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The out of equilibrium behavior of Casimir type fluctuation induced forces for free classical fields

We present a general method to study the non-equilibrium behavior of Casimir type fluctuation induced forces for classical free scalar field theories. In particular we analyze the temporal evolution of the force towards its equilibrium value when the field dynamics is given by a general class of over damped stochastic dynamics (including the model A and model B class). The steady state force is also analyzed for systems which have non-equilibrium steady states, for instance where they are driven by colored noise. The key to the method is that out of equilibrium force is computed by specifying an energy of interaction between the field and the surfaces in the problem. In general we find that there is a mapping of the dynamical problem onto a corresponding static one, and in the case where the latter can be solved the full dynamical behavior of the force can be extracted. The method is used how to compute the non-equilibrium Casimir force induced between two parallel plates by a fluctuating field, in the cases of Dirichlet, Neumann and mixed boundary conditions. Various other examples, such as the fluctuation induced force between inclusions in fluctuating media are discussed.