# May the Hyperforce be with you!

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In the Star Wars saga, “the Force” is an all-encompassing entity which enables creatures sensitive to it to gain extraordinary abilities. In that context “Hyper Forces” refer to the enhanced capability to exert and use the Force. Much in analogy to the Jedi and Sith, the powerful concept of forces also helps physicists to gain new insights into specific problems. The application and generalization of force-based descriptors thereby commonly turns out to uncover underlying physical mechanisms and constraints.  With our publication “Hyperforce Balance via Thermal Noether Invariance of Any Observable” [1] we adopted the idea to statistical mechanics and introduced hyperforces as the physical correlator between a force and another thermodynamic variable.

The story behind hyperforces started with our new Bachelor student, Silas Robitschko. Back then, we have just determined the second order contributions of exploiting Noether’s theorem in statistical mechanics [2]. Considering an invariance with respect to translation yields exact identities (“sum rules”) for correlations between forces and other thermal observables [3,4]. As such correlations have not been studied previously, we asked Silas to sample and investigate them in a simple Lennard-Jones fluid. Indeed, he could verify the predicted Noether identities in simulations.

A usual undergraduate student would have stopped at that point and waited for further instructions. However, intrigued by the correlation functions, Silas took their analysis upon himself and started looking meticulously for systematic features. At some point we focused on one specific correlator 〈F0ext ρ〉, which describes the correlation of the global external force on the system (i.e. integrated over all particle positions) with the spatially resolved density ρ. Silas pointed out that the roots of 〈F0ext ρ〉 coincided with the extrema of the density profile. At that point I had the strong feeling that this phenomenon has no physical origin and that he was trying to interpret mere coincidences. I was wrong. Indeed, using his conviction and intuition Silas discovered within the numerical data the exact relation 〈F0ext ρ〉 = ∇ρ. This sum rule obviously explains his previous observation and we identified it as the inverse Lovett-Mou-Bouff-Wertheim (LMBW) relation. We later derived the identity analytically from Noether’s theorem both as a second order term of the invariant grand potential and as a first order contribution by exploiting the translational invariance of the density.

We were already quite happy with that result, but this turned out to be only the starting point. My coworker Matthias Schmidt has been teaching Nonequilibrium Statistical Mechanics following the book by Zwanzig [5], a classical introduction to the topic with many considerations for general thermodynamic observables Â. Using the inspiration from the book and presumably in a flash of inspiration while listening to some scientific talk, he realized that the translational invariance does not only hold for the density profile but that it can be exploited for any phase space function Â. Hence the inverse LMBW equation evolves to a sum rule for the hyperforce, which is equal to the mean gradient of the observable Â. The achieved generalization from forces to hyperforces is similar in spirit to Hirschfelder’s hypervirial generalization, where he included a dependence on the phase space function Â in the standard virial theorem. We have hence adopted this terminology.

The above considerations also hold for spatially resolved forces, which yields a local version of the sum rule, and for general observables Â(rN,pN), which include particle momenta and which can additionally depend parametrically on a position variable r. An overview of the resulting relations is given in Fig. 1.

Specific choices of Â uncover a variety of sum rules for the hyperforce correlations. In case of the density operator Â=ρ we get the inverse LMBW equation that Silas initially determined numerically. Along with other sum rules, this result offers useful consistency checks in simulations, in numerical and in analytic calculations and it can be used to test and to validate the accuracy of neural networks [6]. Besides these applications exact identities give access to alternative sampling routes. My colleague Florian Sammüller realized that the accuracy of accordance between different observables that are linked by hyperforce sum rules states a valuable indication for the sampling quality and for equilibration, e.g. in slowly evolving systems.  Additionally, considering alternative observables allows to construct reduced-variance sampling schemes, similar to force sampling [7] which can improve the statistical quality of the data.

We are hence confident that the consideration of hyperforces allows for a better understanding and description of correlations and forces acting in fluids. The generality of the hyperforce framework leaves room for many further investigations. To say it with Yoda’s words: “Much to learn, you still have.

References:

[1]  S. Robitschko, F. Sammüller, M. Schmidt, and S. Hermann, Hyperforce balance via thermal Noether invariance of any observable, Commun. Phys. 7, 103  (2024).

[2]  S. Hermann and M. Schmidt, Why Noether's theorem applies to statistical mechanics, J. Phys.: Condens. Matter 34, 213001 (2022).

[3]  F. Sammüller, S. Hermann, D. de las Heras, and M. Schmidt, Noether-constrained correlations in equilibrium liquids, Phys. Rev. Lett. 130, 268203 (2023).

[4]  S. Hermann and M. Schmidt, Variance of fluctuations from Noether invariance, Commun. Phys. 5, 276 (2022).

[5]  R. Zwanzig, Nonequilibrium Statistical Mechanics (Oxford University Press, Oxford, 2001).

[6]  F. Sammüller, S. Hermann, D. de las Heras, and M. Schmidt, Neural functional theory for inhomogeneous fluids: Fundamentals and applications Proc. Nat. Acad. Sci. 120, e2312484120 (2023).

[7]  B. Rotenberg, Use the force! Reduced variance estimators for densities, radial distribution functions, and local mobilities in molecular simulationsJ. Chem. Phys. 153, 150902 (2020).

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