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

On the distribution of surface extrema in several one- and two-dimensional random landscapes

We study here a standard next-nearest-neighbor (NNN) model of ballistic growth on one- and two-dimensional substrates focusing our analysis on the probability distribution function $P(M,L)$ of the number $M$ of maximal points (i.e., local ``peaks'') of growing surfaces. Our analysis is based on two central results: (i) the proof (presented here) of the fact that uniform one--dimensional ballistic growth process in the steady state can be mapped onto ''rise-and-descent'' sequences in the ensemble of random permutation matrices; and (ii) the fact, established in Ref. \cite{ov}, that different characteristics of ``rise-and-descent'' patterns in random permutations can be interpreted in terms of a certain continuous--space Hammersley--type process. For one--dimensional system we compute $P(M,L)$ exactly and also present explicit results for the correlation function characterizing the enveloping surface. For surfaces grown on 2d substrates, we pursue similar approach considering the ensemble of permutation matrices with long--ranged correlations. Determining exactly the first three cumulants of the corresponding distribution function, we define it in the scaling limit using an expansion in the Edgeworth series, and show that it converges to a Gaussian function as $L \to \infty$.Comment: 25 pages, 12 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

Narrow-escape times for diffusion in microdomains with a particle-surface affinity: Mean-field results

We analyze the mean time t_{app} that a randomly moving particle spends in a bounded domain (sphere) before it escapes through a small window in the domain's boundary. A particle is assumed to diffuse freely in the bulk until it approaches the surface of the domain where it becomes weakly adsorbed, and then wanders diffusively along the boundary for a random time until it desorbs back to the bulk, and etc. Using a mean-field approximation, we define t_{app} analytically as a function of the bulk and surface diffusion coefficients, the mean time it spends in the bulk between two consecutive arrivals to the surface and the mean time it wanders on the surface within a single round of the surface diffusion.Comment: 8 pages, 1 figure, submitted to JC

Random patterns generated by random permutations of natural numbers

We survey recent results on some one- and two-dimensional patterns generated by random permutations of natural numbers. In the first part, we discuss properties of random walks, evolving on a one-dimensional regular lattice in discrete time $n$, whose moves to the right or to the left are induced by the rise-and-descent sequence associated with a given random permutation. We determine exactly the probability of finding the trajectory of such a permutation-generated random walk at site $X$ at time $n$, obtain the probability measure of different excursions and define the asymptotic distribution of the number of "U-turns" of the trajectories - permutation "peaks" and "through". In the second part, we focus on some statistical properties of surfaces obtained by randomly placing natural numbers $1,2,3, >...,L$ on sites of a 1d or 2d square lattices containing $L$ sites. We calculate the distribution function of the number of local "peaks" - sites the number at which is larger than the numbers appearing at nearest-neighboring sites - and discuss some surprising collective behavior emerging in this model.Comment: 16 pages, 5 figures; submitted to European Physical Journal, proceedings of the conference "Stochastic and Complex Systems: New Trends and Expectations" Santander, Spai

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