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Numerical Analysis of some Generalized Casimir Pistons

The Casimir force due to a scalar field on a piston in a cylinder of radius $r$ with a spherical cap of radius $R>r$ is computed numerically in the world-line approach. A geometrical subtraction scheme gives the finite interaction energy that determines the Casimir force. The spectral function of convex domains is obtained from a probability measure on convex surfaces that is induced by the Wiener measure on Brownian bridges the convex surfaces are the hulls of. The vacuum force on the piston by a scalar field satisfying Dirichlet boundary conditions is attractive in these geometries, but the strength and short-distance behavior of the force depends crucially on the shape of the piston casing. For a cylindrical casing with a hemispherical head, the force for $a/R\sim 0$ does not depend on the dimension of the casing and numerically approaches $\sim - 0.00326(4)\hbar c/a^2$. Semiclassically this asymptotic force is due to short, closed and non-periodic trajectories that reflect once off the piston near its periphery. The semiclassical estimate $-\hbar c/(96πa^2)(1+2\sqrt{R^2-r^2}/a)$ for the force when $a/r\ll r/R\leq 1$ reproduces the numerical results within statistical errors.