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The specific heat, the energy density and the thermodynamic Casimir force in the neighbourhood of the lambda-transition

We discuss the relation of the specific heat, the energy density and the thermodynamic Casimir effect in the case of thin films in the three dimensional XY universality class. The finite size scaling function $θ(x)$ of the thermodynamic Casimir force can be expressed in terms of the scaling functions h'(x) and h(x) of the excess energy density and the excess free energy density. A priori these quantities depend on the reduced temperature t and the thickness L_0 of the film. However finite size scaling theory predicts that the scaling functions depend only on the combination x=t [L_0/ξ_0]^{1/ν}, where νis the critical exponent and $ξ_0$ the amplitude of the correlation length. We exploit this fact to compute θfrom Monte Carlo data for the excess energy density of the improved two-component ϕ^4 model on the simple cubic lattice with free boundary conditions in the short direction. We repeat this exercise using experimental data for the excess specific heat of 4He films. The finite size scaling behaviour of the excess specific heat is governed by h''(x), which is proportional to the scaling function $f_2$ discussed in the literature. We compare our results with previous work, where the Casimir force has been computed by taking the derivative of the excess free energy with respect to the thickness of the film. As a preparative study we have also computed the scaling functions h'(x) and h(x) for finite L^3 systems with periodic boundary conditions in all directions, where L is the linear extension of the system.