Showing papers in "Physical Review E in 2012"

Nonlinear Schrödinger equation: generalized Darboux transformation and rogue wave solutions.

Abstract: In this paper, we construct a generalized Darboux transformation for the nonlinear Schr\"odinger equation. The associated $N$-fold Darboux transformation is given in terms of both a summation formula and determinants. As applications, we obtain compact representations for the $N$th-order rogue wave solutions of the focusing nonlinear Schr\"odinger equation and Hirota equation. In particular, the dynamics of the general third-order rogue wave is discussed and shown to exhibit interesting structures.

Forces acting on a small particle in an acoustical field in a viscous fluid

Abstract: We calculate the acoustic radiation force from an ultrasound wave on a compressible, spherical particle suspended in a viscous fluid. Using Prandtl-Schlichting boundary-layer theory, we include the kinematic viscosity of the solvent and derive an analytical expression for the resulting radiation force, which is valid for any particle radius and boundary-layer thickness provided that both of these length scales are much smaller than the wavelength of the ultrasound wave (millimeters in water at megahertz frequencies). The acoustophoretic response of suspended microparticles is predicted and analyzed using parameter values typically employed in microchannel acoustophoresis.

Stochastic thermodynamics under coarse graining

Abstract: A general formulation of stochastic thermodynamics is presented for open systems exchanging energy and particles with multiple reservoirs. By introducing a partition in terms of ``mesostates'' (e.g., sets of ``microstates''), the consequence on the thermodynamic description of the system is studied in detail. When microstates within mesostates rapidly thermalize, the entire structure of the microscopic theory is recovered at the mesostate level. This is not the case when these microstates remain out of equilibrium, leading to additional contributions to the entropy balance. Some of our results are illustrated for a model of two coupled quantum dots.

Epidemic spreading on interconnected networks.

Abstract: Many real networks are not isolated from each other but form networks of networks, often interrelated in nontrivial ways. Here, we analyze an epidemic spreading process taking place on top of two interconnected complex networks. We develop a heterogeneous mean-field approach that allows us to calculate the conditions for the emergence of an endemic state. Interestingly, a global endemic state may arise in the coupled system even though the epidemics is not able to propagate on each network separately and even when the number of coupling connections is small. Our analytic results are successfully confronted against large-scale numerical simulations.

Nonrandomness, nonlinear dependence, and nonstationarity of electroencephalographic recordings from epilepsy patients.

Abstract: To derive tests for randomness, nonlinear-independence, and stationarity, we combine surrogates with a nonlinear prediction error, a nonlinear interdependence measure, and linear variability measures, respectively. We apply these tests to intracranial electroencephalographic recordings (EEG) from patients suffering from pharmacoresistant focal-onset epilepsy. These recordings had been performed prior to and independent from our study as part of the epilepsy diagnostics. The clinical purpose of these recordings was to delineate the brain areas to be surgically removed in each individual patient in order to achieve seizure control. This allowed us to define two distinct sets of signals: One set of signals recorded from brain areas where the first ictal EEG signal changes were detected as judged by expert visual inspection ("focal signals") and one set of signals recorded from brain areas that were not involved at seizure onset ("nonfocal signals"). We find more rejections for both the randomness and the nonlinear-independence test for focal versus nonfocal signals. In contrast more rejections of the stationarity test are found for nonfocal signals. Furthermore, while for nonfocal signals the rejection of the stationarity test increases the rejection probability of the randomness and nonlinear-independence test substantially, we find a much weaker influence for the focal signals. In consequence, the contrast between the focal and nonfocal signals obtained from the randomness and nonlinear-independence test is further enhanced when we exclude signals for which the stationarity test is rejected. To study the dependence between the randomness and nonlinear-independence test we include only focal signals for which the stationarity test is not rejected. We show that the rejection of these two tests correlates across signals. The rejection of either test is, however, neither necessary nor sufficient for the rejection of the other test. Thus, our results suggest that EEG signals from epileptogenic brain areas are less random, more nonlinear-dependent, and more stationary compared to signals recorded from nonepileptogenic brain areas. We provide the data, source code, and detailed results in the public domain.

Rogue waves in the Davey-Stewartson I equation.

Abstract: General rogue waves in the Davey-Stewartson-I equation are derived by the bilinear method. It is shown that the simplest (fundamental) rogue waves are line rogue waves which arise from the constant background with a line profile and then disappear into the constant background again. It is also shown that multirogue waves describe the interaction of several fundamental rogue waves. These multirogue waves also arise from the constant background and then decay back to it, but in the intermediate times, interesting curvy wave patterns appear. However, higher-order rogue waves exhibit different dynamics. Specifically, only part of the wave structure in the higher-order rogue waves rises from the constant background and then retreats back to it, and this transient wave possesses patterns such as parabolas. But the other part of the wave structure comes from the far distance as a localized lump, which decelerates to the near field and interacts with the transient rogue wave, and is then reflected back and accelerates to the large distance again.

Community structure and scale-free collections of Erdős-Rényi graphs.

Abstract: Community structure plays a significant role in the analysis of social networks and similar graphs, yet this structure is little understood and not well captured by most models. We formally define a community to be a subgraph that is internally highly connected and has no deeper substructure. We use tools of combinatorics to show that any such community must contain a dense Erd\ifmmode \mbox{\H{o}}\else \H{o}\fi{}s-R\'enyi (ER) subgraph. Based on mathematical arguments, we hypothesize that any graph with a heavy-tailed degree distribution and community structure must contain a scale-free collection of dense ER subgraphs. These theoretical observations corroborate well with empirical evidence. From this, we propose the Block Two-Level Erd\ifmmode \mbox{\H{o}}\else \H{o}\fi{}s-R\'enyi (BTER) model, and demonstrate that it accurately captures the observable properties of many real-world social networks.

Epidemics on Interconnected Networks

Abstract: Populations are seldom completely isolated from their environment. Individuals in a particular geographic or social region may be considered a distinct network due to strong local ties but will also interact with individuals in other networks. We study the susceptible-infected-recovered process on interconnected network systems and find two distinct regimes. In strongly coupled network systems, epidemics occur simultaneously across the entire system at a critical infection strength! c , below which the disease does not spread. In contrast, in weakly coupled network systems, a mixed phase exists below! c of the coupled network system, where an epidemic occurs in one network but does not spread to the coupled network. We derive an expression for the network and disease parameters that allow this mixed phase and verify it numerically. Public health implications of communities comprising these two classes of network systems are also mentioned.

Temporal node centrality in complex networks.

Abstract: Many networks are dynamic in that their topology changes rapidly---on the same time scale as the communications of interest between network nodes. Examples are the human contact networks involved in the transmission of disease, ad hoc radio networks between moving vehicles, and the transactions between principals in a market. While we have good models of static networks, so far these have been lacking for the dynamic case. In this paper we present a simple but powerful model, the time-ordered graph, which reduces a dynamic network to a static network with directed flows. This enables us to extend network properties such as vertex degree, closeness, and betweenness centrality metrics in a very natural way to the dynamic case. We then demonstrate how our model applies to a number of interesting edge cases, such as where the network connectivity depends on a small number of highly mobile vertices or edges, and show that our centrality definition allows us to track the evolution of connectivity. Finally we apply our model and techniques to two real-world dynamic graphs of human contact networks and then discuss the implication of temporal centrality metrics in the real world.

Epidemic thresholds of the susceptible-infected-susceptible model on networks: a comparison of numerical and theoretical results.

Abstract: Recent work has shown that different theoretical approaches to the dynamics of the susceptible-infectedsusceptible (SIS) model for epidemics lead to qualitatively different estimates for the position of the epidemic threshold in networks. Here we present large-scale numerical simulations of the SIS dynamics on various types of networks, allowing the precise determination of the effective threshold for systems of finite size N . We compare quantitatively the numerical thresholds with theoretical predictions of the heterogeneous mean-field theory and of the quenched mean-field theory. We show that the latter is in general more accurate, scaling with N with the correct exponent, but often failing to capture the correct prefactor.

Optimal diffusion coefficient estimation in single-particle tracking.

Abstract: Single-particle tracking is increasingly used to extract quantitative parameters on single molecules and their environment, while advances in spatial and temporal resolution of tracking techniques inspire new questions and avenues of investigation. Correspondingly, sophisticated analytical methods are constantly developed to obtain more refined information from measured trajectories. Here we point out some fundamental limitations of these approaches due to the finite length of trajectories, the presence of localization error, and motion blur, focusing on the simplest motion regime of free diffusion in an isotropic medium (Brownian motion). We show that two recently proposed algorithms approach the theoretical limit of diffusion coefficient uncertainty. We discuss the practical performance of the algorithms as well as some important implications of these results for single-particle tracking.

Multisolitons, breathers, and rogue waves for the Hirota equation generated by the Darboux transformation.

Abstract: The determinant representation of the $n$-fold Darboux transformation of the Hirota equation is given. Based on our analysis, the 1-soliton, 2-soliton, and breathers solutions are given explicitly. Further, the first order rogue wave solutions are given by a Taylor expansion of the breather solutions. In particular, the explicit formula of the rogue wave has several parameters, which is more general than earlier reported results and thus provides a systematic way to tune experimentally the rogue waves by choosing different values for them.

Optimizing controllability of complex networks by minimum structural perturbations

Abstract: To drive a large, complex, networked dynamical system toward some desired state using as few external signals as possible is a fundamental issue in the emerging field of controlling complex networks. Optimal control is referred to the situation where such a network can be fully controlled using only one driving signal. We propose a general approach to optimizing the controllability of complex networks by judiciously perturbing the network structure. The principle of our perturbation method is validated theoretically and demonstrated numerically for homogeneous and heterogeneous random networks and for different types of real networks as well. The applicability of our method is discussed in terms of the relative costs of establishing links and imposing external controllers. Besides the practical usage of our approach, its implementation elucidates, interestingly, the intricate relationship between certain structural properties of the network and its controllability.

Nonequilibrium thermodynamics of feedback control

Abstract: We establish a general theory of feedback control on classical stochastic thermodynamic systems and generalize nonequilibrium equalities such as the fluctuation theorem and the Jarzynski equality in the presence of feedback control with multiple measurements Our results are generalizations of the previous relevant works to the situations with general measurements and multiple heat baths The obtained equalities involve additional terms that characterize the information obtained by measurements or the efficacy of feedback control A generalized Szilard engine and a feedback-controlled ratchet are shown to satisfy the derived equalities

Bridging the rheology of granular flows in three regimes

Abstract: We investigate the rheology of granular materials via molecular dynamics simulations of homogeneous, simple shear flows of soft, frictional, noncohesive spheres. In agreement with previous results for frictionless particles, we observe three flow regimes existing in different domains of particle volume fraction and shear rate, with all stress data collapsing upon scaling by powers of the distance to the jamming point. Though this jamming point is a function of the interparticle friction coefficient, the relation between pressure and strain rate at this point is found to be independent of friction. We also propose a rheological model that blends the asymptotic relations in each regime to obtain a general description for these flows. Finally, we show that departure from inertial number scalings is a direct result of particle softness, with a dimensionless shear rate characterizing the transition.

Absence of influential spreaders in rumor dynamics.

Abstract: Recent research [Kitsak, Gallos, Havlin, Liljeros, Muchnik, Stanley, and Makse, Nature Physics 6, 888 (2010)] has suggested that coreness, and not degree, constitutes a better topological descriptor to identify influential spreaders in complex networks. This hypothesis has been verified in the context of disease spreading. Here, we instead focus on rumor spreading models, which are more suited for social contagion and information propagation. To this end, we perform extensive computer simulations on top of several real-world networks and find opposite results. Namely, we show that the spreading capabilities of the nodes do not depend on their k-core index, which instead determines whether or not a given node prevents the diffusion of a rumor to a system-wide scale. Our findings are relevant both for sociological studies of contagious dynamics and for the design of efficient commercial viral processes.

Two liquid states of matter: a dynamic line on a phase diagram.

Abstract: It is generally agreed that the supercritical region of a liquid consists of one single state (supercritical fluid). On the other hand, we show here that liquids in this region exist in two qualitatively different states: ``rigid'' and ``nonrigid'' liquids. Rigid to nonrigid transition corresponds to the condition $\ensuremath{\tau}\ensuremath{\approx}{\ensuremath{\tau}}_{0}$, where $\ensuremath{\tau}$ is the liquid relaxation time and ${\ensuremath{\tau}}_{0}$ is the minimal period of transverse quasiharmonic waves. This condition defines a new dynamic crossover line on the phase diagram and corresponds to the loss of shear stiffness of a liquid at all available frequencies and, consequently, to the qualitative change in many important liquid properties. We analyze this line theoretically as well as in real and model fluids and show that the transition corresponds to the disappearance of high-frequency sound, to the disappearance of roton minima, qualitative changes in the temperature dependencies of sound velocity, diffusion, viscous flow, and thermal conductivity, an increase in particle thermal speed to half the speed of sound, and a reduction in the constant volume specific heat to 2${k}_{\mathrm{B}}$ per particle. In contrast to the Widom line that exists near the critical point only, the new dynamic line is universal: It separates two liquid states at arbitrarily high pressure and temperature and exists in systems where liquid-gas transition and the critical point are absent altogether. We propose to call the new dynamic line on the phase diagram ``Frenkel line''.

Forcing scheme in pseudopotential lattice Boltzmann model for multiphase flows

Abstract: The pseudopotential lattice Boltzmann (LB) model is a widely used multiphase model in the LB community. In this model, an interaction force, which is usually implemented via a forcing scheme, is employed to mimic the molecular interactions that cause phase segregation. The forcing scheme is therefore expected to play an important role in the pseudoepotential LB model. In this paper, we aim to address some key issues about forcing schemes in the pseudopotential LB model. First, theoretical and numerical analyses will be made for Shan-Chen's forcing scheme [Shan and Chen, Phys. Rev. E 47, 1815 (1993)] and the exact-difference-method forcing scheme [Kupershtokh et al., Comput. Math. Appl. 58, 965 (2009)]. The nature of these two schemes and their recovered macroscopic equations will be shown. Second, through a theoretical analysis, we will reveal the physics behind the phenomenon that different forcing schemes exhibit different performances in the pseudopotential LB model. Moreover, based on the analysis, we will present an improved forcing scheme and numerically demonstrate that the improved scheme can be treated as an alternative approach to achieving thermodynamic consistency in the pseudopotential LB model.

Acoustic radiation- and streaming-induced microparticle velocities determined by microparticle image velocimetry in an ultrasound symmetry plane

Abstract: We present microparticle image velocimetry measurements of suspended microparticles of diameters from 0.6 to 10 μm undergoing acoustophoresis in an ultrasound symmetry plane in a microchannel. The motion of the smallest particles is dominated by the Stokes drag from the induced acoustic streaming flow, while the motion of the largest particles is dominated by the acoustic radiation force. For all particle sizes we predict theoretically how much of the particle velocity is due to radiation and streaming, respectively. These predictions include corrections for particle-wall interactions and ultrasonic thermoviscous effects and match our measurements within the experimental uncertainty. Finally, we predict theoretically and confirm experimentally that the ratio between the acoustic radiation- and streaming-induced particle velocities is proportional to the actuation frequency, the acoustic contrast factor, and the square of the particle size, while it is inversely proportional to the kinematic viscosity.

Size dependence of the propulsion velocity for catalytic Janus-sphere swimmers.

Abstract: The propulsion velocity of active colloids that asymmetrically catalyze a chemical reaction is probed experimentally as a function of their sizes. It is found that over the experimentally accessible range, the velocity decays as a function of size, with a rate that is compatible with an inverse size dependence. A diffusion-reaction model for the concentrations of the fuel and waste molecules that takes into account a two-step process for the asymmetric catalytic activity on the surface of the colloid is shown to predict a similar behavior for colloids at the large size limit, with a saturation for smaller sizes.

Three-dimensional lattice Boltzmann model for immiscible two-phase flow simulations.

Abstract: We present an improved three-dimensional 19-velocity lattice Boltzmann model for immisicible binary fluids with variable viscosity and density ratios. This model uses a perturbation step to generate the interfacial tension and a recoloring step to promote phase segregation and maintain surfaces. A generalized perturbation operator is derived using the concept of a continuum surface force together with the constraints of mass and momentum conservation. A theoretical expression for the interfacial tension is determined directly without any additional analysis and assumptions. The recoloring algorithm proposed by Latva-Kokko and Rothman is applied for phase segregation, which minimizes the spurious velocities and removes lattice pinning. This model is first validated against the Laplace law for a stationary bubble. It is found that the interfacial tension is predicted well for density ratios up to 1000. The model is then used to simulate droplet deformation and breakup in simple shear flow. We compute droplet deformation at small capillary numbers in the Stokes regime and find excellent agreement with the theoretical Taylor relation for the segregation parameter β=0.7. In the limit of creeping flow, droplet breakup occurs at a critical capillary number 0.35

Multiplexity-facilitated cascades in networks.

Abstract: Elements of networks interact in many ways, so modeling them with graphs requires multiple types of edges (or network layers). Here we show that such multiplex networks are generically more vulnerable to global cascades than simplex networks. We generalize the threshold cascade model [Watts, Proc. Natl. Acad. Sci. USA 99, 5766 (2002)] to multiplex networks, in which a node activates if a sufficiently large fraction of neighbors in any layer are active. We show that both combining layers (i.e., realizing other interactions play a role) and splitting a network into layers (i.e., recognizing distinct kinds of interactions) facilitate cascades. Notably, layers unsusceptible to global cascades can cooperatively achieve them if coupled. On one hand, this suggests fundamental limitations on predicting cascades without full knowledge of a system's multiplexity; on the other hand, it offers feasible means to control cascades by introducing or removing sparse layers in an existing network.

Second-order nonlinear Schrödinger equation breather solutions in the degenerate and rogue wave limits.

Abstract: We present an explicit analytic form for the two-breather solution of the nonlinear Schr\"odinger equation with imaginary eigenvalues. It describes various nonlinear combinations of Akhmediev breathers and Kuznetsov-Ma solitons. The degenerate case, when the two eigenvalues coincide, is quite involved. The standard inverse scattering technique does not generally provide an answer to this scenario. We show here that the solution can still be found as a special limit of the general second-order expression and appears as a mixture of polynomials with trigonometric and hyperbolic functions. A further restriction of this particular case, where the two eigenvalues are equal to $i$, produces the second-order rogue wave with two free parameters considered as differential shifts. The illustrations reveal a precarious dependence of wave profile on the degenerate eigenvalues and differential shifts. Thus we establish a hierarchy of second-order solutions, revealing the interrelated nature of the general case, the rogue wave, and the degenerate breathers.

General bounce-back scheme for concentration boundary condition in the lattice-Boltzmann method.

Abstract: In this paper, a general bounce-back scheme is proposed to implement concentration or thermal boundary conditions of convection-diffusion equation with the lattice Boltzmann method (LBM). Using this scheme, the general concentration boundary conditions, i.e., b1(∂Cw/∂n) + b2Cw = b3, can be easily implemented at boundaries with complex geometry structure like that in porous media. The numerical results obtained using the present scheme are in excellent agreement with the analytical solutions of flows with both stationary and moving interfaces. Furthermore, to better understand the halfway bounce-back scheme, an analytical study of the concentration jump is presented. The studies of theoretical analysis and numerical experiments demonstrate that the proposed scheme has second-order accuracy.

Cluster and group synchronization in delay-coupled networks.

Abstract: We investigate the stability of synchronized states in delay-coupled networks where synchronization takes place in groups of different local dynamics or in cluster states in networks with identical local dynamics. Using a master stability approach, we find that the master stability function shows a discrete rotational symmetry depending on the number of groups. The coupling matrices that permit solutions on group or cluster synchronization manifolds show a very similar symmetry in their eigenvalue spectrum, which helps to simplify the evaluation of the master stability function. Our theory allows for the characterization of stability of different patterns of synchronized dynamics in networks with multiple delay times, multiple coupling functions, but also with multiple kinds of local dynamics in the networks' nodes. We illustrate our results by calculating stability in the example of delay-coupled semiconductor lasers and in a model for neuronal spiking dynamics.

Analysis of complex contagions in random multiplex networks.

Abstract: We study the diffusion of influence in random multiplex networks where links can be of $r$ different types, and, for a given content (e.g., rumor, product, or political view), each link type is associated with a content-dependent parameter ${c}_{i}$ in $[0,\ensuremath{\infty}]$ that measures the relative bias type $i$ links have in spreading this content. In this setting, we propose a linear threshold model of contagion where nodes switch state if their ``perceived'' proportion of active neighbors exceeds a threshold $\ensuremath{\tau}$. Namely a node connected to ${m}_{i}$ active neighbors and ${k}_{i}\ensuremath{-}{m}_{i}$ inactive neighbors via type $i$ links will turn active if $\ensuremath{\sum}{c}_{i}{m}_{i}/\ensuremath{\sum}{c}_{i}{k}_{i}$ exceeds its threshold $\ensuremath{\tau}$. Under this model, we obtain the condition, probability and expected size of global spreading events. Our results extend the existing work on complex contagions in several directions by (i) providing solutions for coupled random networks whose vertices are neither identical nor disjoint, (ii) highlighting the effect of content on the dynamics of complex contagions, and (iii) showing that content-dependent propagation over a multiplex network leads to a subtle relation between the giant vulnerable component of the graph and the global cascade condition that is not seen in the existing models in the literature.

Random walks on temporal networks

Abstract: Many natural and artificial networks evolve in time. Nodes and connections appear and disappear at various time scales, and their dynamics has profound consequences for any processes in which they are involved. The first empirical analysis of the temporal patterns characterizing dynamic networks are still recent, so that many questions remain open. Here, we study how random walks, as a paradigm of dynamical processes, unfold on temporally evolving networks. To this aim, we use empirical dynamical networks of contacts between individuals, and characterize the fundamental quantities that impact any general process taking place upon them. Furthermore, we introduce different randomizing strategies that allow us to single out the role of the different properties of the empirical networks. We show that the random walk exploration is slower on temporal networks than it is on the aggregate projected network, even when the time is properly rescaled. In particular, we point out that a fundamental role is played by the temporal correlations between consecutive contacts present in the data. Finally, we address the consequences of the intrinsically limited duration of many real world dynamical networks. Considering the fundamental prototypical role of the random walk process, we believe that these results could help to shed light on the behavior of more complex dynamics on temporally evolving networks.

Distinguishing chaotic and stochastic dynamics from time series by using a multiscale symbolic approach

Abstract: In this paper we introduce a multiscale symbolic information-theory approach for discriminating nonlinear deterministic and stochastic dynamics from time series associated with complex systems. More precisely, we show that the multiscale complexity-entropy causality plane is a useful representation space to identify the range of scales at which deterministic or noisy behaviors dominate the system's dynamics. Numerical simulations obtained from the well-known and widely used Mackey-Glass oscillator operating in a high-dimensional chaotic regime were used as test beds. The effect of an increased amount of observational white noise was carefully examined. The results obtained were contrasted with those derived from correlated stochastic processes and continuous stochastic limit cycles. Finally, several experimental and natural time series were analyzed in order to show the applicability of this scale-dependent symbolic approach in practical situations.

Transition from spatial coherence to incoherence in coupled chaotic systems.

Abstract: We investigate the spatio-temporal dynamics of coupled chaotic systems with nonlocal interactions, where each element is coupled to its nearest neighbors within a finite range. Depending upon the coupling strength and coupling radius, we find characteristic spatial patterns such as wavelike profiles and study the transition from coherence to incoherence leading to spatial chaos. We analyze the origin of this transition based on numerical simulations and support the results by theoretical derivations, identifying a critical coupling strength and a scaling relation of the coherent profiles. To demonstrate the universality of our findings, we consider time-discrete as well as time-continuous chaotic models realized as a logistic map and a R\"ossler or Lorenz system, respectively. Thereby, we establish the coherence-incoherence transition in networks of coupled identical oscillators.

Grand-potential formulation for multicomponent phase transformations combined with thin-interface asymptotics of the double-obstacle potential.

Abstract: In this paper, we describe the derivation of a model for the simulation of phase transformations in multicomponent real alloys starting from a grand-potential functional. We first point out the limitations of a phase-field model when evolution equations for the concentration and the phase-field variables are derived from a free energy functional. These limitations are mainly attributed to the contribution of the grand-chemical-potential excess to the interface energy. For a range of applications, the magnitude of this excess becomes large and its influence on interface profiles and dynamics is not negligible. The related constraint regarding the choice of the interface thickness limits the size of the domain that can be simulated and, hence, the effect of larger scales on microstructure evolution can not be observed. We propose a modification to the model in order to decouple the bulk and interface contributions. Following this, we perform the thin-interface asymptotic analysis of the phase-field model. Through this, we determine the thin-interface kinetic coefficient and the antitrapping current to remove the chemical potential jump at the interface. We limit our analysis to the Stefan condition at lowest order in $\ensuremath{\epsilon}$ (parameter related to the interface width) and apply results from previous literature that the corrections to the Stefan condition (surface diffusion and interface stretching) at higher orders are removed when antisymmetric interpolation functions are used for interpolating the grand-potential densities and the diffusion mobilities.