We analyze limit cycle oscillators under perturbation constructed as a product of two signals, namely, an envelope with a period close to natural period of an oscillator and a high-frequency carrier signal. A theory for obtaining an envelope waveform that achieves the maximal frequency interval of entrained oscillators is presented. The optimization problem for fixed power and maximal allowed amplitude is solved by employing the phase reduction method and the Pontryagin's maximum principle. We have shown that the optimal envelope waveform is a bang-bang-type solution. Also, we have found “inversion” symmetry that relates two signals with different powers but the same interval of entrained frequencies. The theoretical results are confirmed numerically on FitzHugh-Nagumo oscillators.

A crucial result in quantum chaos, which has been established for a long time, is that the spectral properties of classically integrable systems generically are described by Poisson statistics, whereas those of time-reversal symmetric, classically chaotic systems coincide with those of random matrices from the Gaussian orthogonal ensemble (GOE). Does this result hold for two-dimensional Dirac material systems? To address this fundamental question, we investigate the spectral properties in a representative class of graphene billiards with shapes of classically integrable circular-sector billiards. Naively one may expect to observe Poisson statistics, which is indeed true for energies close to the band edges where the quasiparticle obeys the Schrödinger equation. However, for energies near the Dirac point, where the quasiparticles behave like massless Dirac fermions, Poisson statistics is extremely rare in the sense that it emerges only under quite strict symmetry constraints on the straight boundary parts of the sector. An arbitrarily small amount of imperfection of the boundary results in GOE statistics. This implies that, for circular-sector confinements with arbitrary angle, the spectral properties will generically be GOE. These results are corroborated by extensive numerical computation. Furthermore, we provide a physical understanding for our results.

Grid cells in the entorhinal cortex fire when animals that are exploring a certain region of space occupy the vertices of a triangular grid that spans the environment. Different neurons feature triangular grids that differ in their properties of periodicity, orientation, and ellipticity. Taken together, these grids allow the animal to maintain an internal, mental representation of physical space. Experiments show that grid cells are modular, i.e., there are groups of neurons which have grids with similar periodicity, orientation, and ellipticity. We use statistical physics methods to derive a relation between variability of the properties of the grids within a module and the range of space that can be covered completely (i.e., without gaps) by the grid system with high probability. Larger variability shrinks the range of representation, providing a functional rationale for the experimentally observed comodularity of grid cell periodicity, orientation, and ellipticity. We obtain a scaling relation between the number of neurons and the period of a module, given the variability and coverage range. Specifically, we predict how many more neurons are required at smaller grid scales than at larger ones.

Hierarchical structures are very common in nature, but only recently have they been systematically studied in materials science, in order to understand the specific effects they can have on the mechanical properties of various systems. Structural hierarchy provides a way to tune and optimize macroscopic mechanical properties starting from simple base constituents and new materials are nowadays designed exploiting this possibility. This can be true also in the field of tribology. In this paper we study the effect of hierarchical patterned surfaces on the static and dynamic friction coefficients of an elastic material. Our results are obtained by means of numerical simulations using a one-dimensional spring-block model, which has previously been used to investigate various aspects of friction. Despite the simplicity of the model, we highlight some possible mechanisms that explain how hierarchical structures can significantly modify the friction coefficients of a material, providing a means to achieve tunability.

We derive exact scaling laws for a three-dimensional incompressible helical two-fluid plasma, without the assumption of isotropy. For each ideal invariant of the two-fluid model, i.e., the total energy, the electron helicity, and the proton helicity, we derive simple scaling laws in terms of two-point increment correlation functions expressed in terms of the velocity field of each species and the magnetic field. These variables are appropriate for comparison with direct numerical simulation data and with in situ measurements in the near-Earth space over a broad range of spatial scales. Finally, using the exact scaling laws and dimensional analysis we predict the magnetic energy and electron helicity spectra for different ranges of scales.

A point source on a plane constantly emits particles which rapidly diffuse and then stick to a growing cluster. The growth probability of a cluster is presented as a sum over all possible scenarios leading to the same final shape. The classical point for the action, defined as a minus logarithm of the growth probability, describes the most probable scenario and reproduces the Laplacian growth equation, which embraces numerous fundamental free boundary dynamics in nonequilibrium physics. For nonclassical scenarios we introduce virtual point sources, in which presence the action becomes the Kullback-Leibler entropy. Strikingly, this entropy is shown to be the sum of electrostatic energies of layers grown per elementary time unit. Hence the growth probability of the presented nonequilibrium process obeys the Gibbs-Boltzmann statistics, which, as a rule, is not applied out from equilibrium. Each layer's probability is expressed as a product of simple factors in an auxiliary complex plane after a properly chosen conformal map. The action at this plane is a sum of Robin functions, which solve the Liouville equation. At the end we establish connections of our theory with the $\tau$ function of the integrable Toda hierarchy and with the Liouville theory for noncritical quantum strings.

We present a numerical study of the tight-binding approach to overdamped Brownian motion on a tilted periodic potential. In the tight-binding method the probability density is expanded on a basis of Wannier states to transform the Smoluchowski equation to a discrete master equation that can be interpreted in terms of thermal hopping between potential minima. We calculate the Wannier states and hopping rates for a variety of potentials, including tilted cosine and ratchet potentials. For deep potential minima the Wannier states are well localized and the hopping rates between nearest-neighbor states are qualitatively well described by Kramers' escape rate. The next-nearest-neighbor hopping rates are negative and must be negligible compared to the nearest-neighbor rates for the discrete master equation treatment to be valid. We find that the validity of the master equation extends beyond the quantitative applicability of Kramers' escape rate.

By using recent developments for the Langevin dynamics of spatially asymmetric systems, we routinely generalize the Onsager-Machlup fluctuation theory of the second order in time. In this form, it becomes applicable to fluctuating variables, including hydrodynamic currents, in equilibrium as well as nonequilibrium steady states. From the solution of the obtained stochastic equations we derive an analytical expression for the time autocorrelation function of a general fluctuating quantity. This theoretical result is then tested in a study of a shear flow by molecular dynamics simulations. The proposed form of the time autocorrelation function yields an excellent fit to our computational data for both equilibrium and nonequilibrium steady states. Unlike the analogous result of the first-order Onsager-Machlup theory, our expression correctly describes the short-time correlations. Its utility is demonstrated in an application of the Green-Kubo formula for the transport coefficient. Curiously, the normalized time autocorrelation function for the shear flow, which only depends on the deterministic part of the fluctuation dynamics, appears independent of the external shear force in the linear nonequilibrium regime.

The pivotal quality of proximity graphs is connectivity, i.e., all nodes in the graph are connected to one another either directly or via intermediate nodes. These types of graphs are often robust, i.e., they are able to function well even if they are subject to limited removal of elementary building blocks, as may occur for random failures or targeted attacks. Here, we study how the structure of these graphs is affected when nodes get removed successively until an extensive fraction is removed such that the graphs fragment. We study different types of proximity graphs for various node-removal strategies. We use different types of observables to monitor the fragmentation process, simple ones like the number and sizes of connected components and more complex ones like the hop diameter and the backup capacity, which is needed to make a network $N-1$ resilient. The actual fragmentation turns out to be described by a second-order phase transition. Using finite-size scaling analyses we numerically assess the threshold fraction of removed nodes, which is characteristic for the particular graph type and node deletion scheme; this suffices to decompose the underlying graphs.

We study phase transitions of the ferromagnetic $q$-state Potts chain with random nearest-neighbor couplings having a variance ${\mathrm{\Delta }}^2$ and with homogeneous long-range interactions, which decay with distance as a power $r^{-(1+\sigma )}, \sigma >0$. In the large-$q$ limit the free-energy of random samples of length $L\le 2048$ is calculated exactly by a combinatorial optimization algorithm. The phase transition stays first order for $\sigma <{\sigma }_c\left(\mathrm{\Delta }\right)\le 0.5$, while the correlation length becomes divergent at the transition point for ${\sigma }_c\left(\mathrm{\Delta }\right)<\sigma <1$. In the latter regime the average magnetization is continuous for small enough $\mathrm{\Delta }$, but for larger $\mathrm{\Delta }$—according to the numerical results—it becomes discontinuous at the transition point, thus the phase transition is expected of mixed order.

We study the effects of a finite kicked environment (bath) composed of $N$ harmonic oscillators on the particle transport in a weakly dissipative quasisymmetric potential system. The small spatial asymmetry is responsible for the appearance of directed particle transport without a net bias, known as the ratchet transport. The whole dynamics is governed by a generalized map where dissipation in the system emerges due to its interaction with the kicked environment. Distinct spectral densities are imposed to the bath oscillators and play an essential role in such models. By changing the functional form of the spectral density, we observe that the transport can be optimized or even suppressed. We show evidences that the transport optimization is related to stability properties of periodic points of the ratchet system and depends on the bath temperature. In a Markovian approach, transport can be increased or suppressed depending on the bath influence.

We study some statistical properties for the behavior of the average squared velocity—hence the temperature—for an ensemble of classical particles moving in a billiard whose boundary is time dependent. We assume the collisions of the particles with the boundary of the billiard are inelastic, leading the average squared velocity to reach a steady-state dynamics for large enough time. The description of the stationary state is made by using two different approaches: (i) heat transfer motivated by the Fourier law and (ii) billiard dynamics using either numerical simulations and theoretical description.

In multiplex networks, cycles cannot be characterized only by their length, as edges may occur in different layers in different combinations. We define a classification of cycles by the number of edges in each layer and the number of switches between layers. We calculate the expected number of cycles of each type in the configuration model of a large sparse multiplex network. Our method accounts for the full degree distribution including correlations between degrees in different layers. In particular, we obtain the numbers of cycles of length 3 of all possible types. Using these, we give a complete set of clustering coefficients and their expected values. We show that correlations between the degrees of a vertex in different layers strongly affect the number of cycles of a given type, and the number of switches between layers. Both increase with assortative correlations and are strongly decreased by disassortative correlations. The effect of correlations on clustering coefficients is equally pronounced.

We study the firing dynamics of a discrete-state and discrete-time version of an integrate-and-fire neuronal network model with both excitatory and inhibitory neurons. When the integer-valued state of a neuron exceeds a threshold value, the neuron fires, sends out state-changing signals to its connected neurons, and returns to the resting state. In this model, a continuous phase transition from non-ceaseless firing to ceaseless firing is observed. At criticality, power-law distributions of avalanche size and duration with the previously derived exponents, $-3/2$ and $-2$, respectively, are observed. Using a mean-field approach, we show analytically how the critical point depends on model parameters. Our main result is that the combined presence of both inhibitory neurons and integrate-and-fire dynamics greatly enhances the robustness of critical power-law behavior (i.e., there is an increased range of parameters, including both sub- and supercritical values, for which several decades of power-law behavior occurs).

The packing of a genome in virions is a topic of intense current interest in biology and biological physics. The area is dominated by allometric scaling relations that connect, e.g., the length of the encapsulated genome and the size of the corresponding virion capsid. Here we report scaling laws obtained from extensive experiments of packing of a macroscopic wire within rigid three-dimensional spherical and nonspherical cavities that can shed light on the details of the genome packing in virions. We show that these results obtained with crumpled wires are comparable to those from a large compilation of biological data from several classes of virions.

Using analytical approach and Monte Carlo (MC) simulations, we study the elastic behavior of the intrinsically twisted elastic ribbons with bending anisotropy, such as double-stranded DNA (dsDNA), in two-dimensional (2D) confinement. We show that, due to the bending anisotropy, the persistence length of dsDNA in 2D conformations is always greater than three-dimensional (3D) conformations. This result is in consistence with the measured values for DNA persistence length in 2D and 3D in equal biological conditions. We also show that in two dimensions, an anisotropic, intrinsically twisted polymer exhibits an implicit twist-bend coupling, which leads to the transient curvature increasing with a half helical turn periodicity along the bent polymer.

Networks of contacts capable of spreading infectious diseases are often observed to be highly heterogeneous, with the majority of individuals having fewer contacts than the mean, and a significant minority having relatively very many contacts. We derive a two-dimensional diffusion model for the full temporal behavior of the stochastic susceptible-infectious-recovered (SIR) model on such a network, by making use of a time-scale separation in the deterministic limit of the dynamics. This low-dimensional process is an accurate approximation to the full model in the limit of large populations, even for cases when the time-scale separation is not too pronounced, provided the maximum degree is not of the order of the population size.

We find that unjammed packings of frictionless particles with rather weak attraction can always be driven into solidlike states by shear. The structure of shear-driven solids evolves continuously with packing fraction from gel-like to jamminglike, but is almost independent of the shear stress. In contrast, both the density of vibrational states (DOVS) and force network evolve progressively with the shear stress. There exists a packing fraction independent shear stress ${\sigma }_c$, at which the shear-driven solids are isostatic and have a flattened DOVS. Solidlike states induced by a shear stress greater than ${\sigma }_c$ possess properties of marginally jammed solids and are thus strictly defined shear jammed states. Below ${\sigma }_c$, shear-driven solids with rather different structures are all under isostaticity and share common features in the DOVS and force network. Our study leads to a jamming phase diagram for weakly attractive particles, which reveals the significance of the shear stress in determining properties of shear-driven solids and the connections and distinctions between jamminglike and gel-like states.

Understanding the dynamics of fluid-driven sediment transport remains challenging, as it occurs at the interface between a granular material and a fluid flow. Boyer, Guazzelli, and Pouliquen [Phys. Rev. Lett. 107, 188301 (2011)] proposed a local rheology unifying dense dry-granular and viscous-suspension flows, but it has been validated only for neutrally buoyant particles in a confined and homogeneous system. Here we generalize the Boyer, Guazzelli, and Pouliquen model to account for the weight of a particle by addition of a pressure $P_0$ and test the ability of this model to describe sediment transport in an idealized laboratory river. We subject a bed of settling plastic particles to a laminar-shear flow from above, and use refractive-index-matching to track particles' motion and determine local rheology—from the fluid-granular interface to deep in the granular bed. Data from all experiments collapse onto a single curve of friction $\mu$ as a function of the viscous number $I_v$ over the range $3\times {10}^{-5}\le I_v\le 2$, validating the local rheology model. For $I_v<3\times {10}^{-5}$, however, data do not collapse. Instead of undergoing a jamming transition with $\mu \to {\mu }_s$ as expected, particles transition to a creeping regime where we observe a continuous decay of the friction coefficient $\mu \le {\mu }_s$ as $I_v$ decreases. The rheology of this creep regime cannot be described by the local model, and more work is needed to determine whether a nonlocal rheology model can be modified to account for our findings.

Pattern-forming processes, such as electrodeposition, dielectric breakdown, or viscous fingering, are often driven by instabilities. Accordingly, the resulting growth patterns are usually highly branched fractal structures. However, in some of the unstable growth processes the envelope of the structure grows in a highly regular manner, with the perturbations smoothed out over the course of time. In this paper we show that the regularity of the envelope growth can be connected to small-scale instabilities leading to the tip splitting of the fingers at the advancing front of the structure. Whenever the growth velocity becomes too large, the finger splits into two branches. In this way it can absorb an increased flux and thus damp the instability. Hence, somewhat counterintuitively, the instability at a small scale results in a stability at a larger scale. The quantitative analysis of these effects is provided by means of the Loewner equation, which one can use to reduce the problem of the interface motion to that of the evolution of the conformal mapping onto the complex plane. This allows an effective analysis of the multifingered growth in a variety of different geometries. We show how the geometry impacts the shape of the envelope of the growing pattern and compare the results with those observed in natural systems.