NationMaster - Encyclopedia: Degrees of freedom (physics and chemistry)

Degrees of freedom is a general term used in explaining dependence on parameters, and implying the possibility of counting the number of those parameters. The factual accuracy of this article is disputed. ...

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. Mechanics (Greek ) is the branch of physics concerned with the behaviour of physical bodies when subjected to forces or displacements, and the subsequent effect of the bodies on their environment. ... In classical mechanics, momentum (pl. ...

Note that "degrees of freedom" has a different meaning in the context of engineering and machines. In mechanical engineering, aeronautical engineering and robotics, degrees of freedom (DOF) describes flexibility of motion. ...

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. 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. ... The term scalar is used in mathematics, physics, and computing basically for quantities that are characterized by a single numeric value and/or do not involve the concept of direction. ... see also: Entropy (disambiguation) Ice melting - 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. ... see also: Entropy (disambiguation) Ice melting - 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. ...

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. see also: Entropy (disambiguation) Ice melting - 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. ... Phase space of a dynamical system with focal stability. ...

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. Classical mechanics is a branch of physics which studies the deterministic motion of objects. ... In physics, a quantum refers to an indivisible, and perhaps, elementary entity. ... see also: Entropy (disambiguation) Ice melting - 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. ... In classical mechanics, momentum (pl. ... In quantum physics, the Heisenberg uncertainty principle, sometimes called the Heisenberg indeterminacy principle, expresses a limitation on accuracy of (nearly) simultaneous measurement of observables such as the position and the momentum of a particle. ... 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. ...

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. An ideal chain (or freely-jointed chain) is the simplest model to describe a polymer. ...

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. Image File history File links Degrees_of_freedom. ... In classical mechanics, momentum (pl. ...

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. Phase space of a dynamical system with focal stability. ... Phase space of a dynamical system with focal stability. ... Phase space of a dynamical system with focal stability. ...

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 if 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.

In physics and chemistry, monatomic is a combination of the words mono and atomic, and means single atom. ... A computer rendering of the Nitrogen Molecule, which is a diatomic molecule. ...

Independent degrees of freedom

Definition

The set of degrees of freedom $X_1, ldots, X_N$ of a system is independent if the energy associated with the set can be written in the following form:

example: if

X 1

and

X 2

are two degrees of freedom, and

E

is the associated energy:

Properties

If $X_1, ldots, X_N$ is a set of independent degrees of freedom then, at thermodynamic equilibrium, $X_1, ldots, X_n$ are all statistically independent from each other. In thermodynamics, a thermodynamic system is in thermodynamic equilibrium when it is in thermal equilibrium, mechanical equilibrium, and chemical equilibrium. ... 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. ...

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: In physics, the Boltzmann distribution predicts the distribution function for the fractional number of particles Ni / N occupying a set of states i which each has energy Ei: where is the Boltzmann constant, T is temperature (assumed to be a sharply well-defined quantity), is the degeneracy, or number of... In mathematics, a probability density function (pdf) serves to represent a probability distribution in terms of integrals. ...

In this section, and throughout the article the brackets $langle rangle$ denote the mean of the quantity they enclose. In statistics, mean has two related meanings: the average in ordinary English, which is also called the arithmetic mean (and is distinguished from the geometric mean or harmonic mean). ...

The internal energy of the system is the sum of the average energies associated to each of the degrees of freedom: In thermodynamics, the internal energy of a thermodynamic system, or a body with well-defined boundaries, denoted by U, or sometimes E, is the total of the kinetic energy due to the motion of molecules (translational, rotational, vibrational) and the potential energy associated with the vibrational and electric energy of...

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

T

is the system's temperature and

k B

is Boltzman's constant, then the probability density function associated to each micro-state is the following: 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. ... In physics, the thermodynamic limit is the statistical mechanical limit described by a system in which the number of particles approaches infinity. ... In thermodynamics, a thermodynamic system is in thermodynamic equilibrium when it is in thermal equilibrium, mechanical equilibrium, and chemical equilibrium. ... see also: Entropy (disambiguation) Ice melting - 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. ... In physics, the Boltzmann distribution predicts the distribution function for the fractional number of particles Ni / N occupying a set of states i which each has energy Ei: where is the Boltzmann constant, T is temperature (assumed to be a sharply well-defined quantity), is the degeneracy, or number of... Fig. ... The Boltzmann constant (k or kB) is the physical constant relating temperature to energy. ... In mathematics, a probability density function (pdf) serves to represent a probability distribution in terms of integrals. ...

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 $X_1 ldots X_N$ are statistically independent from each other. In mathematics, a probability density function (pdf) serves to represent a probability distribution in terms of integrals. ... 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. ...

Since each function

p i

is normalized, it follows immediately that

p i

is the probability density function of the degree of freedom

X i

, for i from 1 to N. The concept of a normalizing constant arises in probability theory and a variety of other areas of mathematics. ... In mathematics, a probability density function (pdf) serves to represent a probability distribution in terms of integrals. ...

Finally, the internal energy of the system is its mean energy. The energy of a degree of freedom

E i

is a function of the sole variable

X i

. Since $X_1, ldots, X_N$ are independent from each other, the energies $E_1(X_1), ldots, E_N(X_N)$ are also statistically independent from each other. The total internal energy of the system can thus be written as: In thermodynamics, the internal energy of a thermodynamic system, or a body with well-defined boundaries, denoted by U, or sometimes E, is the total of the kinetic energy due to the motion of molecules (translational, rotational, vibrational) and the potential energy associated with the vibrational and electric energy of... In statistics, mean has two related meanings: the average in ordinary English, which is also called the arithmetic mean (and is distinguished from the geometric mean or harmonic mean). ... 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. ... In thermodynamics, the internal energy of a thermodynamic system, or a body with well-defined boundaries, denoted by U, or sometimes E, is the total of the kinetic energy due to the motion of molecules (translational, rotational, vibrational) and the potential energy associated with the vibrational and electric energy of...

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:

where

Y

is a linear combination of other quadratic degrees of freedom. In mathematics, linear combinations are a concept central to linear algebra and related fields of mathematics. ...

example: if

X 1

and

X 2

are two degrees of freedom, and

E

is the associated energy:

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. It has been suggested that this article or section be merged with Classical mechanics. ... In physics, dynamics is the branch of classical mechanics that is concerned with the effects of forces on the motion of objects. ... In mathematics, a linear differential equation is a differential equation Lf = g, where the differential operator L is a linear operator. ... In mathematics, constant coefficients is a term applied to differential operators, and also some difference operators, to signify that they contain no functions of the independent variables, other than constant functions. ...

Quadratic and independent degree of freedom

$X_1, ldots, X_N$ are quadratic and independent degrees of freedom if the energy associated to a microstate of the system they represent can be written as:

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: 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. ... In thermodynamics, a thermodynamic system is in thermodynamic equilibrium when it is in thermal equilibrium, mechanical equilibrium, and chemical equilibrium. ... In thermodynamics, the internal energy of a thermodynamic system, or a body with well-defined boundaries, denoted by U, or sometimes E, is the total of the kinetic energy due to the motion of molecules (translational, rotational, vibrational) and the potential energy associated with the vibrational and electric energy of...

Demonstration

Here, the mean energy associated with a degree of freedom is: In statistics, mean has two related meanings: the average in ordinary English, which is also called the arithmetic mean (and is distinguished from the geometric mean or harmonic mean). ...

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. In thermodynamics, the internal energy of a thermodynamic system, or a body with well-defined boundaries, denoted by U, or sometimes E, is the total of the kinetic energy due to the motion of molecules (translational, rotational, vibrational) and the potential energy associated with the vibrational and electric energy of... In statistics, mean has two related meanings: the average in ordinary English, which is also called the arithmetic mean (and is distinguished from the geometric mean or harmonic mean). ...

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