# Fluctuation-induced forces near continuous phase transitions in and out of equilibrium

In this work we study two different realizations of fluctuation-induced forces near continuous phase transitions. In the first part we focus on thermodynamic Casimir forces in equilibrium systems and calculate the universal finite-size scaling functions which describe these forces near the critical point. We quantitatively calculate the temperature-dependent Casimir force for thin Helium-4 films with Dirichlet boundary conditions using Monte Carlo simulations and compare the results with experiments, finding excellent agreement without adjustable parameters. The dependency of Casimir forces and related free energies on the system shape is systematically investigated within the two- and three-dimensional Ising model, where depending on the system shape both attractive and repulsive forces are found. The analysis is completed by an exact calculation of the Casimir force scaling function for the three-dimensional O(n) model with Dirichlet boundary conditions in film geometry in the large-n limit.

In the second part we investigate the properties of a simple model for fluctuation-induced friction in driven magnetic systems far from equilibrium. In this model certain subsystems of a d-dimensional Ising model are moved against each other with a given velocity v, driving the system into a steady state far from equilibrium. The energy pumped into the system is dissipated into the heat bath and induces a friction force, which is fluctuation induced as it vanishes if the magnetic states at the boundary are translationally invariant. The friction force can be of Coulomb- and of Stokes type depending on the involved time scales.

The model can be investigated in various geometries corresponding to both surface friction as well as shear stress. The driven systems show a continuous nonequilibrium phase transition at a temperature T_c(v) > T_c(0), where the energy dissipation and friction force is maximal. In the limit v to infinity the system properties saturate and the model can be exactly solved by mapping it onto an appropriate equilibrium model, from which the critical temperatures and many other quantities can be computed.

For two-dimensional driven boundaries the phase transition becomes strongly anisotropic, with different critical behavior parallel and perpendicular to the driving direction. The critical exponents nu_parallel = 3/2 and nu_perpendicular = 1/2 calculated using a simple Ginzburg-Landau-Wilson field theory are verified numerically using Monte Carlo simulations. Furthermore, the cross-over from the isotropic equilibrium case at v = 0 to the strongly anisotropic case at high velocities is numerically studied in detail using cross-over scaling. We find that for all finite driving velocities v > 0 the critical behavior becomes strongly anisotropic in the thermodynamic limit.

Finally, the analysis is extended to three-dimensional sheared systems at infinite velocity. We find the critical exponents nu_parallel = 1 and nu_perpendicular = 1/2, with considerable corrections to scaling for the available system sizes. We suppose that also in three dimensions the phase transition becomes strongly anisotropic in the thermodynamic limit for all finite driving velocities v > 0, as in the two-dimensional case.

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