## Abstract

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*.

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## Notes

- 1.
This remark is not meant as a criticism: the diameter of a Brownian particle is of the order of, say, 1 μm, way above atomic size.

- 2.
“Sur la force complémentaire \(X\) nous savons qu’elle est indifféremment positive et négative, et sa grandeur est telle qu’elle maintient l’agitation de la particule que, sans elle, la résistance visqueuse finirait par arrêter.”

- 3.
We see from this relation that time averaging eliminates the contribution to the virial of viscous/frictional force of Stokes type.

- 4.
The case of discrete and continuous material systems with variable mass is studied in [8].

- 5.
This formula was derived by Finger in 1897 (see [9], Sects. 216 and 219).

- 6.
Taking all masses the same entails no substantial loss of generality in the developments to follow. Recall that taking the

*thermodynamic limit*amounts to inspecting the system’s properties for \(N,V\rightarrow \infty\), keeping the \(V/N\) ratio fixed. - 7.
In other words, the box walls are modeled as perfectly reflecting.

- 8.
Recall the well-known distributional relationships between the Heaviside and Dirac functions:

$$ -f(x_{0})= \int_{\mathbb{R}} \theta(x-x_{0})f^{\prime}(x)dx=- \int_{\mathbb{R}} D\theta(x-x_{0})f(x)dx\quad\text{for all test functions } f, $$(here \(D\) denotes distributional differentiation), whence

$$ D\theta(x-x_{0})=\delta(x-x_{0}). $$ - 9.
The fact that the notation we use for the microcanonical density keeps track of the dependence on \(E\) but not of those on \(N\) and \(V\), is not a good reason to overlook it.

- 10.
It is often reported that, around phase changes, MD systems may become sluggish and correlation lengths may increase and even diverge.

By a spatial

*s-correlation function*(see, e.g., [12], Sect. 79, and [10], Chap. 10) we mean a function giving the probability of finding \(s\) particles—say, those of Lagrangian coordinates \((q_{1},q_{2},\ldots,q_{s})\)—in a configurational box \(dq_{1}dq_{2}\ldots dq_{s}\) at a specified time. - 11.
Here a standard lemma of integration by parts is called upon, namely,

$$ \int_{A} \nabla\alpha d(vol_{A})= \int_{\partial A}\alpha \mathbf {n}_{\partial A}d(area_{A}), $$where \(\mathbf {n}_{\partial A}d(area_{A})\) is the oriented area measure over the boundary of the open bounded region \(A\) and where the scalar field \(\alpha\) is conveniently smooth, say, \(\alpha\in C^{0}(clo(A))\cap C^{1}(A)\).

- 12.
We just saw that this result—which, we recall, was first derived and given its name by Clausius in 1870—is nothing but a corollary of the generalization (41) of the Equipartition Theorem, whose first formulation was given by Boltzmann only one year later, in 1871. We read in [9], Sect. 219, that the quantity \(\sum_{i} q_{i} f_{i}\) was introduced in 1837 by Möbius, Schweins, and Jacobi, in that order.

- 13.
For a system confined in a box of finite volume, \(|q|\) stays finite at all times, and so does \(|p|\), because the total energy is conserved.

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## Acknowledgements

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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Podio-Guidugli, P. The Virial Theorem: A Pocket Primer.
*J Elast* **137, **219–235 (2019). https://doi.org/10.1007/s10659-018-09716-6

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### Keywords

- Deterministic virial theorem
- Probabilistic virial theorem
- Equipartition theorem

### Mathematics Subject Classification

- 70-01
- 70H99
- 74-01
- 74A25
- 82-01