|
The phrase "degrees of freedom" is used in three different branches of science: in physics and physical chemistry, in mechanical and aeronautical engineering, and in statistics. The
three usages are linked historically and through the underlying mathematics,
but they are not identical.
Physics and chemistry
In physics and chemistry, each independent mode in which a particle or system may move or be oriented is one degree of
freedom. For a roughly dumbell-shaped hydrogen molecule, three such modes would be rotation (twirling), translation
(hurling through space) and vibration (the two dumbbell "balls" bouncing together and apart). According to thermodynamics, in case of thermal equilibrium each degree of freedom in every particle of a system will contain the same energy
on average (equal to kT, the temperature of the system multiplied by the
fundamental Boltzmann constant). However, thermal equilibrium can
only be reached among interacting particles, a process called thermalization. According to quantum
mechanics and more specifically Heisenberg's uncertainty principle, the amount of energy within any degree of
freedom is never zero, but is always at least equal to the zero-point
energy for that mode.
Engineering
In mechanical and aeronautical engineering, degrees of freedom (DOF) describes flexibility of motion. A
mechanism that has complete freedom of motion (even if only in a limited area, or envelope) has six degrees of freedom. Three
modes are translation - the ability to move in each of three dimensions. Three are rotation, or the ability to change angle
around three perpendicular axes.
To put it in simpler terms, each of the following is one degree of freedom:
- Moving up and down (heaving);
- moving left and right (swaying);
- moving forward and back (surging);
- tilting up and down (pitching);
- turning left and right (yawing);
- tilting side to side (rolling).
See also: Euler angles.
A mechanism that can (for instance) be raised and lowered, which has a pivoting head that can tilt forward or back, left or
right, can be described as having 3 degrees of freedom (colloquially, 3DOF).
Statistics
In statistics, degrees of freedom is a statistical parameter in many important probability distributions. Examples include the chi-square distribution, the F-distribution, Student's
t-distribution, and the beta distribution that underlies
them. See Pearson's chi-square test and
analysis of variance for more information.
In the familiar uses of these distributions, the degrees of freedom takes only integer values (usually low ones). The underlying mathematics do allow for fractional degrees of freedom, which can
arise in more sophisticated uses.
|