5 research outputs found
Numerical Analysis of some Generalized Casimir Pistons
The Casimir force due to a scalar field on a piston in a cylinder of radius
with a spherical cap of radius 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 does not depend on the dimension of the casing and
numerically approaches . 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
for the force when reproduces the numerical results within statistical errors.Comment: 17 pages, 7 figure
Nonstationary random acoustic and electromagnetic fields as wave diffusion processes
We investigate the effects of relatively rapid variations of the boundaries
of an overmoded cavity on the stochastic properties of its interior acoustic or
electromagnetic field. For quasi-static variations, this field can be
represented as an ideal incoherent and statistically homogeneous isotropic
random scalar or vector field, respectively. A physical model is constructed
showing that the field dynamics can be characterized as a generalized diffusion
process. The Langevin--It\^{o} and Fokker--Planck equations are derived and
their associated statistics and distributions for the complex analytic field,
its magnitude and energy density are computed. The energy diffusion parameter
is found to be proportional to the square of the ratio of the standard
deviation of the source field to the characteristic time constant of the
dynamic process, but is independent of the initial energy density, to first
order. The energy drift vanishes in the asymptotic limit. The time-energy
probability distribution is in general not separable, as a result of
nonstationarity. A general solution of the Fokker--Planck equation is obtained
in integral form, together with explicit closed-form solutions for several
asymptotic cases. The findings extend known results on statistics and
distributions of quasi-stationary ideal random fields (pure diffusions), which
are retrieved as special cases.Comment: 54 pages, 8 figures, to appear in J. Phys. A: Math. Theo