand, an extended object may rotate or vibrate and can therefore have more than six degrees of freedom.
In mechanics, a point particle's state can either 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. The specification of all microstates of a system is a point in the system's phase space.
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 three-dimensional 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:
For a general (non-linear) molecule with N > 2 atoms, all 3 rotational degrees of freedom are considered, resulting in the decomposition:
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:
- For a single particle we need 2 coordinates in a 2-D plane to specify its position and 3 coordinates in 3-D plane. Thus its degree of freedom in a 3-D plane is 3.
- For a body consisting of 2 particles (ex. a diatomic molecule) in a 3-D 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 coordinates (x1, y1, z1) and the other has x-coordinate(x2) and y-coordinate (y2). Application of the formula for distance between two coordinates
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 (kBT). In the following table such degrees of freedom are disregarded because of their low effect on total energy. However, at very high[which?] temperatures they cannot be neglected.
|Monatomic||Linear molecules||Non-linear molecules|
|Translation (x, y, and z)||3||3||3|
|Rotation (x, y, and z)||0||2||3|
|Vibration||0||3N − 5||3N − 6|
Independent degrees of freedom
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:
where Ei is a function of the sole variable Xi.
example: if X1 and X2 are two degrees of freedom, and E is the associated energy:
- If , then the two degrees of freedom are independent.
- If , 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.
In this section, and throughout the article the brackets 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:
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A system exchanges energy in the form of heat with its surroundings and the number of particles in the system remains fixed. This corresponds to studying the system in the canonical ensemble. Note that in statistical mechanics, a result that is demonstrated for a system in a particular ensemble remains true for this system at the thermodynamic limit in any ensemble. In the canonical ensemble, at thermodynamic equilibrium, the state of the system is distributed among all micro-states according to the Boltzmann distribution. If T is the system's temperature and kB is Boltzmann's constant, then the probability density function associated to each micro-state is the following:
The denominator in the above expression plays an important role. This expression immediately breaks down into a product of terms depending of a single degree of freedom:
The existence of such a breakdown of the multidimensional probability density function into a product of functions of one variable is enough by itself to demonstrate that X1, … , XN are statistically independent from each other.
Finally, the internal energy of the system is its mean energy. The energy of a degree of freedom Ei is a function of the sole variable Xi. Since X1, … , XN are independent from each other, the energies E1(X1), … , EN(XN) are also statistically independent from each other. The total internal energy of the system can thus be written as:
Quadratic degrees of freedom
A degree of freedom Xi is quadratic if the energy terms associated to this degree of freedom can be written as
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 , then the two degrees of freedom are not independent and non-quadratic.
- If , then the two degrees of freedom are independent and non-quadratic.
- If , then the two degrees of freedom are not independent but are quadratic.
- If , then the two degrees of freedom are independent and quadratic.
Quadratic and independent degree of freedom
X1, … , XN are quadratic and independent degrees of freedom if the energy associated to a microstate of the system they represent can be written as:
Here, the mean energy associated with a degree of freedom is:
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.
- Reif, F. (2009). Fundamentals of Statistical and Thermal Physics. Long Grove, IL: Waveland Press, Inc. p. 51. ISBN 1-57766-612-7.
- Thomas Waldmann, Jens Klein, Harry E. Hoster, R. Jürgen Behm (2012), "Stabilization of Large Adsorbates by Rotational Entropy: A Time-Resolved Variable-Temperature STM Study" (in German), ChemPhysChem: pp. n/a–n/a, doi:10.1002/cphc.201200531
- "Configuration integral (statistical mechanics)".