A central challenge in developmental biology is understanding the creation of robust spatiotemporal heterogeneity. Generally, the mathematical treatments of biological systems have used continuum, mean-field hypotheses for their constituent parts, which ignores any sources of intrinsic stochastic effects. In this paper we consider a stochastic space-jump process as a description of diffusion, i.e., particles are able to undergo a random walk on a discretized domain. By developing analytical Fourier methods we are able to probe this probabilistic framework, which gives us insight into the patterning potential of diffusive systems. Further, an alternative description of domain growth is introduced, with which we are able to rigorously link the mean-field and stochastic descriptions. Finally, through combining these ideas, it is shown that such stochastic descriptions of diffusion on a deterministically growing domain are able to support the nucleation of states that are far removed from the deterministic mean-field steady state.

Statistical properties of forced two-dimensional turbulence generated in two different flow domains are investigated by numerical simulations. The considered geometries are the square domain and the periodic channel domain, both bounded by lateral no-slip sidewalls. The focus is on the direct enstrophy cascade range and how the statistical properties change in the presence of no-slip boundaries. The scaling exponents of the velocity and the vorticity structure functions are compared to the classical Kraichnan-Batchelor-Leith (KBL) theory, which assumes isotropy, homogeneity, and self-similarity for turbulence scales between the forcing and dissipation scale. Our investigation reveals that in the interior of the flow domain, turbulence can be considered statistically isotropic and locally homogeneous for the enstrophy cascade range, but it is weakly intermittent. However, the scaling of the vorticity structure function indicates a steeper slope for the energy spectrum than the KBL theory predicts. Near the walls the turbulence is strongly anisotropic at all flow scales.

This paper presents some experimental results on two-phase flows in model two-dimensional (2D) porous media with different wetting properties. Standard microfluidic techniques are used to fabricate the 2D micromodels that consist of a network of straight microchannels having heterogeneous sizes. The invasion mechanism is analyzed quantitatively for partial and total wetting conditions, and for various stable viscosity ratios and capillary pressure heterogeneity. For capillary numbers ranging from ${10}^{-7}$ to ${10}^{-2}$, we observe a transition between capillary fingering and a stable front. The capillary fingering regime exhibits differences between partial and complete wetting systems: The front width in complete wetting is larger. Simple models are proposed to account for these regimes and indicate that the differences between the systems are likely to be due to the flow of the displaced fluid in the complete wetting situation.

We distinguish between trivial and nontrivial differences in retrieving the real or imaginary parts of the Green's function. Trivial differences come from different Green's function definitions. The energy and Lagrangian forms constitute nontrivial differences. Magnetic noise sources suffice to extract the quasistatic electromagnetic-field Earth impulse response in the Lagrangian form. This is of interest for Earth subsurface imaging. A numerical example demonstrates that all source vector components are necessary to extract a single-field vector component.

Stress and strain relaxation in randomly folded paper sheets under axial compression is studied both experimentally and theoretically. Equations providing the best fit to the experimental data are found. Our findings suggest that, in an axially compressed ball folded from an elastic or elasto-plastic material, the relaxation dynamics is ruled by activated processes of an energy foci rearrangement in the crumpling network. The dynamics of relaxation is discussed within a framework of Edwards's statistical mechanics. The functional forms of the activation barrier between admissible jammed folding configurations of the crumpling network under axial compression are derived. It is shown that relaxation kinetics can be mapped to activated dynamics of depinning and creep of elastic interface in a disordered medium.

We show that a harmonic lattice model with amplifying and attenuating elements, when coupled to two thermal baths, exhibits unique heat transport properties. Some of these novel features include anomalous nonequilibrium steady-state heat currents, negative differential thermal conductance, as well as nonreciprocal heat transport. We find that when these elements are arranged in a $\mathcal{P}\mathcal{T}$-symmetric manner, the domain of existence of the nonequilibrium steady state is maximized. We propose an electronic experimental setup based on resistive-inductive-capacitive (RLC) transmission lines, where our predictions can be tested.

Discrete uniform Poisson-Voronoi tessellations of two-dimensional triangular tilings resulting from the Kolmogorov-Johnson-Mehl-Avrami (KJMA) growth of triangular islands have been studied. This shape of tiles and islands, rarely considered in the field of random tessellations, is prompted by the birth-growth process of Ir(210) faceting. The growth mode determines a triangular metric different from the Euclidean metric. Kinetic characteristics of tessellations appear to be metric sensitive, in contrast to area distributions. The latter have been studied for the variant of nuclei growth to the first impingement in addition to the conventional case of complete growth. Kiang conjecture works in both cases. The averaged number of neighbors is six for all studied densities of random tessellations, but neighbors appear to be mainly different in triangular and Euclidean metrics. Also, the applicability of the obtained results for simulating birth-growth processes when the 2D nucleation and impingements are combined with the 3D growth in the particular case of similar shape and the same orientation of growing nuclei is briefly discussed.

We introduce a method, based on symbolic analysis, to characterize the temporal correlations of the spiking activity exhibited by excitable systems. The technique is applied to the experimentally observed dynamics of a semiconductor laser with optical feedback operating in the low-frequency fluctuations regime, where the laser intensity displays irregular trains of sudden dropouts that can be interpreted as excitable pulses. Symbolic analysis transforms the series of interdropout time intervals into sequences of words, which represent the local ordering of a certain (small) number of those intervals. We then focus on the transition probabilities between pairs of words, showing that certain transitions are overrepresented (resulting in others being underrepresented) with respect to the surrogate series, provided the laser injection current is above a critical value. These experimental observations are in very good agreement with numerical simulations of the delay-differential Lang-Kobayashi model that is commonly used to describe this laser system, which supports the fact that the language organization reported here is generic and not a particular feature of the specific laser employed or the experimental time series analyzed. We also present results of simulations of the phenomenological nondelayed Eguia-Mindlin-Giudici(EMG) model and find that in this model the agreement between the experiments and the simulations is good at a qualitative, but not at a quantitative, level.

We report three-dimensional numerical simulations of the flow of an electrically conducting fluid in a spherical shell when a magnetic field is applied. Different spherical Couette configurations are investigated by varying the rotation ratio between the inner and the outer sphere, the geometry of the imposed field, and the magnetic boundary conditions on the inner sphere. Either a Stewartson layer or a Shercliff layer, accompanied by a radial jet, can be generated depending on the rotation speeds and the magnetic-field strength, and various nonaxisymmetric destabilizations of the flow are observed. We show that instabilities arising from the presence of boundaries present striking similarities with the magnetorotational instability (MRI). To this end, we compare our numerical results to experimental observations of the Maryland experiment [D. R. Sisan et al., Phys. Rev. Lett. 93, 114502 (2004)], which claimed to observe MRI in a similar setup.

Experimental and numerical studies of thermal convection have shown that sufficiently vigorous convective flows exhibit a large-scale thermal wind component sweeping along small-scale thermal boundary layer instabilities. A characteristic feature of these flows is an intermittent behavior in the form of irregular reversals in the orientation of the large-scale circulation. There have been several attempts toward a better understanding and description of the phenomenon of flow reversals, but so far most of these models are based on a statistical analysis of few-point measurements or on simplified theoretical assumptions. The analysis of long-term data sets ($>5\times {10}^5$ turnover times ${\tau }_t=d/u_{\mathrm{rms}}$) obtained by numerical simulations of turbulent two-dimensional Rayleigh-Bénard convection allows us to get a more comprehensive view of the spatio-temporal flow behavior. By means of a global statistical analysis of the characteristic spatial modes of the flow we extract information about the stability of dominant large-scale modes as well as the reversal paths in state subspace. We examine probability density functions and drift vector fields of two-dimensional state subspaces spanned by different large-scale spatial modes. This also provides information about the coexistence of dominant modes.

We study numerically the suppression of the soliton self-frequency shift as well as the compression in the presence of bandwidth-limited optical amplification (BLA). Our results confirm the existence of equilibrium points (stable focuses) for the soliton amplitude and speed identified by the soliton perturbation theory. Analyzing the equilibrium amplitudes as a function of physical parameters, maximum compression factor in the amplification of short pulses is revealed. The results of linear stability analysis allow estimation of the necessary distance for the appearance of equilibrium states from different initial conditions. It has been shown that the shape of the perturbed stationary solution in the presence of intrapulse Raman scattering and BLA can be described by the earlier derived analytic expressions.

To ensure fast gene activation, transcription factors (TFs) use a mechanism known as facilitated diffusion to find their DNA promoter site. Here we analyze such a process where a TF alternates between three- and one-dimensional diffusion. In the latter (TF bound to the DNA), the TF further switches between a fast translocation state dominated by interaction with the DNA backbone, and a slow examination state where interaction with DNA base pairs (bp) is predominant. We derive a formula for the mean search time, and show that it is faster and less sensitive to the binding-energy fluctuations as compared to the case with a single sliding state. We find that for an optimal search, the time spent bound to the DNA is larger compared to the three-dimensional time, in agreement with recent experimental data.

Starting from first principles, we formulate a theory of wave-packet propagation in a nonlinear, disordered medium of any dimension, through the derivation of a Fokker-Planck transport equation. Our theory is based on a diagrammatic expansion of the wave packet’s density, and is supported by a heuristic picture that involves a Boltzmann equation with an effective, external potential. Our approach also confirms results obtained in previous work for two-dimensional, nonlinear disordered media.

We study the dynamics of the single-atom maser, or micromaser, by means of the recently introduced method of thermodynamics of quantum jump trajectories. We find that the dynamics of the micromaser displays multiple space-time phase transitions, i.e., phase transitions in ensembles of quantum jump trajectories. This rich dynamical phase structure becomes apparent when trajectories are classified by dynamical observables that quantify dynamical activity, such as the number of atoms that have changed state while traversing the cavity. The space-time transitions can be either first order or continuous, and are controlled not just by standard parameters of the micromaser but also by nonequilibrium “counting” fields. We discuss how the dynamical phase behavior relates to the better known stationary-state properties of the micromaser.

The one-dimensional continuous time anomalous diffusion in composite media consisting of a finite number of layers in immediate contact is investigated. The diffusion process itself is described with the help of two probability density functions (PDFs), one of which is an arbitrary jump-length PDF, and the other is a long-tailed waiting-time PDF characterized by the waiting-time index $\beta \in (0,1)$. The former is assumed to be a function of the space coordinate $x$ and the time coordinate $t$ while the latter is a function of $x$ and the time interval. For such an environment a very general form of the diffusion equation is derived which describes the continuous time anomalous diffusion in a composite medium. This result is then specialized to two particular forms of the jump-length PDF, namely the continuous time Lévy flight PDF and the continuous time truncated Lévy flight PDF. In both cases the PDFs are characterized by the Lévy index $\alpha \in (0,2)$ which is regarded to be a function of $x$ and $t$. It is possible to demonstrate that for particular choices of the indices $\alpha$ and $\beta$ other equations for anomalous diffusion, well known from the literature, follow immediately. This demonstrates the very general applicability of the derivation and of the resulting fractional differential equation discussed here.

We present a master-equation approach to the narrow-escape-time (NET) problem, i.e., the time needed for a particle contained in a confining domain with a single small or narrow opening to exit the domain. In this paper we introduce an alternative type of confining domain (to the usually spherical one) and we consider the diffusion process on a lattice rather than in continuous space. We have obtained analytic results for the basic quantity studied in the NET problem, the mean first-passage time, and we have studied its dependence in terms of the transition (desorption) probability over (from) the surface boundary and the confining domain dimensions. In addition to our analytical approach, we have implemented Monte Carlo simulations, finding excellent agreement between the theoretical results and simulations.

The physical process of confined comminution is investigated within the framework of complex networks. We first characterize the topology of the unweighted contact networks as generated by the confined comminution process. We find this process gives rise to an ultimate contact network which exhibits a scale-free degree distribution and small world properties. In particular, if viewed in the context of networks through which information travels along shortest paths, we find that the global average of the node vulnerability decreases as the comminution process continues, with individual node vulnerability correlating with grain size. A possible application to the design of synthetic networks (e.g., sensor networks) is highlighted. Next we turn our attention to the physics of the granular comminution process and examine force transmission with respect to the weighted contact networks, where each link is weighted by the inverse magnitude of the normal force acting at the associated contact. We find that the strong forces (i.e., force chains) are transmitted along pathways in the network which are mainly following shortest-path routing protocols, as typically found, for example, in communication systems. Motivated by our earlier studies of the building blocks for self-organization in dense granular systems, we also explore the properties of the minimal contact cycles. The distribution of the contact strain energy intensity of 4-cycle motifs in the ultimate state of the confined comminution process is shown to be consistent with a scale-free distribution with infinite variance, thereby suggesting that 4-cycle arrangements of grains are capable of storing vast amounts of energy in their contacts without breaking.

The coefficient of restitution of colliding viscoelastic spheres is known analytically as a complete series expansion in terms of the impact velocity where all (infinitely many) coefficients are known. While being analytically exact, this result is not suitable for applications in efficient event-driven molecular dynamics (eMD) or direct simulation Monte Carlo (DSMC) methods. Based on the analytic result, here we derive expressions for the coefficient of restitution that allow for application in efficient eMD and DSMC simulations of granular systems.

We analyze the scaling properties of the Hertz-Kuwabara-Kono (HKK) model, which is commonly used in numerical simulations to describe the collision of macroscopic noncohesive viscoelastic spherical particles. Parameters describing the elastic and viscous properties of the material, its density, and the size of the colliding particles affect the restitution coefficient $\varepsilon$ and collision time $\tau$ only via appropriate rescaling but do not change the shape of $\varepsilon \left(v\right)$ and $\tau \left(v\right)$ curves, where $v$ is the impact velocity. We have measured the restitution coefficient experimentally for relatively large (1 cm) particles of microcrystalline cellulose to deduce material parameters and then to predict collision properties for smaller microcrystalline cellulose (MCC) particles by assuming the scaling properties of the HKK model. In particular, we demonstrate that the HKK model predicts the restitution coefficient of microscopic particles of about 100 $\mu$m to be considerably smaller than that of the macroscopic particles. In fact, the energy dissipation is so large that only completely inelastic collisions occur for weakly attractive particles. We propose a straightforward self-consistent extension to the Johnson-Kendall-Roberts (JKR) model to include dissipative forces and discuss the implications of our findings for the behavior of experimental powder systems.

We report on the interaction of cationic azobenzene-containing surfactant with DNA investigated by absorption and fluorescence spectroscopy, dynamic light scattering, and atomic force microscopy. The properties of the surfactant can be controlled with light by reversible switching of the azobenzene unit, incorporated into the surfactant tail, between a hydrophobic trans (visible irradiation) and a hydrophilic cis (UV irradiation) configuration. The influence of the trans-cis isomerization of the azobenzene on the compaction process of DNA molecules and the role of both isomers in the formation and colloidal stability of DNA-surfactant complexes is discussed. It is shown that the trans isomer plays a major role in the DNA compaction process. The influence of the cis isomer on the DNA coil configuration is rather small. The construction of a phase diagram of the DNA concentration versus surfactant/DNA charge ratio allows distancing between three major phases: colloidally stable and unstable compacted globules, and extended coil conformation. There is a critical concentration of DNA above which the compacted globules can be hindered from aggregation and precipitation by adding an appropriate amount of the surfactant in the trans configuration. This is because of the compensation of hydrophobicity of the globules with an increasing amount of the surfactant. Below the critical DNA concentration, the compacted globules are colloidally stable and can be reversibly transferred with light to an extended coil state.