Phase Separation and Interfaces. Exact Results

We will devote Chapter 1 to a short review of traditional approaches to interfacial phenomena. This starts with an overview on phenomenological descriptions and terminates with a discussion on mean field theories of interfaces. In Chapter 2 we recall some essential notions of scattering theory in two dimensions on which we will rely in the rest of the thesis. In Chapter 3 we will pose the basis of the exact field-theoretic approach to phase separation in two dimensions. In particular, we will develop the formalism for the study of interfaces in a strip geometry. Drops on a flat substrate and the corresponding wetting transition will be discussed in Chapter 4. In Chapter 5 we will analyze phase separation in presence of a wedge-shaped substrate and its field-theoretical implications. The exposition will cover phase separation both with and without the occurrence of intermediate phases. These two regimes will be discussed in detail for the strip, half-plane and wedge geometries. Our study is based on universal properties of the scaling limit and accounts exactly for the properties of the different universality classes. The field-theoretical approach to near-critical behavior does not exhaust its applications to interfacial phenomena. We will conclude in Chapter 6 with a further application in which we will consider the thermal Casimir e\u21b5ect, i.e. the analogue of the quantum Casimir e\u21b5ect for statistical systems near criticality. We will show how bulk and boundary e\u21b5ects, jointly with the symmetry of boundary conditions, play a role in the determination of the long-distance decay of the Casimir force

Multipoint correlation functions at phase separation. Exact results from field theory

We consider near-critical two-dimensional statistical systems with boundary conditions inducing phase separation on the strip. By exploiting low-energy properties of two-dimensional field theories, we compute arbitrary $n$-point correlation of the order parameter field. Finite-size corrections and mixed correlations involving the stress tensor trace are also discussed. As an explicit illustration of the technique, we provide a closed-form expression for a three-point correlation function and illustrate the explicit form of the long-ranged interfacial fluctuations as well as their confinement within the interfacial region.Comment: 49 pages, 3 figure

Shape and interfacial structure of droplets. Exact results and simulations

We consider the fluctuating interface of a droplet pinned on a flat wall. For such a system we compare results obtained within the exact field theory of phase separation in two dimensions and Monte Carlo (MC) simulations for the Ising model. The interface separating coexisting phases splits and hosts drops whose effect is to produce subleading corrections to the order parameter profile and correlation functions. In this paper we establish the first direct measurement of interface structure effects by means of high-performance MC simulations which successfully confirm the field-theoretical results. Simulations are found to corroborate the theoretical predictions for interface structure effects whose analytical expression has recently been obtained. It is thanks to these higher-order corrections that we are able to correctly resettle a longstanding discrepancy between simulations and theory for the order parameter profile. In addition, our results clearly establish the long-ranged decay of interfacial correlations in the direction parallel to the interface and their spatial confinement within the interfacial region also in the presence of a wall from which the interface is entropically repelled

Inhomogeneous surface tension of chemically active fluid interfaces

We study the dependence of the surface tension of a fluid interface on the density profile of a third suspended phase. By means of an approximated model for the binary mixture and of a perturbative approach we derive close formulas for the free energy of the system and for the surface tension of the interface. Our results show a remarkable non-monotonous dependence of the surface tension on the peak of the density of the suspended phase. Our results also predict the local value of the surface tension in the case in which the density of the suspended phase is not homogeneous along the interface.Comment: 12 pages, 5 figure

Multiple phases and vicious walkers in a wedge

We consider a statistical system in a planar wedge, for values of the bulk parameters corresponding to a first order phase transition and with boundary conditions inducing phase separation. Our previous exact field theoretical solution for the case of a single interface is extended to a class of systems, including the Blume-Capel model as the simplest representative, allowing for the appearance of an intermediate layer of a third phase. We show that the interfaces separating the different phases behave as trajectories of vicious walkers, and determine their passage probabilities. We also show how the theory leads to a remarkable form of wedge covariance, i.e. a relation between properties in the wedge and in the half plane, which involves the appearance of self-Fourier functions

Ensemble dependence of Critical Casimir Forces in Films with Dirichlet Boundary Conditions

In a recent study [Phys. Rev. E \textbf{94}, 022103 (2016)] it has been shown that, for a fluid film subject to critical adsorption, the resulting critical Casimir force (CCF) may significantly depend on the thermodynamic ensemble. Here, we extend that study by considering fluid films within the so-called ordinary surface universality class. We focus on mean-field theory, within which the OP profile satisfies Dirichlet boundary conditions and produces a nontrivial CCF in the presence of external bulk fields or, respectively, a nonzero total order parameter within the film. Our analytical results are supported by Monte Carlo simulations of the three-dimensional Ising model. We show that, in the canonical ensemble, i.e., when fixing the so-called total mass within the film, the CCF is typically repulsive instead of attractive as in the grand canonical ensemble. Based on the Landau-Ginzburg free energy, we furthermore obtain analytic expressions for the order parameter profiles and analyze the relation between the total mass in the film and the external bulk field.Comment: 22 pages, 15 figures. Version 2: minor corrections; added Journal referenc

Passive advection of fractional Brownian motion by random layered flows

We study statistical properties of the process $Y(t)$ of a passive advection by quenched random layered flows in situations when the inter-layer transfer is governed by a fractional Brownian motion $X(t)$ with the Hurst index $H \in (0,1)$. We show that the disorder-averaged mean-squared displacement of the passive advection grows in the large time $t$ limit in proportion to $t^{2 - H}$, which defines a family of anomalous super-diffusions. We evaluate the disorder-averaged Wigner-Ville spectrum of the advection process $Y(t)$ and demonstrate that it has a rather unusual power-law form $1/f^{3 - H}$ with a characteristic exponent which exceed the value $2$. Our results also suggest that sample-to-sample fluctuations of the spectrum can be very important.Comment: 18 pages, 4 figure

Power spectral density of a single Brownian trajectory: what one can and cannot learn from it

The power spectral density (PSD) of any time-dependent stochastic processXt is ameaningful feature of its spectral content. In its text-book definition, the PSD is the Fourier transform of the covariance function of Xt over an infinitely large observation timeT, that is, it is defined as an ensemble-averaged property taken in the limitT  ¥.Alegitimate question iswhat information on the PSDcan be reliably obtained from single-trajectory experiments, if one goes beyond the standard definition and analyzes thePSD of a single trajectory recorded for a finite observation timeT. In quest for this answer, for a d-dimensionalBrownian motion (BM) we calculate the probability density function of a single-trajectory PSDfor arbitrary frequency f, finite observation timeTand arbitrary number k of projections of the trajectory on different axes.We show analytically that the scaling exponent for the frequency-dependence of the PSDspecific to an ensemble ofBMtrajectories can be already obtained from a single trajectory, while the numerical amplitude in the relation between the ensemble-averaged and single-trajectory PSDs is afluctuating property which varies from realization to realization. The distribution of this amplitude is calculated exactly and is discussed in detail.Our results are confirmed by numerical simulations and single-particle tracking experiments, with remarkably good agreement. In addition we consider a truncatedWiener representation ofBM, and the case of a discrete-time lattice randomwalk.Wehighlight some differences in the behavior of a single-trajectory PSDforBMand for the two latter situations.The framework developed herein will allow formeaningful physical analysis of experimental stochastic trajectories

Casimir Contribution to the Interfacial Hamiltonian for 3D Wetting

Previous treatments of three-dimensional (3D) short-ranged wetting transitions have missed an entropic or low-temperature Casimir contribution to the binding potential describing the interaction between the unbinding interface and wall. This we determine by exactly deriving the interfacial model for 3D wetting from a more microscopic Landau-Ginzburg-Wilson Hamiltonian. The Casimir term changes the interpretation of fluctuation effects occurring at wetting transitions so that, for example, mean-field predictions are no longer obtained when interfacial fluctuations are ignored. While the Casimir contribution does not alter the surface phase diagram, it significantly increases the adsorption near a first-order wetting transition and changes completely the predicted critical singularities of tricritical wetting, including the nonuniversality occurring in 3D arising from interfacial fluctuations. Using the numerical renormalization group, we show that, for critical wetting, the asymptotic regime is extremely narrow with the growth of the parallel correlation length characterized by an effective exponent in quantitative agreement with Ising model simulations, resolving a long-standing controversy.Junta de Andalucía US-1380729, P20_00816Der Wissenschaftsfonds M 3300-