The Virial Theorem: A Pocket Primer
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The force virial is a construct combining information about the current configuration of a mechanical system in motion with information about the acting forces; its long-time average turns out to be proportional to the long-time average of the system’s kinetic energy, that is, the virial theorem holds true; a version of this theorem is obtained when a connection between kinetic energy and temperature is established, as it happens in the kinetic theory of gases or in classical equilibrium statistical mechanics. In fact, a virial theorem holds in whatever mechanics. This paper is an exposition of its various formulations, from the simplest deterministic formulation for a single massy particle in Newtonian motion to the fairly more complicated statistical formulation, which makes the theorem a corollary of the equipartition theorem.
KeywordsDeterministic virial theorem Probabilistic virial theorem Equipartition theorem
Mathematics Subject Classification70-01 70H99 74-01 74A25 82-01
I presented part of this material in two lectures given at OIST (Okinawa Institute of Science and Technology, Okinawa, Japan) during the Summer School on “Hierarchical multiscale methods using the Anderson-Parinello-Rahman formulation of molecular dynamics”, April 3–8, 2017. It is a pleasant duty for me to acknowledge OIST’s support and hospitality.
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