Critical Casimir Interactions Between Spherical Particles in the Presence of the Bulk Ordering Fields

The spatial suppression of order parameter fluctuations in a critical media produces Critical Casimir forces acting on confining surfaces. This scenario is realized in a critical binary mixture near the demixing transition point that corresponds to the second order phase transition of the Ising universality class. Due to these critical interactions similar colloids, immersed in a critical binary mixture near the consolute point, exhibit attraction. The numerical method for computation of the interaction potential between two spherical particles using Monte Carlo simulations for the Ising model is proposed. This method is based on the integration of the local magnetization over the applied local magnetic field. For the stronger interaction the concentration of the component of the mixture that does not wet colloidal particles, should be larger, than the critical concentration. The strongest amplitude of the interactions is observed below the critical point.Comment: 7 pages, 4 figure

Critical Casimir Interactions and Percolation: the quantitative description of critical fluctuations

Casimir forces in a critical media are produced by spatial suppression of order parameter fluctuations. In this paper we address the question how fluctuations of a critical media relates the magnitude of critical Casimir interactions. Namely, for the Ising model we express the potential of critical Casimir interactions in terms of Fortuin-Kasteleyn site-bond correlated percolation clusters. These clusters are quantitative representation of fluctuations in the media. New Monte Carlo method for the computation of the Casimir force potential which is based on this relation is proposed. We verify this method by computation of Casimir interactions between two disks for 2D Ising model. The new method is also applied to the investigation of non-additivity of the critical Casimir potential. The non-additive contribution to three-particles interaction is computed as a function of the temperature.Comment: 13 pages, 4 figure

Critical Casimir Forces for Films with Bulk Ordering Fields

The confinement of long-ranged critical fluctuations in the vicinity of second-order phase transitions in fluids generates critical Casimir forces acting on confining surfaces or among particles immersed in a critical solvent. This is realized in binary liquid mixtures close to their consolute point $T_{c}$ which belong to the universality class of the Ising model. The deviation of the difference of the chemical potentials of the two species of the mixture from its value at criticality corresponds to the bulk magnetic filed of the Ising model. By using Monte Carlo simulations for this latter representative of the corresponding universality class we compute the critical Casimir force as a function of the bulk ordering field at the critical temperature $T=T_{c}$. We use a coupling parameter scheme for the computation of the underlying free energy differences and an energy-magnetization integration method for computing the bulk free energy density which is a necessary ingredient. By taking into account finite-size corrections, for various types of boundary conditions we determine the universal Casimir force scaling function as a function of the scaling variable associated with the bulk field. Our numerical data are compared with analytic results obtained from mean-field theory.Comment: 12 pages, 4 figure

Action at a distance in classical uniaxial ferromagnetic arrays

We examine in detail the theoretical foundations of striking long-range couplings emerging in arrays of fluid cells connected by narrow channels by using a lattice gas (Ising model) description of a system. We present a reexamination of the well known exact determination of the two-point correlation function along the edge of a channel using the transfer matrix technique and a new interpretation is provided. The explicit form of the correlation length is found to grow exponentially with the cross section of the channels at the bulk two-phase coexistence. The aforementioned result is recaptured by a refined version of the Fisher-Privman theory of first order phase transitions in which the Boltzmann factor for a domain wall is decorated with a contribution stemming from the point tension originated at its endpoints. The Boltzmann factor for a domain wall together with the point tension is then identified exactly thanks to two independent analytical techniques, providing a critical test of the Fisher-Privman theory. We then illustrate how to build up the network model from its elementary constituents, the cells and the channels. Moreover, we are able to extract the strength of the coupling between cells and express them in terms of the length and width and coarse grained quantities such as surface and point tensions. We then support our theoretical investigation with a series of corroborating results based on Monte Carlo simulations. We illustrate how the long range ordering occurs and how the latter is signaled by the thermodynamic quantities corresponding to both planar and three-dimensional Ising arrays.Comment: 36 pages, 19 figure

Current-mediated synchronization of a pair of beating non-identical flagella

The basic phenomenology of experimentally observed synchronization (i.e., a stochastic phase locking) of identical, beating flagella of a biflagellate alga is known to be captured well by a minimal model describing the dynamics of coupled, limit-cycle, noisy oscillators (known as the noisy Kuramoto model). As demonstrated experimentally, the amplitudes of the noise terms therein, which stem from fluctuations of the rotary motors, depend on the flagella length. Here we address the conceptually important question which kind of synchrony occurs if the two flagella have different lengths such that the noises acting on each of them have different amplitudes. On the basis of a minimal model, too, we show that a different kind of synchrony emerges, and here it is mediated by a current carrying, steady-state; it manifests itself via correlated "drifts" of phases. We quantify such a synchronization mechanism in terms of appropriate order parameters $Q$ and $Q_{\cal S}$ - for an ensemble of trajectories and for a single realization of noises of duration ${\cal S}$, respectively. Via numerical simulations we show that both approaches become identical for long observation times ${\cal S}$. This reveals an ergodic behavior and implies that a single-realization order parameter $Q_{\cal S}$ is suitable for experimental analysis for which ensemble averaging is not always possible.Comment: 10 pages, 2 figure