Degrees of freedom (physics and chemistry)

This article is about physics and chemistry. For otherelds, see Degrees of freedom.

In physics, a degree of freedom is an independent phys-ical parameter in the formal description of the state of aphysical system. The set of all dimensions of a systemis known as a phase space, and degrees of freedom aresometimes referred to as its dimensions.

1 DenitionA degree of freedom of a physical system refers to a (typ-ically real) parameter that is necessary to characterize thestate of a physical system.Consider a point particle that is free to move in threedimensions. The location of any particle in three-dimensional space can be specied by three position co-ordinates: x, y, and z. The direction and speed at which aparticle moves can be described in terms of three veloc-ity components, e.g. vx, vy, and vz. If the time evolutionof the system is deterministic, where the state at one in-stant uniquely determines its past and future position andvelocity as a function of time, such a system will havesix degrees of freedom. If the motion of the particle isconstrained to a lower number of dimensions if, for ex-ample, the particle must move along a wire or on a xedsurface then the systemwill have less than six degrees offreedom. On the other hand, a system with an extendedobject that may rotate or vibrate can have more than sixdegrees of freedom. A force on the particle that dependsonly upon time and the particles position and velocity tsthis description.In mechanics, a point particle's state at any given time canbe described with position and velocity coordinates in theLagrangian formalism, or with position and momentumcoordinates in the Hamiltonian formalism.Similarly, in statistical mechanics, a degree of freedomis a single scalar number describing the microstate of asystem.[1] The specication of all microstates of a systemis a point in the systems phase space.A degree of freedom may be any useful property that isnot dependent on other variables. For example, in the 3Dideal chain model, two angles are necessary to describeeach monomers orientation.In statistical mechanics and thermodynamics, it is oftenuseful to specify quadratic degrees of freedom. These

are degrees of freedom that contribute in a quadratic wayto the energy of the system. They are also variables thatcontribute quadratically to the Hamiltonian.

2 Degrees of freedom of gasmolecules

Dierent ways of visualizing the 6 degrees of freedom of adiatomic molecule. (CM: center of mass of the system, T:translational motion, R: rotational motion, V: vibrational mo-tion.)

In three-dimensional space, three degrees of freedom areassociated with the movement of a particle. A diatomicgas molecule thus has 6 degrees of freedom. This set maybe decomposed in terms of translations, rotations, and vi-brations of the molecule. The center of mass motion ofthe entire molecule accounts for 3 degrees of freedom.In addition, the molecule has two rotational degrees ofmotion and one vibrational mode. The rotations occuraround the two axes perpendicular to the line betweenthe two atoms. The rotation around the atomatom bondis not a physical rotation. This yields, for a diatomicmolecule, a decomposition of:

3N = 6 = 3 + 2 + 1:

For a general (non-linear) molecule with N > 2 atoms, all3 rotational degrees of freedom are considered, resultingin the decomposition:

1

2 4 QUADRATIC DEGREES OF FREEDOM

3N = 3 + 3 + (3N 6)which means that an N-atom molecule has 3N 6 vibra-tional degrees of freedom for N > 2. In special cases,such as adsorbed large molecules, the rotational degreesof freedom can be limited to only one.[2]

As dened above one can also count degrees of freedomusing the minimum number of coordinates required tospecify a position. This is done as follows:

1. For a single particle we need 2 coordinates in a 2-Dplane to specify its position and 3 coordinates in 3-Dplane. Thus its degree of freedom in a 3-D plane is3.

2. For a body consisting of 2 particles (ex. a diatomicmolecule) in a 3-D plane with constant distance be-tween them (lets say d) we can show (below) its de-grees of freedom to be 5.

Lets say one particle in this body has coordinate (x1, y1,z1) and the other has coordinate (x2, y2, z2) with z2 un-known. Application of the formula for distance betweentwo coordinates

d =p(x2 x1)2 + (y2 y1)2 + (z2 z1)2

results in one equation with one unknown, in which wecan solve for z2. One of x1, x2, y1, y2, z1, or z2 can beunknown.Contrary to the classical equipartition theorem, at roomtemperature, the vibrational motion of molecules typi-cally makes negligible contributions to the heat capac-ity. This is because these degrees of freedom are frozenbecause the spacing between the energy eigenvalues ex-ceeds the energy corresponding to ambient temperatures(kBT). In the following table such degrees of freedomare disregarded because of their low eect on total en-ergy. However, at very high temperatures they cannot beneglected.

3 Independent degrees of freedomThe set of degrees of freedom X1, ... , XN of a system isindependent if the energy associated with the set can bewritten in the following form:

E =NXi=1

Ei(Xi);

where E is a function of the sole variable X.example: if X1 and X2 are two degrees of freedom, andE is the associated energy:

If E = X41 +X42 , then the two degreesof freedom are independent.

If E = X41 +X1X2+X42 , then the twodegrees of freedom are not independent.The term involving the product ofX1 andX2 is a coupling term, that describes aninteraction between the two degrees offreedom.

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

pi(Xi) =e EikBTR

dXi e EikBT

In this section, and throughout the article the brackets hidenote the mean of the quantity they enclose.The internal energy of the system is the sum of the aver-age energies associated to each of the degrees of freedom:

hEi =NXi=1

hEii:

4 Quadratic degrees of freedomA degree of freedom X is quadratic if the energy termsassociated to this degree of freedom can be written as

E = i X2i + i XiY

where Y is a linear combination of other quadratic de-grees of freedom.example: if X1 and X2 are two degrees of freedom, andE is the associated energy:

IfE = X41 +X31X2+X42 , then the twodegrees of freedom are not independentand non-quadratic.

If E = X41 + X42 , then the two de-grees of freedom are independent andnon-quadratic.

IfE = X21+X1X2+2X22 , then the twodegrees of freedom are not independentbut are quadratic.

If E = X21 + 2X22 , then the two de-grees of freedom are independent andquadratic.

For example, in Newtonian mechanics, the dynamics ofa system of quadratic degrees of freedom are controlledby a set of homogeneous linear dierential equations withconstant coecients.

34.1 Quadratic and independent degree offreedom

X1, ... , XN are quadratic and independent degrees offreedom if the energy associated to a microstate of thesystem they represent can be written as:

E =

NXi=1

iX2i

4.2 Equipartition theorem

In the classical limit of statistical mechanics, atthermodynamic equilibrium, the internal energy of a sys-tem of N quadratic and independent degrees of freedomis:

U = hEi = N kBT2

Here, the mean energy associated with a degree of free-dom is:

hEii =Z

dXi iX2i pi(Xi) =

RdXi iX

2i e

iX2i

kBTRdXi e

iX2i

kBT

hEii = kBT2

Rdx x2 e

x2

2Rdx e

x2

2

=kBT

2

Since the degrees of freedom are independent, theinternal energy of the system is equal to the sum of themean energy associated with each degree of freedom,which demonstrates the result.

5 Generalizations

The description of a systems state as a point in its phasespace, although mathematically convenient, is thought tobe fundamentally inaccurate. In quantum mechanics, themotion degrees of freedom are superseded with the con-cept of wave function, and operators which correspondto other degrees of freedom have discrete spectra. Forexample, intrinsic angular momentum operator (whichcorresponds to the rotational freedom) for an electron orphoton have only two eigenvalues. This discreteness be-comes apparent when action has an order of magnitudeof the Planck constant, and individual degrees of free-dom can be distinguished.

6 References[1] Reif, F. (2009). Fundamentals of Statistical and Thermal

Physics. Long Grove, IL: Waveland Press, Inc. p. 51.ISBN 1-57766-612-7.

[2] Thomas Waldmann, Jens Klein, Harry E. Hoster, R.Jrgen Behm (2012), Stabilization of Large Adsor-bates by Rotational Entropy: A Time-Resolved Variable-Temperature STM Study (in German), ChemPhysChem:pp. n/an/a, doi:10.1002/cphc.201200531

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