We present a theoretical framework to analyze the dynamics of gene expression with stochastic bursts. Beginning with an individual-based model which fully accounts for the messenger RNA (mRNA) and protein populations, we propose an expansion of the master equation for the joint process. The resulting coarse-grained model reduces the dimensionality of the system, describing only the protein population while fully accounting for the effects of discrete and fluctuating mRNA population. Closed form expressions for the stationary distribution of the protein population and mean first-passage times of the coarse-grained model are derived and large-scale Monte Carlo simulations show that the analysis accurately describes the individual-based process accounting for mRNA population, in contrast to the failure of commonly proposed diffusion-type models.

Chiral symmetry breaking is ubiquitous in biological systems, from DNA to bacterial suspensions. A key unresolved problem is how chiral structures may spontaneously emerge from achiral interactions. We study a simple model of active swimmers in three dimensions that effectively incorporates hydrodynamic interactions. We perform large-scale molecular dynamics simulations (up to ${10}^6$ particles) and find long-lived metastable collective states that exhibit chiral organization although the interactions are achiral. We elucidate under which conditions these chiral states will emerge and grow to large scales. To explore the complex phase space available to the system, we perform nonequilibrium quenches on a one-dimensional Lebwohl-Lasher model with periodic boundary conditions to study the likelihood of formation of chiral structures.

Understanding how nanoparticles (NPs) diffuse, stick, and assemble into larger structures within polymers is key to the design and fabrication of NP-polymer composites. Here we describe an approach for inferring the dynamic parameters of NP assembly from spatially and temporally disjointed images of composites. The approach involves iterative adjustment of the parameters of a kinetic model of assembly until the computed size statistics of NP clusters match those obtained from high-throughput analysis of the experimental images. Application of this approach to the assembly of shaped, metal NPs in polymer films suggests that NP structures grow via a cluster-cluster aggregation mechanism, where NPs and their clusters diffuse with approximately Stokes-Einstein diffusivity and stick to other NPs or clusters with a probability that depends strongly on the size and shape of the NPs and the molecular weight of the polymer.

When pulling a probe particle in a many-particle system at a fixed velocity, the probe's effective friction, defined as the average external force over its velocity, ${\gamma }_{\mathrm{eff}}:=\langle F_{\mathrm{ex}}\rangle /u$, first stays constant (linear response), then decreases (thinning), and, finally, increases again (thickening). We propose a three-time-scale picture to describe the thinning and thickening behavior. There are three distinct time scales for the bath particles: diffusion, damping, and single probe-bath collision. The dominating time scales are controlled by the pulling velocity and determine the behavior of the probe's friction. We test and confirm this description with a Langevin dynamics simulation. Microscopically, we find that for computing the effective friction, the Maxwellian distribution of bath particles' velocities fails in the regime of high Reynolds and Peclet numbers. This can be understood based on the microscopic mechanism of thickening obtained in the $T=0$ limit. The dynamic regimes defined by the ratio of different time scales can explain several observations of thinning and thickening in literature.

The Richtmyer-Meshkov instability (RMI) of single-mode air-${\mathrm{SF}}_6$ interfaces is studied numerically and the emphasis is placed on the effect of the principal curvature on the early evolution of the shocked interface. Two three-dimensional initial interfaces with opposite ($3\mathrm{D}-$) and identical (3D+) principal curvatures and a traditional two-dimensional interface (2D) are considered. The weighted essentially nonoscillatory scheme and the Level-Set method combined with the real ghost fluid method are adopted. For comparison, perturbations on the initial interfaces with the same wavelength and amplitude in the symmetry plane are employed. The numerical results confirm the experimental finding that the growth rate of perturbations in the symmetry plane at the linear stage in the $3\mathrm{D}-$ case is much smaller than that in the 2D and 3D+ cases. The difference among them can be ascribed to the different pressure and vorticity distributions associated with the principal curvatures of the initial interface. On the one hand, the high-pressure zones in the vicinity of the deformed interface are significantly different for three cases especially in the very beginning. The shock convergence and divergence at the interface are more severe in the 3D+ case than those in the 2D case, while the wave pattern in the $3\mathrm{D}-$ case is more complex. On the other hand, the baroclinic vorticity distribution plays a leading role in the interface deformation of the 3D RMI after the passage of the planar shock. The accumulated vorticity changes the movement of the deformed interface and makes the local growth of perturbations different among three cases.

When two oppositely charged drops come into light contact, a liquid meniscus bridge with double-cone geometry forms between the drops. Recent experiments have demonstrated the existence of a critical cone angle above which the meniscus bridge pinches off and the drops do not coalesce. This striking behavior—which has implications for processes ranging from the coarsening of emulsions to electrospray ionization in mass spectrometry—has been studied theoretically and experimentally for inertial liquid drops. Little is known, however, about the influence of the liquid viscosity on the critical cone angle. Here, we use high-fidelity numerical simulations to gain insight into the coalescence dynamics of conical drops at intermediate Reynolds numbers. The simulations, which account for viscous, inertial, and surface tension effects, predict that the critical cone angle increases as the viscosity of the drops decreases. When approaching the inertial regime, however, the predicted critical angle quickly stabilizes at approximately ${27}^{\circ }$, as observed in experiments.

We present a model to calculate temperature-relaxation rates in dense plasmas. The electron-ion interaction potential and the thermodynamic data of interest are provided by an average-atom model. This approach allows the study of the temperature relaxation in a two-temperature electron-ion system.

We present a theoretical treatment of overdamped Brownian motion on a time-independent bichromatic periodic potential with spatially fast- and slow-changing components. In our approach, we generalize the Wannier basis commonly used in the analysis of periodic systems to define a basis of S states that are localized at local minima of the potential. We demonstrate that the S states are orthonormal and complete on the length scale of the periodicity of the fast-changing potential, and we use the S-state basis to transform the continuous Smoluchowski equation for the system to a discrete master equation describing hopping between local minima. We identify the parameter regime where the master equation description is valid and show that the interwell hopping rates are well approximated by Kramers' escape rate in the limit of deep potential minima. Finally, we use the master equation to explore the system dynamics and determine the drift and diffusion for the system.

Given an arbitrary elliptic partial differential equation (PDE), a procedure for obtaining its solution is proposed based on the method of Ritz: the solution is written as a linear combination of plane waves and the coefficients are obtained by variational minimization. The PDE to be solved is cast as a system of linear equations $\mathbf{A}\mathbf{x}=\mathbf{b}$, where the matrix $\mathbf{A}$ is not sparse, which prevents the straightforward application of standard iterative methods in order to solve it. This sparseness problem can be circumvented by means of a recursive bisection approach based on the fast Fourier transform, which makes it possible to implement fast versions of some stationary iterative methods (such as Gauss-Seidel) consuming $O(NlogN)$ memory and executing an iteration in $O\left(N{log}^{2}N\right)$ time, $N$ being the number of plane waves used. In a similar way, fast versions of Krylov subspace methods and multigrid methods can also be implemented. These procedures are tested on Poisson's equation expressed in adaptive coordinates. It is found that the best results are obtained with the GMRES method using a multigrid preconditioner with Gauss-Seidel relaxation steps.

We study a universal structure of two- and three-dimensional self-gravitating systems in the quasiequilibrium state. It is shown numerically that the two-dimensional self-gravitating system in the quasiequilibrium state has the same kind of density profile as the three-dimensional one, especially when null virial conditions are fulfilled. It is unveiled why the conditions are necessary for the universal structure by the envelope equation. We develop a phenomenological model to describe this universal structure by using a special Langevin equation with a distinctive random noise to self-gravitating systems. We find that the density profile derived theoretically is very consistent with results of observations and simulations.

The recently discovered chimera state involves the coexistence of synchronized and desynchronized states for a group of identical oscillators. In this work, we show the existence of (inwardly) rotating spiral wave chimeras in the three-component reaction-diffusion systems where each element is locally coupled by diffusion. A transition from spiral waves with the smooth core to spiral wave chimeras is found as we change the local dynamics of the system or as we gradually increase the diffusion coefficient of the activator. Our findings on the spiral wave chimera in the reaction-diffusion systems suggest that spiral chimera states may be found in chemical and biological systems that can be modeled by a large population of oscillators indirectly coupled via a diffusive environment.

Thunderclouds can produce bremsstrahlung gamma-ray emission, and sometimes even positrons. At 00:27:00 (UT) on 13 January 2012, an intense burst of gamma rays from a thundercloud was detected by the GROWTH experiment, located in Japan, facing the Sea of Japan. The event started with a sharp gamma-ray flash with a duration of $<300$ ms coincident with an intracloud discharge, followed by a decaying longer gamma-ray emission lasting for $\sim 60$ s. The spectrum of this prolonged emission reached $\sim 10$ MeV, and contained a distinct line emission at $508\pm 3(\mathrm{stat}.)\pm 5(\mathrm{sys}.)$ keV, to be identified with an electron-positron annihilation line. The line was narrow within the instrumental energy resolution $(\sim 80\text{keV})$, and contained $520\pm 50$ photons which amounted to $\sim 10\%$ of the total signal photons of $5340\pm 190$ detected over 0.1–10 MeV. As a result, the line equivalent width reached $280\pm 40$ keV, which implies a nontrivial result. The result suggests that a downward positron beam produced both the continuum and the line photons.

We investigate the behavior of nonequilibrium phase transitions under the influence of disorder that locally breaks the symmetry between two symmetrical macroscopic absorbing states. In equilibrium systems such “random-field” disorder destroys the phase transition in low dimensions by preventing spontaneous symmetry breaking. In contrast, we show here that random-field disorder fails to destroy the nonequilibrium phase transition of the one- and two-dimensional generalized contact process. Instead, it modifies the dynamics in the symmetry-broken phase. Specifically, the dynamics in the one-dimensional case is described by a Sinai walk of the domain walls between two different absorbing states. In the two-dimensional case, we map the dynamics onto that of the well studied low-temperature random-field Ising model. We also study the critical behavior of the nonequilibrium phase transition and characterize its universality class in one dimension. We support our results by large-scale Monte Carlo simulations, and we discuss the applicability of our theory to other systems.

The surface exponents, scaling behavior, and bulk porosity of a generalized ballistic deposition (GBD) model are studied. In nature, there exist particles with varying degrees of stickiness ranging from completely nonsticky to fully sticky. Such particles may adhere to any one of the successively encountered surfaces, depending on a sticking probability that is governed by the underlying stochastic mechanism. The microscopic configurations possible in this model are much larger than those allowed in existing models of ballistic deposition and competitive growth models that seek to mix ballistic and random deposition processes. In this article, we find the scaling exponents for surface width and porosity for the proposed GBD model. In terms of scaled width $\tilde{W}$ and scaled time $\tilde{t}$, the numerical data collapse onto a single curve, demonstrating successful scaling with sticking probability $p$ and system size $L$. Similar scaling behavior is also found for the porosity.

We propose a quantum Otto cycle based on the properties of a two-level system in a realistic out-of-thermal-equilibrium electromagnetic field acting as its sole reservoir. This steady configuration is produced without the need of active control over the state of the environment, which is a noncoherent thermal radiation, sustained only by external heat supplied to macroscopic objects. Remarkably, even for nonideal finite-time transformations, it largely over-performs the standard ideal Otto cycle and asymptotically achieves unit efficiency at finite power.

This study is concerned with ensembles of continuous-time Markov chains evolving independently under a common transition rate matrix in some finite state space. In this context, our prior work [Phys. Rev. E 91, 062116 (2015)] has formulated an approximation scheme, called the minimal diffusion formulation, to deduce how the number of chains in a prescribed relevant state evolves in time. The formulation consists of two specifically coupled Ornstein-Uhlenbeck processes in a stochastic differential equation representation; it is minimal in the sense that its structure does not change with the state space size or the transition matrix density, and it requires no matrix square-root operations. In the present study, we first calculate the autocorrelation function of the relevant state density in the minimal diffusion formulation, which is fundamental to the identification of the ensemble dynamics. The obtained autocorrelation function is then employed to develop two diffusion formulations that reduce the structural complexity of the minimal diffusion formulation without significant loss of accuracy in the dynamics. One of these variant formulations includes one less noise term than the minimal diffusion formulation and still satisfies the above-mentioned autocorrelation function in its dynamics. The second variant is in the form of a one-dimensional Langevin equation, therefore it is the simplest possible diffusion formulation one can obtain for the problem, yet its autocorrelation function is first-order accurate in time gap. Numerical simulations supporting the theoretical analysis are delivered.

The dynamics of many natural systems is dominated by nonlinear waves propagating through the medium. We show that in any extended system that supports nonlinear wave fronts with positive surface tension, the asymptotic wave-front dynamics can be formulated as a gradient system, even when the underlying evolution equations for the field variables cannot be written as a gradient system. The variational potential is simply given by a linear combination of the occupied volume and surface area of the wave front and changes monotonically over time.

We present an alternative approach to the analysis of nonlinear systems with long-term memory that is based on the Koopman operator and a Lévy transformation in time. Memory effects are considered to be the result of interactions between a system and its surrounding environment. The analysis leads to the decomposition of a nonlinear system with memory into modes whose temporal behavior is anomalous and lacks a characteristic scale. On average, the time evolution of a mode follows a Mittag-Leffler function, and the system can be described using the fractional calculus. The general theory is demonstrated on the fractional linear harmonic oscillator and the fractional nonlinear logistic equation. When analyzing data from an ill-defined (black-box) system, the spectral decomposition in terms of Mittag-Leffler functions that we propose may uncover inherent memory effects through identification of a small set of dynamically relevant structures that would otherwise be obscured by conventional spectral methods. Consequently, the theoretical concepts we present may be useful for developing more general methods for numerical modeling that are able to determine whether observables of a dynamical system are better represented by memoryless operators, or operators with long-term memory in time, when model details are unknown.

We propose a coupled system of fast and slow phase oscillators. We observe two-step transitions to quasiperiodic motions by direct numerical simulations of this coupled oscillator system. A low-dimensional equation for order parameters is derived using the Ott-Antonsen ansatz. The applicability of the ansatz is checked by the comparison of numerical results of the coupled oscillator system and the reduced low-dimensional equation. We investigate further several interesting phenomena in which mutual interactions between the fast and slow oscillators play an essential role. Fast oscillations appear intermittently as a result of excitatory interactions with slow oscillators in a certain parameter range. Slow oscillators experience an oscillator-death phenomenon owing to their interaction with fast oscillators. This oscillator death is explained as a result of saddle-node bifurcation in a simple phase equation obtained using the temporal average of the fast oscillations. Finally, we show macroscopic synchronization of the order $1:m$ between the slow and fast oscillators.