This applet simulates the evolution of a simple pendulum, a small bob of mass m attached to a frictionless pivot by a massless rod of length l. The state of a pendulum is characterized by a single coordinate, the angle of deviation from the vertical θ. To simplify the algebra, we set all the physical constants to unity, m = l = g = 1. As a result, the period of the pendulum is 2π in the limit of small amplitude of oscillations. For larger amplitudes the period becomes longer, diverging as the amplitude approaches π.
The applet shows the time evolution in phase space, which in this case is two-dimensional. The horizontal axis is the pendulum's coordinate θ, the vertical axis is momentum p = dθ/dt. The pendulum has two distinct dynamical regimes. If its energy is low then it oscillates back and forth (finite motion). If the energy exceeds a threshold then the pendulum rotates in one direction and the angle keeps increasing indefinitely (infinite motion). These regions in phase space are separated by the red curve, on which the energy is such that the amplitude of oscillations is exactly π. For convenience, we show the angle modulo 2π; thus a pendulum whose angle exceeds π will exit the phase space on the right and reenter it on the left.
In fact, we track down the evolution of not one but many (more than a thousand) identical pendula that begin their motion with slightly different initial conditions (both the initial angle and velocity). The pendula within a small box with sides dθ and dp and follow their trajectories through phase space. If the energies are low then the motion of the pendula can be approximated as harmonic. All of them will have phase-space orbits with a period 2π. Thus the box of phase-space points merely rotates about the origin preserving both its size and shape.
However, as we shift the initial states to higher energies we begin to notice the effects of anharmonicity. Pendula with higher energies have a slightly longer period of motion than ones with lower energies. Consequently, the outer edge of the box will lag behind the inner edge and the initially rectangular box will gradually deform into a parallelogram. (The area will stay the same as guaranteed by Liouville's theorem.) At still higher energies the effects of anharmonicity set in rather quickly and we observe how the box turns into a long sausage winding inside a ring of phase space whose boundaries are set by the minimal and maximal energies present in the ensemble of pendula.
If you wait long enough the points corresponding to the 1600 pendula will fill the ring more or less uniformly. This is a remarkable thing. Even though we started with an ensemble of pendula whose coordinates and momenta were almost the same, they have now spread all over the phase space! The only restriction is imposed by energy conservation. One may say that the pendula are uniformly spread over the available phase space. They can have any coordinates and momenta with the same probability as long as they have the specified energy. A collection of identical physical systems whose only restriction is the requirement of a fixed energy is known as a microcanonical ensemble.
The energy constraint provides a fairly tight restriction in a two-dimensional phase space (it leaves only one degree of freedom, so to speak). However, in more complex systems, with N coordinates and N momenta, the restriction is less onerous and the definition of a microcanonical ensemble remains unchanged: the system's coordinates and momenta are uniformly distributed in its 2N-dimensional phase space, subject to the constraint on energy.
You should note that the spreading of the initially localized packet in phase space all over the slice of constant energy requires anharmonicity in the case of a pendulum. In the limit of small energy, when it can be treated as a harmonic oscillator, the packet will not spread. More precisely, anharmonicity is small and it will take a long time for the points to disperse throughout the available phase space. In a purely harmomic oscillator the spreading will never happen. Then statistical mechanics won't work! Fortunately, physical oscillators always have a degree of anharmonicity, so statistical mechanics is safe!
For further mathematical details, consult this note.
This material is based upon work supported by the National Science Foundation under Grant No. DMR-1104753. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author and do not necessarily reflect the views of the National Science Foundation.
Johns Hopkins University