The Virial Theorem: A Pocket Primer

  • Paolo Podio-GuidugliEmail author
Classroom Note


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.


Deterministic virial theorem Probabilistic virial theorem Equipartition theorem 

Mathematics Subject Classification

70-01 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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© Springer Nature B.V. 2018

Authors and Affiliations

  1. 1.Accademia Nazionale dei Lincei and Department of MathematicsUniversity of Rome TorVergataRomeItaly

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