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Information about Degrees Of Freedom (physics And Chemistry)

For information on degrees of freedom in other sciences, see degrees of freedom. For other uses of degree, see Degree


Degrees of freedom is a general term used in explaining dependence on parameters, and implying the possibility of counting the number of those parameters.

Degrees of freedom in mechanics (physics)

In mechanics, for each particle belonging to a system, and for each independent direction in which movement is possible, two degrees of freedom are defined, one describing the particle's momentum in that direction, the other describing the particle's position along an axis defined by that direction.

Note that "degrees of freedom" has a different meaning in the context of engineering and machines.

A more general definition

In statistical mechanics, a degree of freedom is a single scalar number describing the classical micro-state of a system. The micro-state of a system is completely described by the set of all values of all its degrees of freedom.

If the system studied can be described as a set of mechanical particles, then degrees of freedom are defined in the same manner as above. Thus, a micro-state of the system is a point in the system's phase space.

It must be noted that for a system, a micro-state defined by using degrees of freedom is intrinsically a classical state. This is because for a quantum micro-state, defining a precise value of both the position and momentum of a particle violates the Heisenberg uncertainty principle. The description of a system through a set of degrees of freedom is thus only valid in the classical (or high temperature) limit of statistical mechanics.

In some cases, when the system is not appropriately described as a set of mechanical particles, other types of degrees of freedom have to be defined. For example, in the 3D ideal chain model, two angles are necessary to describe each monomer's orientation. The value of each of these angles can each be a degree of freedom.

Example: classical ideal diatomic gas



In 3D, there are 6 degrees of freedom associated to the movement of a mechanical particle, 3 for its position, and 3 for its momentum.

There are 6 degrees of freedom in total. Another way to justify this figure is to consider that the movement of the molecule will be described by the movement of the two mechanical particles representing its two atoms, and 6 degrees of freedom are attached to each particle, as above. With this alternative breakdown, it appears that different sets of degrees of freedom can be defined to describe the movement of the molecule. In fact a set of degrees of freedom for a mechanical system is a set of independent axes in the phase space of the system, and that allows the generation of the whole phase space. For a multidimensional space like phase space, there is more than one possible set of axes.

It is notable that not all degrees of freedom of the hydrogen molecule participate in the above expression of its energy. For example, those degrees of freedom associated to the position of the center of mass of the particle do not weigh in the energy.

In the table below the degrees which are disregarded are like this because of their low effect on total energy, unless they are at very very high temperatures or energies. The diatomic rotation is disregarded due to rotation about the molecules axis. Monatomic rotation is disregarded for the same reason as diatomic, but this effect continues into the other 2 directions.

Monatomic Linear molecules Non-Linear molecules
Position (x, y and z)333
Rotation (x, y and z)023
Vibration03N - 53N - 6
Total33N3N

Independent degrees of freedom

Definition

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:


where is a function of the sole variable .

example: if and are two degrees of freedom, and 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 and is a coupling term, that describes an interaction between the two degrees of freedom.

Properties

If is a set of independent degrees of freedom then, at thermodynamic equilibrium, are all statistically independent from each other.

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


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:

Demonstrations

We will assume that our system exchanges energy in the form of heat with the outside, and that its number of particles 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 is the system's temperature and is Boltzman's constant, then the probability density function associated to each micro-state is the following:
,


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 are statistically independent from each other.

Since each function is normalized, it follows immediately that is the probability density function of the degree of freedom , for i from 1 to N.

Finally, the internal energy of the system is its mean energy. The energy of a degree of freedom is a function of the sole variable . Since are independent from each other, the energies 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 is quadratic if the energy terms associated to this degree of freedom can be written as:
,


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

example: if and are two degrees of freedom, and 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 degrees of freedom in mechanics

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

are quadratic and independent degrees of freedom if the energy associated to a microstate of the system they represent can be written as:

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:

Demonstration

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


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

See also

Degrees of freedom can mean:
  • Degrees of freedom (physics and chemistry)
  • Degrees of freedom (engineering)
  • Degrees of freedom (statistics)
  • Degrees of freedom (numerics)

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Degree, in the sense of measurement, is used to indicate:
  • Comparative degree ("good", "better") and superlative degree ("best")
  • Severity of similar crimes (in some jurisdictions) — for example first degree murder
  • Intensity of a burn (from first degree

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Parameters, in the plural form, has recently become popular with non-technical users to mean limits, but this should not be confused with the word's technical meaning.

In mathematics, statistics, and the mathematical sciences, parameters (L: auxiliary measure
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For other uses, see Mechanic (disambiguation).


Mechanics (Greek Μηχανική
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momentum (pl. momenta; SI unit kg m/s, or, equivalently, N•s) is the product of the mass and velocity of an object. For more accurate measures of momentum, see the section "modern definitions of momentum" on this page.
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degrees of freedom (DOF) are the set of independent displacements that specify completely the displaced or deformed position of the body or system. This is a fundamental concept relating to systems of moving bodies in mechanical engineering, aeronautical engineering, robotics,
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Statistical mechanics is the application of probability theory, which includes mathematical tools for dealing with large populations, to the field of mechanics, which is concerned with the motion of particles or objects when subjected to a force.
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A scalar is a variable that only has magnitude, e.g. a speed of 40 km/h. Compare it with vector, a quantity comprising both magnitude and direction, e.g. a velocity of 40km/h north.
  • A scalar (mathematics), a quantity which is independent of viewpoint, a non-tensor.

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Ice melting - a classic example of entropy increasing[1] described in 1862 by Rudolf Clausius as an increase in the disgregation of the molecules of the body of ice.
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Ice melting - a classic example of entropy increasing[1] described in 1862 by Rudolf Clausius as an increase in the disgregation of the molecules of the body of ice.
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Ice melting - a classic example of entropy increasing[1] described in 1862 by Rudolf Clausius as an increase in the disgregation of the molecules of the body of ice.
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phase space, introduced by Willard Gibbs in 1901, is a space in which all possible states of a system are represented, with each possible state of the system corresponding to one unique point in the phase space.
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Classical mechanics (commonly confused with Newtonian mechanics, which is a subfield thereof) is used for describing the motion of macroscopic objects, from projectiles to parts of machinery, as well as astronomical objects, such as spacecraft, planets, stars, and galaxies.
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“Quanta” redirects here. For other uses, see Quantum (disambiguation).

Development of quantum theory

Quantum theory, the branch of physics which is based on quantization, began in 1900 when Max Planck published his theory explaining the
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Ice melting - a classic example of entropy increasing[1] described in 1862 by Rudolf Clausius as an increase in the disgregation of the molecules of the body of ice.
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momentum (pl. momenta; SI unit kg m/s, or, equivalently, N•s) is the product of the mass and velocity of an object. For more accurate measures of momentum, see the section "modern definitions of momentum" on this page.
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Heisenberg uncertainty principle, or HUP, gives a lower bound on the product of the standard deviations of position and momentum for a system, implying that it is impossible to have a particle that has an arbitrarily well-defined position and momentum simultaneously.
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Statistical mechanics is the application of probability theory, which includes mathematical tools for dealing with large populations, to the field of mechanics, which is concerned with the motion of particles or objects when subjected to a force.
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An ideal chain (or freely-jointed chain) is the simplest model to describe a polymer. It only assumes a polymer as a random walk and neglects any kind of interactions among monomers. Although it is simple, its generality gives us some insights about the physics of polymers.
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momentum (pl. momenta; SI unit kg m/s, or, equivalently, N•s) is the product of the mass and velocity of an object. For more accurate measures of momentum, see the section "modern definitions of momentum" on this page.
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phase space, introduced by Willard Gibbs in 1901, is a space in which all possible states of a system are represented, with each possible state of the system corresponding to one unique point in the phase space.
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phase space, introduced by Willard Gibbs in 1901, is a space in which all possible states of a system are represented, with each possible state of the system corresponding to one unique point in the phase space.
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phase space, introduced by Willard Gibbs in 1901, is a space in which all possible states of a system are represented, with each possible state of the system corresponding to one unique point in the phase space.
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energy (from the Greek ενεργός, energos, "active, working")[1] is a scalar physical quantity that is a property of objects and systems of objects which is conserved by nature.
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In physics and chemistry, monatomic is a combination of the words "mono" and "atomic," and means "single atom." It is usually applied to gases: a monatomic gas is one in which atoms are not bound to each other.
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thermodynamic equilibrium when it is in thermal equilibrium, mechanical equilibrium, and chemical equilibrium. The local state of a system at thermodynamic equilibrium is determined by the values of its intensive parameters, as pressure, temperature, etc.
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In probability theory, to say that two events are independent, intuitively means that the occurrence of one event makes it neither more nor less probable that the other occurs.
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Boltzmann distribution predicts the distribution function for the fractional number of particles Ni / N occupying a set of states i which each respectively possess energy Ei:


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In mathematics, a probability density function (pdf) is a function that represents a probability distribution in terms of integrals.

Formally, a probability distribution has density f, if f
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In statistics, mean has two related meanings:
  • the arithmetic mean (and is distinguished from the geometric mean or harmonic mean).
  • the expected value of a random variable, which is also called the population mean.

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