Casimir Forces and Boundary Conditions in One Dimension : Attraction , Repulsion , Planck Spectrum , and Entropy

Quantities associated with Casimir forces are calculated in a model wave system of one spatial dimension where the physical ideas are transparent and the calculations allow easy numerical evaluation. The calculations show strong dependence upon fixedor free-end (Dirichlet or Neumann) boundary conditions for waves on a one-dimensional string, analogous to infinitely-conducting or infinitely-permeable materials for electromagnetic waves. 1) Due to zero-point fluctuations, a partition in a one-dimensional box is found to be attracted to the walls if the wave boundary conditions are alike for the partition and the walls, but is repelled if the conditions are different. 2) The use of Casimir energies in the presence of zero-point radiation introduces a natural maximum-entropy principle which is satisfied only by the Planck spectrum for both like and unlike boundary conditions between the box and partition. 3)The one-dimensional Casimir forces increase or decrease with increasing temperature depending upon like or unlike boundary conditions. The Casimir forces are attractive and increasing with temperature for like boundary conditions, but are repusive and decreasing with temperature for unlike boundary conditions. 4) In the high-temperature limit, there is a temperature-independent Casimir entropy for like boundary conditions, but a vanishing Casimir entropy for unlike boundary conditions. These one-dimensional results have counterparts for electromagnetic Casimir forces in three dimensions.