The Casimir effect: from quantum to critical fluctuations

The Casimir effect in quantum electrodynamics (QED) is perhaps the best-known example of fluctuation-induced long-ranged force acting on objects (conducting plates) immersed in a fluctuating medium (quantum electromagnetic field in vacuum). A similar effect emerges in statistical physics, where the force acting, e.g., on colloidal particles immersed in a binary liquid mixture is affected by the classical thermal fluctuations occurring in the surrounding medium. The resulting Casimir-like force acquires universal features upon approaching a critical point of the medium and becomes long-ranged at criticality. In turn, this universality allows one to investigate theoretically the temperature dependence of the force via representative models and to stringently test the corresponding predictions in experiments. In contrast to QED, the Casimir force resulting from critical fluctuations can be easily tuned with respect to strength and sign by surface treatments and temperature control. We present some recent advances in the theoretical study of the universal properties of the critical Casimir force arising in thin films. The corresponding predictions compare very well with the experimental results obtained for wetting layers of various fluids. We discuss how the Casimir force between a colloidal particle and a planar wall immersed in a binary liquid mixture has been measured with femto-Newton accuracy, comparing these experimental results with the corresponding theoretical predictions.Comment: Talk delivered at the International Workshop "60 Years of Casimir Effect", Brasilia, 23-27 June 2008 (17 pages, 7 figures

Normal and lateral critical Casimir forces between colloids and patterned substrates

We study the normal and lateral effective critical Casimir forces acting on a spherical colloid immersed in a critical binary solvent and close to a chemically structured substrate with alternating adsorption preference. We calculate the universal scaling function for the corresponding potential and compare our results with recent experimental data [Soyka F., Zvyagolskaya O., Hertlein C., Helden L., and Bechinger C., Phys. Rev. Lett., 101, 208301 (2008)]. The experimental potentials are properly captured by our predictions only by accounting for geometrical details of the substrate pattern for which, according to our theory, critical Casimir forces turn out to be a sensitive probe.Comment: 6 pages, 3 figure

Effective temperatures in a simple model of non-equilibrium, non-Markovian dynamics

The concept of effective temperatures in nonequilibrium systems is studied within an exactly solvable model of non-Markovian diffusion. The system is coupled to two heat baths which are kept at different temperatures: one ('fast') bath associated with an uncorrelated Gaussian noise and a second ('slow') bath with an exponential memory kernel. Various definitions of effective temperatures proposed in the literature are evaluated and compared. The range of validity of these definitions is discussed. It is shown in particular, that the effective temperature defined from the fluctuation-dissipation relation mirrors the temperature of the slow bath in parameter regions corresponding to a separation of time scales. On the contrary, quasi-static and thermodynamic definitions of an effective temperature are found to display the temperature of the fast bath in most parameter regions

Dynamic crossover in the global persistence at criticality

We investigate the global persistence properties of critical systems relaxing from an initial state with non-vanishing value of the order parameter (e.g., the magnetization in the Ising model). The persistence probability of the global order parameter displays two consecutive regimes in which it decays algebraically in time with two distinct universal exponents. The associated crossover is controlled by the initial value m_0 of the order parameter and the typical time at which it occurs diverges as m_0 vanishes. Monte-Carlo simulations of the two-dimensional Ising model with Glauber dynamics display clearly this crossover. The measured exponent of the ultimate algebraic decay is in rather good agreement with our theoretical predictions for the Ising universality class.Comment: 5 pages, 2 figure

Critical Casimir Effect in superfluid wetting films

Recent experimental data for the complete wetting behavior of pure 4He and of 3He-4He mixtures exposed to solid substrates show that there is a change of the corresponding film thicknesses L upon approaching thermodynamically the lambda-transition and the tricritical end point, respectively, which can be attributed to critical Casimir forces f_C. We calculate the scaling functions vartheta of f_C within models representing the corresponding universality classes. For the mixtures our analysis provides an understanding of the rich behavior of vartheta deduced from the experimental data and predicts the crossover behavior between the tricritical point and the lambda-transition of pure 4He which are connected by a line of critical points. The formation of a 'soft-mode' phase within the wetting films gives rise to a pronounced maximum of f_C below the tricritical point as observed experimentally. Near the tricritical point we find logarithmic corrections ~L^(-3)(ln L)^(1/2) for the leading behavior of vartheta dominating the contributions from the background dispersion forces.Comment: 32 pages, 12 figure

The bulk correlation length and the range of thermodynamic Casimir forces at Bose-Einstein condensation

The relation between the bulk correlation length and the decay length of thermodynamic Casimir forces is investigated microscopically in two three-dimensional systems undergoing Bose-Einstein condensation: the perfect Bose gas and the imperfect mean-field Bose gas. For each of these systems, both lengths diverge upon approaching the corresponding condensation point from the one-phase side, and are proportional to each other. We determine the proportionality factors and discuss their dependence on the boundary conditions. The values of the corresponding critical exponents for the decay length and the correlation length are the same, equal to 1/2 for the perfect gas, and 1 for the imperfect gas

Critical Langevin dynamics of the O(N)-Ginzburg-Landau model with correlated noise

We use the perturbative renormalization group to study classical stochastic processes with memory. We focus on the generalized Langevin dynamics of the \phi^4 Ginzburg-Landau model with additive noise, the correlations of which are local in space but decay as a power-law with exponent \alpha in time. These correlations are assumed to be due to the coupling to an equilibrium thermal bath. We study both the equilibrium dynamics at the critical point and quenches towards it, deriving the corresponding scaling forms and the associated equilibrium and non-equilibrium critical exponents \eta, \nu, z and \theta. We show that, while the first two retain their equilibrium values independently of \alpha, the non-Markovian character of the dynamics affects the dynamic exponents (z and \theta) for \alpha < \alpha_c(D, N) where D is the spatial dimensionality, N the number of components of the order parameter, and \alpha_c(x,y) a function which we determine at second order in 4-D. We analyze the dependence of the asymptotic fluctuation-dissipation ratio on various parameters, including \alpha. We discuss the implications of our results for several physical situations

Universal scaling functions of critical Casimir forces obtained by Monte Carlo simulations

Effective Casimir forces induced by thermal fluctuations in the vicinity of bulk critical points are studied by means of Monte Carlo simulations in three-dimensional systems for film geometries and within the experimentally relevant Ising and XY universality classes. Several surface universality classes of the confining surfaces are considered, some of which are relevant for recent experiments. A novel approach introduced previously EPL 80, 60009 (2007), based inter alia on an integration scheme of free energy differences, is utilized to compute the universal scaling functions of the critical Casimir forces in the critical range of temperatures above and below the bulk critical temperature. The resulting predictions are compared with corresponding experimental data for wetting films of fluids and with available theoretical results.Comment: 21 pages, 17 figure

Ageing in the contact process: Scaling behavior and universal features

We investigate some aspects of the ageing behavior observed in the contact process after a quench from its active phase to the critical point. In particular we discuss the scaling properties of the two-time response function and we calculate it and its universal ratio to the two-time correlation function up to first order in the field-theoretical epsilon-expansion. The scaling form of the response function does not fit the prediction of the theory of local scale invariance. Our findings are in good qualitative agreement with recent numerical results.Comment: 20 pages, 3 figure

Exact thermodynamic Casimir forces for an interacting three-dimensional model system in film geometry with free surfaces

The limit n to infinity of the classical O(n) phi^4 model on a 3d film with free surfaces is studied. Its exact solution involves a self-consistent 1d Schr\"odinger equation, which is solved numerically for a partially discretized as well as for a fully discrete lattice model. Numerically exact results are obtained for the scaled Casimir force at all temperatures. Obtained via a single framework, they exhibit all relevant qualitative features of the thermodynamic Casimir force known from wetting experiments on Helium-4 and Monte Carlo simulations, including a pronounced minimum below the bulk critical point.Comment: 5 pages, 2 figure