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Archive for December 21st, 2008

Equipartition Theorem

I am awed by, and feel sorry for, anyone who understands this – 

Figure 1. Thermal motion of an α-helical peptide. The jittery motion is random and complex, and the energy of any particular atom can fluctuate wildly. Nevertheless, the equipartition theorem allows the average kinetic energy of each atom to be computed, as well as the average potential energies of many vibrational modes. The grey, red and blue spheres represent atoms of carbon, oxygen and nitrogen, respectively; the smaller white spheres represent atoms of hydrogen.

The equipartition theorem is a formula from statistical mechanics that relates the temperature of a system with its average energies. The original idea of equipartition was that, in thermal equilibrium, energy is shared equally among its various forms; for example, the average kinetic energy in the translational motion of a molecule should equal the average kinetic energy in its rotational motion. Like the virial theorem, the equipartition theorem gives the total average kinetic and potential energies for a system at a given temperature, from which the system’s heat capacity can be computed. However, equipartition also gives the average values of individual components of the energy. It can be applied to any classical system in thermal equilibrium, no matter how complicated. The equipartition theorem can be used to derive the classical ideal gas law, and the Dulong–Petit law for the specific heat capacities of solids. It can also be used to predict the properties of stars, even white dwarfs and neutron stars, since it holds even whenrelativistic effects are considered. Although the equipartition theorem makes very accurate predictions in certain conditions, it becomes inaccurate when quantum effects are significant, namely at low enough temperatures.

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