Statistical field theory with constraints: application to critical Casimir forces in the canonical ensemble

Abstract

The effect of imposing a constraint on a fluctuating scalar order parameter field in a system of finite volume is studied within statistical field theory. The canonical ensemble, corresponding to a fixed total integrated order parameter, is obtained as a special case of the theory. A perturbative expansion is developed which allows one to systematically determine the constraint-induced finite-volume corrections to the free energy and to correlation functions. In particular, we focus on the Landau-Ginzburg model in a film geometry (i.e., a rectangular parallelepiped with a small aspect ratio) with periodic, Dirichlet, or Neumann boundary conditions in the transverse direction and periodic boundary conditions in the remaining, lateral directions. Within the expansion in terms of $\epsilon =4-d$, where $d$ is the spatial dimension of the bulk, the finite-size contribution to the free energy and the associated critical Casimir force are calculated to leading order in $\epsilon$ and are compared to the corresponding expressions for an unconstrained (grand canonical) system. The constraint restricts the fluctuations within the system and it accordingly modifies the residual finite-size free energy. The resulting Casimir force is shown to depend on whether it is defined by assuming a fixed transverse area or a fixed total volume. In the former case, the constraint is typically found to significantly enhance the attractive character of the force as compared to the grand canonical case. In contrast to the grand canonical Casimir force, which, for supercritical temperatures, vanishes in the limit of thick films, the canonical Casimir force defined for fixed transverse area attains for thick films a negative value for all boundary conditions studied here. Typically, the dependence of the Casimir force both on the temperature- and on the field-like scaling variables is different in the two ensembles.

46 pages, 11 figures

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• Fluctuation-induced Interactions in Micro- and Nano-systems: Survey of Some Basic Results
• Exact expressions for the partition function of the one-dimensional Ising model in the fixed-$M$ ensemble
• Surface-induced non-equilibrium dynamics and critical Casimir forces for model B in film geometry
• Dynamics and steady states of a tracer particle in a confined critical fluid
• Casimir versus Helmholtz forces: Exact results
• Dynamics of the critical Casimir force for a conserved order parameter after a critical quench
• Casimir and Helmholtz forces in one-dimensional Ising model with Dirichlet (free) boundary conditions
• Repulsive to Attractive Fluctuation-Induced Forces in Disordered Landau-Ginzburg Model
• Critical surface adsorption of confined binary liquids with locally conserved mass and composition
• Fluctuations of the critical Casimir force
• Variance of the Casimir force in an ideal Bose gas
• Casimir-like effect from thermal field fluctuations
• Constraint correlation functions of the one-dimensional Ising model in the scaling limit