# Casimir Forces

H. Casimir proved in 1948 that two metallic parallel plates attract each other due to the quantum fluctuations induced by the electric field of the vacuum between the plates [1]. The dependence of this force with the temperature was calculated by Lifschitz [2]. Three decades after, Fisher and De Gennes [3] showed that the fluctuation-induced forces have to be present in systems which are close to the critical point, where the dynamics is governed by the fluctuations. Later, these forces have been derived for any system with fluctuations where the correlation has medium or long range: nematic fluids, wetting phenomena in fluids or superfluids, biological systems, charged fluids, etc. (see, e.g. [4]).

The research on this topic has increased exponentially in the last fifty years. One of the reasons is the possibility to access to the nanometric scale experimentally, where Casimir forces are relevant. In fact, this forces are fundamental in the design and construction of MEMS and NEMS (Micro and Nano Electro Mechanical Systems). Because of that, there is a growing interest in calculating Casimir Forces for complicated geometries,

Our group has developed a method to calculate Casimir Forces in Classical and Quantum systems, both in Equilibrium and Non Equilibrium situation. The method takes as an starting point the Langevin Equation, that describes a system subjected to fluctuations. The solution of the Langevin equation is used to evaluate the stress tensor, which gives the Fluctuation induced forces. Usual regularization techniques are required. We have applied such formalism to Granular fluids [5], and other systems in and out of equilibrium [6].

In collaboration with the group of Emig, we have applied a scattering matrix method to a two perfect metal sphere geometry close to a perfect metal plane [7], i.e. to a three-body system. The importance of this work lies in the fact that the Casimir force is not additive, so the presence of a third body modifies the force between the other two object. In addition to that, we derived the Pairwise Summation Approximation from such multi-scattering formalism [8], and found that the superposition of Casimir forces is recovered in the diluted limit.

[1] H.B.G. Casimir, On the Attraction between Two Perfectly Conducting Plates. Proc. Kon. Nederland. Akad. Wetensch. B51, 793 (1948).

[2] E.M. Lifshitz, The Theory of Molecular Attractive Forces between Solids. Soviet Physics Jetp-Ussr 2, 73-83 (1956).

[3] M.E. Fisher and P.G.D. Gennes, Wall Phenomena in a Critical Binary Mixture. Comptes Rendus Hebdomadaires Des Seances De L Academie Des Sciences Serie B 287, 207-209 (1978).

[4] M. Kardar and R. Golestanian, The “Friction” of Vacuum, and Other Fluctuation-Induced Forces. Reviews of Modern Physics 71, 1233-1245 (1999).

[5] C. Cattuto, R. Brito, U. Marini Bettolo Marconi, F. Nori and R. Soto, Fluctuation-Induced Casimir Forces in Granular Fluids, Physical Review Letters 96, 178001 (2006).

[6] P. Rodríguez-Lopez, R. Brito, and R. Soto, Physical Review E 83, 031102 (2011).

[7] P. Rodriguez-Lopez, Pairwise Summation Approximation of Casimir Energy from First Principles. Physical Review E 80, 061128 (2009).

[8] P. Rodriguez-Lopez, S.J. Rahi, and T. Emig, Three-Body Casimir Effects and Nonmonotonic Forces. Physical Review A 80, 022519 (2009).