Analysis of invasion front has been widely used to decipher biological properties, as well as the growth dynamics of the corresponding populations. Likewise, the invasion front of tumors has been investigated, from which insights into the biological mechanisms of tumor growth have been gained. We develop a model to study how tumors' invasion front depends on the relevant properties of a cellular environment. To do so, we develop a model based on a nonlinear reaction-diffusion equation, the Fisher-Kolmogorov-Petrovsky-Piskunov equation, to model tumor growth. Our study aims to understand how heterogeneity in the cellular environment's stiffness, as well as spatial correlations in its morphology, the existence of both of which has been demonstrated by experiments, affects the properties of tumor invasion front. It is demonstrated that three important factors affect the properties of the front, namely the spatial distribution of the local diffusion coefficients, the spatial correlations between them, and the ratio of the cells' duplication rate and their average diffusion coefficient. Analyzing the scaling properties of tumor invasion front computed by solving the governing equation, we show that, contrary to several previous claims, the invasion front of tumors and cancerous cell colonies cannot be described by the well-known models of kinetic growth, such as the Kardar-Parisi-Zhang equation.

The mean ($\kappa$) and Gaussian ($\overline{\kappa }$) bending rigidities of liquid-liquid interfaces, of importance for shape fluctuations and topology of interfaces, respectively, are not yet established: Even their signs are debated. Using the Scheutjens Fleer variant of the self-consistent field theory, we implemented a model for a symmetric L-L interface and obtained high-precision (mean-field) results in the grand-canonical $(\mu ,V,T)$ ensemble. We report positive values for both moduli when the system is close to critical where the rigidities show the same scaling behavior as the interfacial tension $\gamma$. At strong segregation, when the interfacial width becomes of the order of the segment size, $\overline{\kappa }$ turns negative. The length scale $\lambda \equiv \sqrt{\kappa /\gamma }$ remains of the order of segment size for all strengths of interaction; yet the $1/\sqrt{N}$ chain length correction reduces $\lambda$ significantly when the chain length $N$ is small.

Turbulence structures with the magnetic Prandtl number larger than unity are studied by means of direct numerical simulations of homogeneous, isotropic, and incompressible Hall magnetohydrodynamic (MHD) turbulence driven by a random force. Spectral and spatial structures on the scales smaller than the ion skin depth are focused upon in this numerical paper. The numerical simulations reveal the emergence of a new power law in the velocity field whereas it is not observed in MHD turbulence simulation without the Hall term. An order estimate of the energy budget in the spectral space shows that this new power law appears in association with the Hall effect and that a balance between the Lorentz force and the viscous dissipation is crucial for formation of the power law. A resemblance to an elastic turbulence is found in the power-law-formation mechanism. Frequent eruptions of strong current ribbons accompanying strong palinstrophy density are observed, showing generation of the palinstrophy density by the Lorentz force at the scales below the ion skin depth. These properties in spectral and spatial structures characterize a high magnetic Prandtl number Hall MHD turbulence at the scales smaller than the ion skin depth.

We develop and implement a finite difference lattice Boltzmann scheme to study multicomponent flows on curved surfaces, coupling the continuity and Navier-Stokes equations with the Cahn-Hilliard equation to track the evolution of the binary fluid interfaces. The standard lattice Boltzmann method relies on regular Cartesian grids, which makes it generally unsuitable to study flow problems on curved surfaces. To alleviate this limitation, we use a vielbein formalism to write the Boltzmann equation on an arbitrary geometry, and solve the evolution of the fluid distribution functions using a finite difference method. Focusing on the torus geometry as an example of a curved surface, we demonstrate drift motions of fluid droplets and stripes embedded on the surface of the torus. Interestingly, they migrate in opposite directions: fluid droplets to the outer side while fluid stripes to the inner side of the torus. For the latter we demonstrate that the global minimum configuration is unique for small stripe widths, but it becomes bistable for large stripe widths. Our simulations are also in agreement with analytical predictions for the Laplace pressure of the fluid stripes, and their damped oscillatory motion as they approach equilibrium configurations, capturing the corresponding decay timescale and oscillation frequency. Finally, we simulate the coarsening dynamics of phase separating binary fluids in the hydrodynamics and diffusive regimes for tori of various shapes, and compare the results against those for a flat two-dimensional surface. Our finite difference lattice Boltzmann scheme can be extended to other surfaces and coupled to other dynamical equations, opening up a vast range of applications involving complex flows on curved geometries.

We develop a method to investigate the effect of noise timescales on the first-passage time of nonlinear oscillators. Using Fredholm theory, we derive an exact integral equation for the mean event rate of a leaky integrate-and-fire oscillator that receives constant input and temporally correlated noise. Furthermore, we show that Fredholm theory provides a unified framework to determine the system scaling behavior for small and large noise timescales. In this framework, the leading-order and higher-order asymptotic corrections for slow and fast noise are naturally emerging. We show the scaling behavior in the both limits is not reciprocal. We discuss further how this approach can be extended to study the first-passage time in a general class of nonlinear oscillators driven by colored noise at arbitrary timescales.

A many-body system of charged particles interacting via a pairwise Yukawa potential, the so-called Yukawa one-component plasma (YOCP), is a good approximation for a variety of physical systems. Such systems are completely characterized by two parameters: the screening parameter, $\kappa$, and the nominal coupling strength, $\mathrm{\Gamma }$. It is well known that the collective spectrum of the YOCP is governed by a longitudinal acoustic mode, both in the weakly and strongly coupled regimes. In the long-wavelength limit, the linear term in the dispersion (i.e., $\omega =sk)$ defines the sound speed $s$. We study the evolution of this latter quantity from the weak- through the strong-coupling regimes by analyzing the dynamic structure function $S(\mathbf{k},\omega )$ in the low-frequency domain. Depending on the values of $\mathrm{\Gamma }$ and $\kappa$ and $w=s/v_{\mathrm{th}}$ (i.e., the ratio between the phase velocity of the wave and the thermal speed of the particles), we identify five domains in the $(\kappa ,\mathrm{\Gamma })$ parameter space in which the physical behavior of the YOCP exhibits different features. The competing physical processes are the collective Coulomb-like versus binary-collision-dominated behavior and the individual particle motion versus quasilocalization. Our principal tool of investigation is molecular dynamics (MD) computer simulation from which we obtain $S(\mathbf{k},\omega )$. Recent improvements in the simulation technique have allowed us to obtain a large body of high-quality data in the range $\mathrm{\Gamma }=\{0.1-10000\}$ and $\kappa =\{0.5-5\}$. The theoretical results based on various models are compared in order to see which one provides the most cogent physical description and the best agreement with MD data in the different domains.

Thermodynamic observables of mesoscopic systems can be expressed as integrated empirical currents. Their fluctuations are bound by thermodynamic uncertainty relations. We introduce the hyperaccurate current as the integrated empirical current with the least fluctuations in a given nonequilibrium system. For steady-state systems described by overdamped Langevin equations, we derive an equation for the hyperaccurate current by means of a variational principle. We show that the hyperaccurate current coincides with the entropy production if and only if the latter saturates the thermodynamic uncertainty relation, and it can be substantially more precise otherwise. The hyperaccurate current can be used to improve estimates of entropy production from experimental data.

We compare the stochastic closure theory (SCT) to the Townsend-Perry constants as estimated from measurements in the Flow Physic Facility (FPF) at the University of New Hampshire. First, we explain the derivation of the Townsend-Perry constants, which were originally formulated by Meneveau and Marusic, in analogy with a Gaussian distribution. However, this was not supported by the data. Instead, the data show a sub-Gaussian relation that was explained by Birnir and Chen. We show herein how the SCT can be used to compute the constants, which explains their sub-Gaussian relations. We then compare the SCT theory predictions, including Reynolds-number-dependent corrections, with the data, showing good agreement.

Working in and out of equilibrium and using state-of-the-art techniques we have computed the dynamic critical exponent of the three-dimensional Heisenberg model. By computing the integrated autocorrelation time at equilibrium, for lattice sizes $L\le 64$, we have obtained $z=2.033\left(5\right)$. In the out-of-equilibrium regime we have run very large lattices ($L\le 250$) obtaining $z=2.04\left(2\right)$ from the growth of the correlation length. We compare our values with that previously computed at equilibrium with relatively small lattices ($L\le 24$), with that provided by means a three-loops calculation using perturbation theory and with experiments. Finally we have checked previous estimates of the static critical exponents, $\eta$ and $\nu$, in the out-of-equilibrium regime.

The Voronoi construction is ubiquitous across the natural sciences and engineering. In statistical mechanics, however, only its dual, the Delaunay triangulation, has been considered in the investigation of critical phenomena. In this paper we set to fill this gap by studying three prominent systems of classical statistical mechanics, the equilibrium spin-1/2 Ising model, the nonequilibrium contact process, and the conserved stochastic sandpile model on two-dimensional random Voronoi graphs. Particular motivation comes from the fact that these graphs have vertices of constant coordination number, making it possible to isolate topological effects of quenched disorder from node-intrinsic coordination number disorder. Using large-scale numerical simulations and finite-size scaling techniques, we are able to demonstrate that all three systems belong to their respective clean universality classes. Therefore, quenched disorder introduced by the randomness of the lattice is irrelevant and does not influence the character of the phase transitions. We report the critical points to considerable precision and, for the Ising model, also the first correction-to-scaling exponent.

In phase space, we analytically obtain the characteristic functions (CFs) of a forced harmonic oscillator [Talkner et al., Phys. Rev. E 75, 050102(R) (2007)], a time-dependent mass and frequency harmonic oscillator [Deffner and Lutz, Phys. Rev. E 77, 021128 (2008)], and coupled harmonic oscillators under driving forces in a simple and unified way. For general quantum systems, a numerical method that approximates the CFs to ${\hslash }^2$ order is proposed. We exemplify the method with a time-dependent frequency harmonic oscillator and a family of quantum systems with time-dependent even power-law potentials.

We revisit the problem on the inner structure of shock waves in simple gases modelized by the Boltzmann kinetic equation. In a paper by Pomeau [Y. Pomeau, Transp. Theory Stat. Phys. 16, 727 (1987)], a self-similarity approach was proposed for infinite total cross section resulting from a power-law interaction, but this self-similar form does not have finite energy. Motivated by the work of Pomeau [Y. Pomeau, Transp. Theory Stat. Phys. 16, 727 (1987)] and Bobylev and Cercignani [A. V. Bobylev and C. Cercignani, J. Stat. Phys. 106, 1039 (2002)], we started the research on the rigorous study of the solutions of the spatial homogeneous Boltzmann equation, focusing on those which do not have finite energy. However, infinite energy solutions do not have physical meaning in the present framework of kinetic theory of gases with collisions conserving the total kinetic energy. In the present work, we provide a correction to the self-similar form, so that the solutions are more physically sound in the sense that the energy is no longer infinite and that the perturbation brought by the shock does not grow at large distances of it on the cold side in the soft potential case.

We present a systematic analysis of diffusion-controlled interaction and collapse of two identical spatially separated $d$-dimensional $A$-particle islands in the $B$-particle sea at propagation of the sharp reaction front $A+B\to 0$ at equal species diffusivities. We show that at a sufficiently large initial distance between the centers of islands $2\ell$ compared to their characteristic initial size and a relatively large initial ratio of island to sea concentrations, the evolution dynamics of the island-sea-island system is determined unambiguously by the dimensionless parameter $\mathrm{\Lambda }={\mathcal{N}}_0/{\mathcal{N}}_{\mathrm{\Omega }}$, where ${\mathcal{N}}_0$ is the initial particle number in the island and ${\mathcal{N}}_{\mathrm{\Omega }}$ is the initial number of sea particles in the volume $\mathrm{\Omega }={\left(2\ell \right)}^d$. It is established that (a) there is a $d$-dependent critical value ${\mathrm{\Lambda }}_{★}$ above which island coalescence occurs; (b) regardless of $d$ the centers of each of the islands move toward each other along a universal trajectory merging in a united center at the $d$-dependent critical value ${\mathrm{\Lambda }}_s\ge {\mathrm{\Lambda }}_{★}$; (c) in one-dimensional systems ${\mathrm{\Lambda }}_{★}={\mathrm{\Lambda }}_s$, therefore, at $\mathrm{\Lambda }<{\mathrm{\Lambda }}_{★}$ each of the islands dies individually, whereas at $\mathrm{\Lambda }>{\mathrm{\Lambda }}_{★}$ coalescence is completed by collapse of a single-centered island in the system center; (d) in two- and three-dimensional systems in the range ${\mathrm{\Lambda }}_{★}<\mathrm{\Lambda }<{\mathrm{\Lambda }}_s$ coalescence is accompanied by subsequent fragmentation of a two-centered island and is completed by individual collapse of each of the islands. We discuss a detailed picture of coalescence, fragmentation, and collapse of islands focusing on evolution of their shape and on behavior of the relative width of the reaction front at the final collapse stage and in the vicinity of starting coalescence and fragmentation points. We demonstrate that in a wide range of parameters, the front remains sharp up to a narrow vicinity of the coalescence, fragmentation, and collapse points.

We explore the intriguing spatial patterns that emerge in a two-dimensional spatially inhomogeneous Katz-Lebowitz-Spohn (KLS) driven lattice gas with attractive nearest-neighbor interactions. The domain is split into two regions with hopping rates governed by different temperatures $T>T_c$ and $T_c$, respectively, where $T_c$ indicates the critical temperature for phase ordering, and with the temperature boundaries oriented perpendicular to the drive. In the hotter region, the system behaves like the (totally) asymmetric exclusion processes (TASEP), and experiences particle blockage in front of the interface to the critical region. To explain this particle density accumulation near the interface, we have measured the steady-state current in the KLS model at $T>T_c$ and found it to decay as $1/T$. In analogy with TASEP systems containing “slow” bonds, we argue that transport in the high-temperature subsystem is impeded by the lower current in the cooler region, which tends to set the global stationary particle current value. This blockage is induced by the extended particle clusters, growing logarithmically with system size, in the critical region. We observe the density profiles in both high- and low-temperature subsystems to be similar to the well-characterized coexistence and maximal-current phases in (T)ASEP models with open boundary conditions, which are respectively governed by hyperbolic and trigonometric tangent functions. Yet if the lower temperature is set to $T_c$, we detect marked fluctuation corrections to the mean-field density profiles, e.g., the corresponding critical KLS power-law density decay near the interfaces into the cooler region.

The bacterial cell wall is primarily composed of a mesh of glycan strands cross-linked by peptide bridges and is essential for safeguarding the cell. The structure of the cell wall has to be stiff enough to bear the high turgor pressure and sufficiently tough to ensure protection against failure. Here we explore the role of various design features of the cell in enhancing the toughness of the cell wall. We explain how the glycan strand length distribution, the degree of cross-linking and the placement of the cross-links on the glycan strands can act in tandem to ensure that the cell wall offers sufficient resistance to propagation of cracks. Further, we suggest a possible mechanism by which peptide bond hydrolysis, via judicious cleaving of peptide cross-links, can act to mitigate this risk of failure. We also study the reinforcing effect of MreB cytoskeleton, which can offer a degree of safety to the cell wall. However, we show that the cross-linked structure of the cell wall is its primary line of defense against mechanical failure.

Origami-inspired design has recently emerged as a major thrust area of research in the fields of science and engineering. One such design utilizes Kresling-pattern origami to construct nonlinear springs that can act as mechanical bit memory switches, wave guides, fluidic muscles, and vibration isolators. The main objective of this work is to characterize the static equilibria of such springs, their stability, and bifurcations as the geometric parameters of the Kresling pattern are varied. To this end, a mathematical model which assumes that the different panels can be represented by axially deformable truss elements is adopted. The adopted model demonstrates that the shape of the potential energy of the spring is very sensitive to changes in its geometric parameters. This causes the static configuration to undergo several bifurcations as one or more of the geometrical parameters are varied. In particular, it is shown that the geometric parameter space of the Kresling pattern can be divided into five regions, each of which results in a qualitatively different spring behavior. Results of the axial truss model are verified experimentally demonstrating that, for the most part, the model is capable of predicting the loci and bifurcations of the spring's equilibria. Nevertheless, it is also observed that, away from the equilibrium points, the quasistatic behavior of the spring is not well-approximated by the axial truss model. To overcome this issue, a modified model is developed which accounts for (i) the rotary stiffness of the creases, (ii) self avoidance due to panel contact at small angles between the panels, and (iii) buckling of the creases under compressive loads. It is shown that the modified model is capable of providing a better overall qualitative approximation of the quasistatic behavior.

The phenomenon of localized surface plasmon resonance (LSPR) provides high sensitivity in detecting biomolecules through shifts in resonance frequency when a target is present. Computational studies in this field have used the full Maxwell equations with simplified models of a sensor-analyte system, or they neglected the analyte altogether. In the long-wavelength limit, one can simplify the theory via an electrostatics approximation while adding geometrical detail in the sensor and analytes (at moderate computational cost). This work uses the latter approach, expanding the open-source PyGBe code to compute the extinction cross section of metallic nanoparticles in the presence of any target for sensing. The target molecule is represented by a surface mesh, based on its crystal structure. PyGBe is research software for continuum electrostatics, written in python with computationally expensive parts accelerated on GPU hardware, via PyCUDA. It is also accelerated algorithmically via a treecode that offers $O(NlogN)$ computational complexity. These features allow PyGBe to handle problems with half a million boundary elements or more. In this work, we demonstrate the suitability of PyGBe, extended to compute LSPR response in the electrostatic limit, for biosensing applications. Using a model problem consisting of an isolated silver nanosphere in an electric field, our results show grid convergence as $1/N$, and accurate computation of the extinction cross section as a function of wavelength (compared with an analytical solution). For a model of a sensor-analyte system, consisting of a spherical silver nanoparticle and a set of bovine serum albumin (BSA) proteins, our results again obtain grid convergence as $1/N$ (with respect to the Richardson extrapolated value). Computing the LSPR response as a function of wavelength in the presence of BSA proteins captures a redshift of 0.5 nm in the resonance frequency due to the presence of the analytes at 1-nm distance. The final result is a sensitivity study of the biosensor model, obtaining the shift in resonance frequency for various distances between the proteins and the nanoparticle. All results in this paper are fully reproducible, and we have deposited in archival data repositories all the materials needed to run the computations again and recreate the figures. PyGBe is open source under a permissive license and openly developed. Documentation is available at http://pygbe.github.io/pygbe/docs/.

We present a random-interface representation of the three-dimensional (3D) Ising model based on thermal fluctuations of a uniquely defined geometric spin cluster in the 3D model and its 2D cross section. Extensive simulations have been carried out to measure the global interfacial width as a function of temperature for different lattice sizes which is shown to signal the criticality of the model at $T_c$ by forming a size-independent cusp in 3D, along with an emergent super-roughening at its 2D cross section. We find that the super-rough state is accompanied by an intrinsic anomalous scaling behavior in the local properties characterized by a set of geometric exponents which are the same as those for a pure 2D Ising model.

Phase-field theory is widely used to model multiphase flow. The fact that a drop can shrink or grow spontaneously due to the redistribution of interface and bulk energies to minimize the system energy may produce ill effects on the simulation. In this Rapid Communication, the spontaneous behavior of a drop on a partially wetting surface is investigated. It is found that there exists a critical radius dependent on the contact angle, the domain size, and the interface width, below which the drop will eventually disappear. In particular, the critical radius can be very large when the surface becomes very hydrophilic. The theoretical prediction of the critical radius is verified numerically by simulating a drop on a surface with various contact angles, the domain sizes, and the interface widths.

We study the relationship between avalanche criticality and the number of orientational domains in ferroelastic transitions. To this end, we use a general Ginzburg-Landau model appropriate for displacive transitions of the square lattice. The model includes disorder as a quenched distribution of local transition temperatures. We focus on the square-to-rectangle and the square-to-oblique ferroelastic transitions, which have two and four orientational domains, respectively, which in turn determine the corresponding degeneracy of the ground state of the system. The phase transitions are driven by temperature under the assumption of a strict athermal behavior. That is, we assume that thermal fluctuations do not play any role. Numerical results are obtained using a purely relaxational dynamics, and it is shown that both the square-to-rectangle and the square-to-oblique transitions occur intermittently in the form of avalanches. Avalanche sizes and avalanche energies are found to display power-law distributions, which corroborates avalanche criticality. We compare and contrast the dependence of avalanche criticality on the number of orientational domains of the low-symmetry phase. It is found that the critical exponents depend on that number, in agreement with recent experimental results.