We consider a particle performing run-and-tumble dynamics with space-dependent speed. The model has biological relevance as it describes motile bacteria or cells in heterogeneous environments. We give exact expression for the probability density function in the case of free motion in unbounded space. We then analyze the case of a particle moving in a confined interval in the presence of partially absorbing boundaries, reporting the probability density in the Laplace (time) domain and the mean time to absorption. We also discuss the relaxation to the steady state in the case of confinement with reflecting boundaries and drift effects due to direction-dependent tumbling rates, modeling taxis phenomena of cells. The case of diffusive particles with spatially variable diffusivity is obtained as a limiting case.

We study transitions between the Floquet states of a periodically driven oscillator caused by the coupling of the oscillator to a thermal reservoir. The analysis refers to the oscillator that is driven close to triple its eigenfrequency and displays resonant period tripling. The interstate transitions result in a random “walk” over the states. We find the transition rates and show that the walk is nonlocal in the state space: the stationary distribution over the states is formed by the transitions between remote states. This is to be contrasted with systems in thermal equilibrium, where the distribution is usually formed by transitions between nearby states. The analysis of period tripling allows us to explore generic features of the dynamics of resonantly driven systems related to the multiplicity of the states, including those missing in the previously explored models of driven oscillators. We use the results to study switching between the period-3 states of the oscillator due to quantum fluctuations and find the scaling of the switching rates with the parameters.

The generalized Einstein relation (GER) for nonergodic processes is investigated within the framework of the generalized Langevin equation. The conditions for anomalous relaxation such as long-tail decay and non-vanishing velocity autocorrelation function (VAF) are proposed and distinguished. For the stationary nonergodic process, if the initial preparation of the particle velocity is non-thermal, an asymptotic GER occurs in a departure from the usual result. It is shown that the GER holding is a necessary condition rather than a full condition for the system being close to equilibrium. For the nonergodic process of the second type due to cutoff of high frequencies, the VAF oscillates with time, the GER holds but the equilibrium fails in the long-time limit. Applications to some practical examples confirm the present theoretical findings.

The concept of geometrical frustration in condensed matter physics refers to the fact that a system has a locally preferred structure with an energy density lower than the infinite ground state. This notion is, however, often used in a qualitative sense only. In this article, we discuss a quantitative definition of geometrical frustration in the context of lattice models of binary spins. To this aim, we introduce the framework of local energy landscapes, within which frustration can be quantified as the discrepancy between the energy of locally preferred structures and the ground state. Our definition is scale dependent and involves an optimization over a gauge class of equivalent local energy landscapes, related to one another by local energy displacements. This ensures that frustration depends only on the physical Hamiltonian and its range, and not on unphysical choices in how it is written. Our framework shows that a number of popular frustrated models, including the antiferromagnetic Ising model on a triangular lattice, only have finite-range frustration: geometrical incompatibilities are local and can be eliminated by an exact coarse graining of the local energies.

The double-periodic solutions of the focusing nonlinear Schrödinger equation have been previously obtained by the method of separation of variables. We construct these solutions by using an algebraic method with two eigenvalues. Furthermore, we characterize the Lax spectrum for the double-periodic solutions and analyze rogue waves arising on the background of such solutions. Magnification of the rogue waves is studied numerically.

Many networks must maintain synchrony despite the fact that they operate in noisy environments. Important examples are stochastic inertial oscillators, which are known to exhibit fluctuations with broad tails in many applications, including electric power networks with renewable energy sources. Such non-Gaussian fluctuations can result in rare network desynchronization. Here we build a general theory for inertial oscillator network desynchronization by non-Gaussian noise. We compute the rate of desynchronization and show that higher moments of noise enter at specific powers of coupling: either speeding up or slowing down the rate exponentially depending on how noise statistics match the statistics of a network's slowest mode. Finally, we use our theory to introduce a technique that drastically reduces the effective description of network desynchronization. Most interestingly, when instability is associated with a single edge, the reduction is to one stochastic oscillator.

Complex networks with directed, local interactions are ubiquitous in nature and often occur with probabilistic connections due to both intrinsic stochasticity and disordered environments. Sparse non-Hermitian random matrices arise naturally in this context and are key to describing statistical properties of the nonequilibrium dynamics that emerges from interactions within the network structure. Here we study one-dimensional (1D) spatial structures and focus on sparse non-Hermitian random matrices in the spirit of tight-binding models in solid state physics. We first investigate two-point eigenvalue correlations in the complex plane for sparse non-Hermitian random matrices using methods developed for the statistical mechanics of inhomogeneous two-dimensional interacting particles. We find that eigenvalue repulsion in the complex plane directly correlates with eigenvector delocalization. In addition, for 1D chains and rings with both disordered nearest-neighbor connections and self-interactions, the self-interaction disorder tends to decorrelate eigenvalues and localize eigenvectors more than simple hopping disorder. However, remarkable resistance to eigenvector localization by disorder is provided by large cycles, such as those embodied in 1D periodic boundary conditions under strong directional bias. The directional bias also spatially separates the left and right eigenvectors, leading to interesting dynamics in excitation and response. These phenomena have important implications for asymmetric random networks and highlight a need for mathematical tools to describe and understand them analytically.

Cholesteric liquid crystals, subject to externally applied magnetic fields and confined between two parallel planar surfaces with strong homeotropic anchoring conditions, are found to undergo transitions to different types of helicoidal configurations with disclinations. Analytical and numerical studies are performed in order to characterize their properties. In particular, we produce a phase diagram for the transitions from the nematic state to the helicoidal phases in terms of the molecular chirality and the strength of the applied magnetic field.

We report experimental results of the orientational order parameter, the apparent tilt angle, and the field-induced tilt angle for three chiral smectic liquid crystalline materials investigated using infrared (IR) polarized spectroscopy. The common feature in these materials is use of the core 5-methyl-2- pyrimidine benzoate as the central part of the mesogen. This core is terminated by siloxane or carbosilane chains on one of the ends and by the chiral alkoxy chains on the opposite. These compounds exhibit low concomitant layer shrinkage at the smectic $A^{*}$ ($\mathrm{Sm}{A}^{*}$) to smectic $C^{*}$ ($\mathrm{Sm}{C}^{*}$) transition temperature and within the $\mathrm{Sm}{C}^{*}$ phase itself. The maximum layer shrinkage in $\mathrm{Sm}{C}^{*}$ is observed as ∼1.5%. We calculate the apparent orientational order parameter, $S_{\mathrm{app}}$ in the laboratory reference frame from the observed IR absorbance for homeotropic aligned samples, and the true order parameter, S, is calculated using the measured tilt angle and is also interpolated from Iso-$\mathrm{Sm}{A}^{*}$ transition temperature closer to $\mathrm{Sm}{C}^{*}$ phase. The apparent tilt angle in the $\mathrm{Sm}{A}^{*}$ phase calculated from a comparison of order parameters S and $S_{\mathrm{app}}$ is found to be significantly large. A low magnitude of $S_{\mathrm{app}}$ found for homeotropic aligned samples in the $\mathrm{Sm}{A}^{*}$ phase indicates that the order parameter plays a vital role in determining the de Vries characteristics, especially of exhibiting larger apparent tilt angles. Furthermore there is a significant increase in the true order parameter at temperatures close to $\mathrm{Sm}{A}^{*}$ to $\mathrm{Sm}{C}^{*}$ transition temperature in all three compounds. The planar-aligned samples are used to study the dependence of induced tilt angle on the applied electric field. The generalized Langevin–Debye model given by Shen et al. reasonably fits the experimental data on the field-induced tilt angle. The results show that the dipole moment of the tilt correlated domain in $\mathrm{Sm}{A}^{*}$ diverges as temperature is lowered to the $\mathrm{Sm}{A}^{*}-\mathrm{Sm}{C}^{*}$ transition temperature. The generalized Langevin-Debye model is also found to be extremely effective in confirming some of the conclusions of the de Vries behavior.

The quantum hydrodynamic model is used to study the linear and nonlinear properties of small amplitude magnetosonic shock waves in dissipative plasma with degenerate inertialess spin-up and spin-down electrons and inertial classical ions. Spin effects are considered via spin pressure and macroscopic spin magnetization current. A linear dispersion relation is derived analytically and plotted numerically for different plasma parameters such as spin density, polarization ratio, plasma beta, quantum diffraction, spin magnetization energy, and magnetic diffusivity. Employing the standard reductive perturbation technique, a Korteweg–de Vries–Burgers-type equation is derived for small amplitude waves and studied numerically. We have observed that an oscillatory and monotonic shock waves are generated depending upon the plasma configurations. The phase portraits of both oscillatory and monotonic shock waves are also presented. Interestingly, different plasma parameters are found to play a significant role in the transition of oscillatory to monotonic shock waves or vice versa. Most importantly it is found that, the magnetosonic excitations obtained with spin-up and spin-down electrons are significantly different from the usual electron ion quantum plasma. The work presented is related to magnetosonic waves in dense astrophysical environments such as a pulsar magnetosphere and neutron stars.

By introducing preplasma truncation to cases with an initial preplasma scale length larger than $0.2\lambda$, the efficiency of high-order harmonics generated from relativistic laser-solid interactions can be enhanced by more than one order of magnitude and the angular spread can be confined into near-diffraction-limited divergence. Numerical simulations show that density truncation results in more compact oscillation of the surface electron sheet and the curvature of the reflection surface for the driving laser is greatly reduced. This leads to an overall improvement in the harmonic beam quality. More importantly, density truncation makes the harmonic generation weakly dependent on the preplasma scale length, which provides a way to relax the extremely high requirement on the temporal contrast of the driving laser pulse. A feasible scheme to realize the required preplasma truncation is also proposed and demonstrated by numerical simulations.

Translation is a key step in the synthesis of proteins. Accordingly, cells have evolved an intricate array of control mechanisms to regulate this process. By constructing a multicomponent mathematical framework we uncover how translation may be controlled via interacting feedback loops. Our results reveal that this interplay gives rise to a remarkable range of protein synthesis dynamics, including oscillations, step change, and bistability. This suggests that cells may have recourse to a much richer set of control mechanisms than was previously understood.

We study the non-Markovian random continuous processes described by the Mori-Zwanzig equation. As a starting point, we use the Markovian Gaussian Ornstein-Uhlenbeck process and introduce an integral memory term depending on the past of the process into an expression for the higher-order transition probability function and the stochastic differential equation. We show that the proposed processes can be considered as continuous-time interpolations of discrete-time higher-order autoregressive sequences. An equation connecting the memory function (the kernel of integral term) and the two-point correlation function is obtained. A condition for stationarity of the process is established. We suggest a method to generate stationary continuous stochastic processes with a prescribed pair correlation function. As an illustration, some examples of numerical simulation of the processes with nonlocal memory are presented.

The long-term dynamics of long-range interacting $N$-body systems can generically be described by the Balescu-Lenard kinetic equation. However, for one-dimensional homogeneous systems, this collision operator exactly vanishes by symmetry. These systems undergo a kinetic blocking, and cannot relax as a whole under $1/N$ resonant effects. As a result, these systems can only relax under $1/N^2$ effects, and their relaxation is drastically slowed down. In the context of the homogeneous Hamiltonian mean field model, we present a closed and explicit kinetic equation describing self-consistently the very long-term evolution of such systems, in the limit where collective effects can be neglected, i.e., for dynamically hot initial conditions. We show in particular how that kinetic equation satisfies an $H$ theorem that guarantees the unavoidable relaxation to the Boltzmann equilibrium distribution. Finally, we illustrate how that kinetic equation quantitatively matches with the measurements from direct $N$-body simulations.

We present an analytical approach for measuring the temperature of nonlinear dynamical systems in the microcanonical ensemble. Via the self-consistent phonon theory, one can analytically obtain the temperature with respect to the internal energy density in a canonical way. We show how that provides a measure of temperature in the microcanonical ensemble, under the hypothesis of ensemble equivalence. Two models, the FPU-$\beta$ and ${\varphi }^4$ lattices, are studied obtaining results consistent with those derived from time averages along trajectories in the phase space. Furthermore, our approach is corroborated by the fact that temperature obtained in terms of the average energy density after thermalization agrees with the thermostat temperature. The hypothesis is validated via examining the energy distribution for different numbers of particles in the canonical ensemble. Further, we have quantified the corresponding finite size effects. Unlike other existing methods, which require time-consuming computations, our analytical approach performance improves with the number of particles.

We study reaction-diffusion systems beyond the Markovian approximation to take into account the effect of memory on the formation of spatiotemporal patterns. Using a non-Markovian Brusselator model as a paradigmatic example, we show how to use reductive perturbation to investigate the formation and stability of patterns. Focusing in detail on the Hopf instability and short-term memory, we derive the corresponding complex Ginzburg-Landau equation that governs the amplitude of the critical mode and we establish the explicit dependence of its parameters on the memory properties. Numerical solution of this memory-dependent complex Ginzburg-Landau equation as well as direct numerical simulation of the non-Markovian Brusselator model illustrates that memory changes the properties of the spatiotemporal patterns. Our results indicate that going beyond the Markovian approximation might be necessary to study the formation of spatiotemporal patterns even in systems with short-term memory. At the same time, our work opens up a new window into the control of these patterns using memory.

We present a feedback mechanism for dissipative solitons in the cubic complex Ginzburg-Landau (CGL) equation with a nonlinear gradient term. We are making use of a mechanical analog containing contributions from a potential and from a nonlinear viscous term. The feedback mechanism relies on the continuous supply of energy as well as on dissipation of the stable pulse. Our picture is corroborated by the numerical solution of the full equation. A quintic contribution is not necessary for stabilization in the presence of a suitable nonlinear gradient term as we show by using a linear stability analysis for the stationary pulses. We find that the limit of vanishing magnitude of the nonlinear gradient term is singular: For exactly vanishing nonlinear gradient terms stable pulses do not exist. This situation is qualitatively different from that found for the cubic-quintic complex Ginzburg-Landau equation with a nonlinear gradient term: In this case the limit of a vanishing nonlinear gradient contribution is completely smooth. We demonstrate that, for small magnitude of the nonlinear gradient term simple types of scaling behavior are found for the amplitude, the full width at half maximum (FWHM), the velocity, and the effective frequency of the stable pulse as a function of the magnitude of the nonlinear gradient term.

Recent years have witnessed a growing interest in using machine learning to predict and identify critical dynamical phase transitions in physical systems (e.g., many-body quantum systems). The underlying lattice structures in these applications are generally regular. While machine learning has been adopted to complex networks, most existing works concern about the structural properties. To use machine learning to detect phase transitions and accurately identify the critical transition points associated with dynamical processes on complex networks thus stands out as an open and significant problem. Here we develop a framework combining supervised and unsupervised learning, incorporating proper sampling of training data set. In particular, using epidemic spreading dynamics on complex networks as a paradigmatic setting, we start from supervised learning alone and identify situations that degrade the performance. To overcome the difficulties leads to the idea of exploiting confusion scheme, effectively a combination of supervised and unsupervised learning. We demonstrate that the scheme performs well for identifying phase transitions associated with spreading dynamics on homogeneous networks, but the performance deteriorates for heterogeneous networks. To strive to meet this challenge leads to the realization that sampling the training data set is necessary for heterogeneous networks, and we test two sampling methods: one based on the hub nodes together with their neighbors and another based on $k$-core of the network. The end result is a general comprehensive machine learning framework for detecting phase transition and accurately identifying the critical transition point, which is robust, computationally efficient, and universally applicable to complex networks of arbitrary size and topology. Extensive tests using synthetic and empirical networks verify the virtues of the articulated framework, opening the door to exploiting machine learning for understanding, detection, prediction, and control of complex dynamical systems in general.

Proximity networks are time-varying graphs representing the closeness among humans moving in a physical space. Their properties have been extensively studied in the past decade as they critically affect the behavior of spreading phenomena and the performance of routing algorithms. Yet the mechanisms responsible for their observed characteristics remain elusive. Here we show that many of the observed properties of proximity networks emerge naturally and simultaneously in a simple latent space network model, called dynamic-${\mathbb{S}}^1$. The dynamic-${\mathbb{S}}^1$ does not model node mobility directly but captures the connectivity in each snapshot—each snapshot in the model is a realization of the ${\mathbb{S}}^1$ model of traditional complex networks, which is isomorphic to hyperbolic geometric graphs. By forgoing the motion component the model facilitates mathematical analysis, allowing us to prove the contact, intercontact, and weight distributions. We show that these distributions are power laws in the thermodynamic limit with exponents lying within the ranges observed in real systems. Interestingly, we find that network temperature plays a central role in network dynamics, dictating the exponents of these distributions, the time-aggregated agent degrees, and the formation of unique and recurrent components. Further, we show that paradigmatic epidemic and rumor-spreading processes perform similarly in real and modeled networks. The dynamic-${\mathbb{S}}^1$ or extensions of it may apply to other types of time-varying networks and constitute the basis of maximum likelihood estimation methods that infer the node coordinates and their evolution in the latent spaces of real systems.

Epigenetics is a driving force of important and ubiquitous phenomena in nature such as cell differentiation or even metamorphosis. Opposite to its widespread role, understanding the biophysical principles that allow epigenetics to control and rewire gene regulatory networks remains an open challenge. In this work we study the effects of epigenetic modifications on the spatial folding of chromosomes—and hence on the expression of the underlying genes—by mapping the problem to a class of models known as magnetic polymers. In this work we show that a first order phase transition underlies the simultaneous spreading of certain epigenetic marks and the conformational collapse of a chromosome. Further, we describe Brownian dynamics simulations of the model in which the topology of the polymer and thermal fluctuations are fully taken into account and that confirm our mean field predictions. Extending our models to allow for nonequilibrium terms yields new stable phases which qualitatively agrees with observations in vivo. Our results show that statistical mechanics techniques applied to models of magnetic polymers can be successfully exploited to rationalize the outcomes of experiments designed to probe the interplay between a dynamic epigenetic landscape and chromatin organization.