According to literature, the Casimir effect refers to attraction of two parallel neutral metal plates due to the wavelength cutoff of quantum fluctuations between the two plates.

I come from condensed matter / soft condensed matter / classical statistical mechanical background and I am not able to dive into the underlying QED / QFT formalism to have a good insight about this effect. But many works consider the classical analog which refers to the case of a confined medium (such as a fluid confined between two walls) and subject to thermal fluctuations. So I would like to understand this in more detail.

According to the highly cited work of Fischer and de Gennes, if a binary liquid mixture close to the critical point of phase-separation is confined between two walls then a non-zero force between the two walls will result due to concentration fluctuations of the liquid. Away from the critical point the force would be non-zero if the two walls are closer than the correlation length in the liquid. This correlation length diverges close to the critical point. The study of Fischer and de Gennes was made in the context of stability of colloids in binary mixtures.

But intuitively, how can a thermally fluctuating fluid imply an attractive force between the confining walls? What I find difficult is that in the literature we find many different methods for formalizing / modeling / calculating the classical Casimir force, often with contradicting results.

I briefly summarize :

The intuitive picture presented in the review (on fluctuations induced forces by Kardar in Rev. Mod. Phys) is that the Casimir force stems from the deficient number of normal modes of the fluctuating medium implied by the confining walls. This deficiency favors larger separations between the confining walls due to the rise of entropy. To calculate this, we find in the literature two options :

  • With molecular dynamics or Ising lattice simulations define a fluid at thermal equilibrium confined between two walls, calculate the free energy density, and finally differentiate with respect to the distance between the walls to get the force. (Example https://journals.aps.org/pre/abstract/10.1103/PhysRevE.92.042315)

  • With hydrodynamic equations (momentum conservation and mass conservation, i.e Navier-Stokes and Euler-equations) solve for a fluid confined between two walls with stochastic force due to thermal fluctuations, and calculate the mean force between the two walls. (Example https://pubs.rsc.org/en/content/articlelanding/2016/sm/c5sm02346g)

Here are my questions :

  • Most molecular dynamics or lattice simulations are done in periodic boundary conditions. But imposing periodic boundaries also implies restrictions on the allowed modes of thermal fluctuations, right ? So the "Casimir effect" should be generically expected in any kind of molecular dynamics or lattice simulations with periodic boundaries ? Not just with walls or (so called) symmetry breaking boundary conditions ?

  • For the second method I do not see how the entropy difference can be taken into account in the calculation. This method seem to be only "mechanical" in the sense that free energy (or entropic terms) is not calculated. So how can this be ? How can this approach quantify the qualitative picture suggested above ?

  • Finally, can the change of free energy as function of box length in molecular dynamics simulations possible be expected to imply a change in other (more commonly used) structural or dynamical properties, such as diffusivity, viscosity, and so on ?

  • $\begingroup$ The mechanism how any fluctuations that have energy can give a force can be seen by variation of the volume, similar to the recent discussion here: physics.stackexchange.com/questions/804570/… And it doesn't have to be a binary mixture; the Casimir force equivalent is even found in ordinary water waves. $\endgroup$ Commented Mar 7 at 9:04
  • $\begingroup$ I guess that depends on the source of energy. If water waves or acoustic waves are used to force the system (which is done in few published works), then fluid does not have to be a critical binary mixture. But the case of thermal fluctuations is different. $\endgroup$ Commented Mar 10 at 16:29


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