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''Degrees of freedom'' is a quite 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

At this stage it must be noted that for a system, a micro-state defined using degrees of freedom is intrinsically a Classical state. 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 though 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 play the role of a degree of freedom.


EXAMPLE: CLASSICAL IDEAL DIATOMIC GAS




In this section, and throughout the article the brackets \langle angle 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 angle = \sum_{i=1}^N \langle E_i angle


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-state s 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:
: P(X_1, \ldots, X_N) = rac{e^{- rac{E}{k_B T}}}{\int dX_1\,dX_2 \ldots dX_N e^{- rac{E}{k_B T}}},

This expression immediately breaks down into a product of terms depending of a single degree of freedom:
:P(X_1, \ldots, X_N) = p_1(X_1) \ldots p_N(x_N)

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.

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''.

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:
: U = \langle E angle = \langle \sum_{i=1}^N E_i angle = \sum_{i=1}^N \langle E_i angle


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:


Quadratic degrees of freedom in mechanics


In Newtonian Mechanics , the Dynamic s of a system of quadratic degrees of freedom are controlled by a set of homogeneous Linear Differential Equation s with Constant Coefficients .


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:
: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 angle = N\, rac{k_B T}{2}


DEMONSTRATION


Here, the Mean energy associated with a degree of freedom is:
:\langle E_i angle = \int dX_i\,\,\alpha_i X_i^2\,\, p_i(X_i) = rac{\int dX_i\,\,\alpha_i X_i^2\,\, e^{- rac{\alpha_i X_i^2}{k_B T}}}{\int dX_i\,\, e^{- rac{\alpha_i X_i^2}{k_B T}}}
:\langle E_i angle = rac{k_B T}{2} rac{\int dx\,\,x^2\,\, e^{- rac{x^2}{2}}}{\int dx\,\, e^{- rac{x^2}{2}}} = rac{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 to each degree of freedom, which demonstrates the result.


SEE ALSO