Asymmetric momentum transmission by diffraction and its exact cancellation in equilibrium: a multi-slit wall facing a single-slit wall

A structure made of a wall carrying a double slit (two slits of width 𝑎) and a wall carrying a single slit of width 2𝑎, placed facing each other at a distance 𝑑 and joined together, exhibits an asymmetric diffraction effect for molecules that behave as de Broglie waves. If, when 𝑑 is taken of the order of the mean free path, the probability of entering on the double-slit side and leaving through the single slit without a collision differs from the probability of the reverse process, the structure appears to receive a net force in an isotropic gas. This paper examines this question, generalizing the double slit to a multiple slit, with a two-dimensional scalar Helmholtz model, locally reacting impedance walls (𝛽 = 0 to 1), and the boundary element method. First, asymmetric momentum transmission exists. The net momentum delivered to the structure by a pair of counter-propagating beams of equal strength is not zero; in the model with 𝜆/𝑎 = 0.5 it reaches 0.4 to 12% of the momentum delivered by the flux from one side, and it is nonzero even under conditions in which a geometric-optics control without diffraction gives exactly zero. Second, in an isotropic gas in equilibrium, the force generated by this asymmetry cancels exactly. The cancellation can be proved as a chain of four identities — the vanishing of the flux of the incident field, a momentum version of the extinction theorem, a reciprocity lemma, and Kirchhoff's law — of which the first three are general rules of linear scattering and only Kirchhoff's law is specific to thermal equilibrium. Numerically, the sum of the absorption-stage force and the re-emission recoil converges to zero at the level of 10^(−5) over resolution series of 7 to 9 rungs for 𝛽∈{0.25,0.5,1}. Third, the asymmetry survives neither in the net force nor in the first moment of the force, but only in the structure of the angular distribution — two lobes of opposite sign and fringes of width 𝜆/𝑊. In non-thermal isotropic fields in which Kirchhoff's law does not act — a diffuse sound field, or water waves with an isotropic directional distribution — a net force remains on structures with asymmetric dissipation, as the isotropic sum of the known drift forces, and none remains on lossless structures. This paper is a wave version of the Smoluchowski–Feynman ratchet, and the passive propulsion by diffraction originally proposed by the author is retracted.