We present an efficient method for estimating variables and parameters of a given system of ordinary differential equations by adapting the model output to an observed time series from the (physical) process described by the model. The proposed method is based on (unconstrained) nonlinear optimization exploiting the particular structure of the relevant cost function. To illustrate the features and performance of the method, simulations are presented using chaotic time series generated by the Colpitts oscillator, the three-dimensional Hindmarsh-Rose neuron model, and a nine-dimensional extended Rössler system.

The secular variation of the geomagnetic field as observed at the Earth's surface results from the complex magnetohydrodynamics taking place in the fluid core of the Earth. One way to analyze this system is to use the data in concert with an underlying dynamical model of the system through the technique of variational data assimilation, in much the same way as is employed in meteorology and oceanography. The aim is to discover an optimal initial condition that leads to a trajectory of the system in agreement with observations. Taking the Earth's core to be an electrically conducting fluid sphere in which convection takes place, we develop the continuous adjoint forms of the magnetohydrodynamic equations that govern the dynamical system together with the corresponding numerical algorithms appropriate for a fully spectral method. These adjoint equations enable a computationally fast iterative improvement of the initial condition that determines the system evolution. The initial condition depends on the three dimensional form of quantities such as the magnetic field in the entire sphere. For the magnetic field, conservation of the divergence-free condition for the adjoint magnetic field requires the introduction of an adjoint pressure term satisfying a zero boundary condition. We thus find that solving the forward and adjoint dynamo system requires different numerical algorithms. In this paper, an efficient algorithm for numerically solving this problem is developed and tested for two illustrative problems in a whole sphere: one is a kinematic problem with prescribed velocity field, and the second is associated with the Hall-effect dynamo, exhibiting considerable nonlinearity. The algorithm exhibits reliable numerical accuracy and stability. Using both the analytical and the numerical techniques of this paper, the adjoint dynamo system can be solved directly with the same order of computational complexity as that required to solve the forward problem. These numerical techniques form a foundation for ultimate application to observations of the geomagnetic field over the time scale of centuries.

The hydrodynamic interaction between two nonspherical capsules suspended in a simple shear flow is studied numerically using a front-tracking method. The capsules are enclosed by thin shells which develop in-plane tensions and bending moments due to a preferred three-dimensional unstressed configuration. Computations are performed for capsules with spherical, oblate spheroidal, and biconcave unstressed shapes for a wide range of dimensionless shear rates and initial separation distances between the two capsules. The bending modulus and viscosity ratio between the internal and surrounding fluids are chosen to be those of healthy red blood cells. Depending on the initial separation distance, we find that two spherical capsules in shear flow either cross over each other or undergo spiraling motion. In addition, the long-time interaction behavior of capsules also depends strongly on the unstressed shapes. More oblate or biconcave capsules exhibit two additional type of interactions, namely swapping and continuous rotation, which occur only when each capsule undergoes tumbling motion.

Compound drops comprise two or more immiscible phases, one of which entirely or partially engulfs the others. In this work we consider the thermocapillary-induced motion of partially engulfed compound drops, composed of two phases, in an immiscible fluid. If the capillary number is negligibly small, Ca $\ll$ 1, the partially engulfed compound drop is composed of three spherical surface segments, intersecting at contact angles that are determined by the three interfacial tensions associated with the three fluid phases that make up the compound drop and the ambient medium. Corrections to the shapes of the undeformable case at Ca $=$ 0 are expected to be of the order Ca. However, as the drop propagates through the nonisothermal fluid, the temperature at the three-phase contact line and, hence, the contact angles, may considerably change, resulting in a dramatic change of the compound drop shape. Moreover, the changes in the interfacial tensions may be so significant that the partially engulfed configuration may become impossible and either two immiscible parts of the compound drop separate or one of them becomes completely engulfed by the other.

We study the system of equations for the canonically conjugate variables $p$ and $q$ specified by the one-dimensional Hamiltonian $H=H\left(p,q,{\Lambda }_1,...,{\Lambda }_N\right)$ dependent on $N$self-consistent slightly changing parameters obeying the equations: ${\dot{\Lambda }}_n=\varepsilon f_n({\Lambda }_1,...,{\Lambda }_N,p,q)$. A broad range of oscillatory and wave processes with weak dissipation is described by analogous systems. The general method of adiabatic invariant construction for this system is proposed. Self-consistent averaged equations for the evolution of the action integral and the parameters ${\Lambda }_n$ are obtained. The constructed theory is applied to a generalized model of the nonlinear resonance. The autoresonance (phase locking) regime of decay parametric instability in a dissipative medium is revealed.

The scattered wave formalism developed for a quantum subsystem interacting with reservoirs through open boundaries is applied to one- or two-dimensional barrier scattering and quantum transistors. The total wave function is divided into incident and scattered components. Markovian outgoing wave boundary conditions are imposed on the scattered or total wave function by either the ratio or polynomial methods. For barrier scattering problems, accurate time-dependent transmission probabilities are obtained through the integration of the modified time-dependent Schrödinger equations for the scattered wave function. For quantum transistors, the time-dependent transport is studied for a quantum wave packet propagating through the conduction channel of a field effect transistor. This study shows that the scattered wave formalism significantly reduces computational effort relative to other open boundary methods and demonstrates wide applications to quantum dynamical processes.

The diffusive non-Markovian motion over a single-well potential barrier in the presence of a weak sinusoidal time modulation is studied. We found nonmonotonic dependence of the mean escape time from the barrier on a frequency of the periodic modulation that is analogous to the stochastic resonance phenomenon. The resonant increase of diffusion over the barrier occurs at the frequency inversely proportional to the mean first-passage time for the motion in the absence of the time modulation.

We have studied dynamical responses and the Jarzynski equality (JE) of classical open systems described by the generalized Caldeira-Leggett model with the nonlocal system-bath coupling. In the derived non-Markovian Langevin equation, the nonlinear nonlocal dissipative term and state-dependent diffusion term yielding multiplicative colored noise satisfy the fluctuation-dissipation relation. Simulation results for harmonic oscillator systems have shown the following: (a) averaged responses of the system $\langle x\rangle$ to applied sinusoidal and step forces significantly depend on model parameters of magnitudes of additive and multiplicative noises and the relaxation time of colored noise, although stationary marginal probability distribution functions are independent of them, (b) a combined effect of nonlinear dissipation and multiplicative colored noise induces enhanced fluctuations $\langle {[x-\langle x\rangle ]}^2\rangle$ for an applied sinusoidal force, and (c) the JE holds for an applied ramp force independently of the model parameters with a work distribution function which is (symmetric) Gaussian and (asymmetric) non-Gaussian for additive and multiplicative noises, respectively. It has been shown that the non-Markovian Langevin equation in the local and overdamped limits is quite different from the widely adopted, phenomenological Markovian Langevin equation subjected to additive and multiplicative noises.

Probability distributions that emerge from the formalism of nonextensive statistical mechanics have been applied to a variety of problems. In this article we unite modeling of such distributions with the model of widespread $1/f$ noise. We propose a class of nonlinear stochastic differential equations giving both the $q$-exponential or $q$-Gaussian distributions of signal intensity, revealing long-range correlations and $1/f^{\beta }$ behavior of the power spectral density. The superstatistical framework to get $1/f^{\beta }$ noise with $q$-exponential and $q$-Gaussian distributions of the signal intensity is proposed, as well.

We study photon diffusion in a two-dimensional random packing of monodisperse disks as a simple model of granular material. We apply ray optics approximation to set up a persistent random walk for the photons. We employ Fresnel's intensity reflectance with its rich dependence on the incidence angle and polarization state of the light. We present an analytic expression for the transport-mean-free path $l^{*}$ in terms of the refractive indices of grains and host medium, grain radius, and packing fraction. We perform numerical simulations to examine our analytical result.

Ice crystal growth experiments in heavy water were carried out under microgravity to investigate the morphological transition from a disk crystal to a dendrite. Surprisingly, however, no transition was observed, namely, the disk crystal or dendrite maintained its shape throughout the experiments, unlike the results obtained on the ground. Therefore, we introduce a growth model to understand disk growth. The Gibbs-Thomson effect is taken into account as a stabilization mechanism. The model is numerically solved by varying both an interfacial tension of the prism plane and supercooling so that the final sizes of the crystals can become almost the same to determine the interfacial tension. The results are compared with the typical experimental ones and thus the interfacial tension is estimated to be 20 mJ/m${}^2$. Next, the model is solved under two supercooling conditions by using the estimated interfacial tension to understand stable growth. Comparisons between the numerical and experimental results show that our model explains well the microgravity experiments. It is also found that the experimental setup has the capability of controlling temperature on the order of 1/100 K.

We report three-dimensional tomographic reconstruction of optical parameters for the mesoscopic light-scattering regime from experimentally obtained datasets by employing angularly selective data acquisition. The approach is based on the assumption that the transport coefficient of a scattering medium differs by an order of magnitude for weakly and highly scattering regions. Datasets were obtained by imaging a weakly scattering phantom, which embeds a highly scattering cylinder of two to three photons' mean path length in diameter containing light-absorbing inclusions. Reconstruction results are presented and discussed.

Strongly nonlinear degrade-and-fire (DF) oscillations may emerge in genetic circuits having a delayed negative feedback loop as their core element. Here we study the synchronization of DF oscillators coupled through a common repressor field. For weak coupling, initially distinct oscillators remain desynchronized. For stronger coupling, oscillators can be forced to wait in the repressed state until the global repressor field is sufficiently degraded, and then they fire simultaneously forming a synchronized cluster. Our analytical theory provides necessary and sufficient conditions for clustering and specifies the maximum number of clusters that can be formed in the asymptotic regime. We find that in the thermodynamic limit a phase transition occurs at a certain coupling strength from the weakly clustered regime with only microscopic clusters to a strongly clustered regime where at least one giant cluster has to be present.

In this paper, we investigate rotating electro-osmotic (EO) flow over an infinite plate or in a channel formed by two parallel plates. The analysis is based on the Debye-Hückel approximation for charge distributions and the Navier-Stokes equation for a transport electrolyte in the rotating frame. It is shown that, for the single plate, the nondimensional speed of system rotation $\omega$ is the singly most important parameter, while for the channel, in addition to $\omega$, the nondimensional electrokinetic width $K$ also plays an important role. However, the parameter $\omega \equiv {\eta }^2$ has different natural appearances in the respective cases of a single plate (SP) and two plates (TPs). More precisely, $\eta$(SP) measures the ratio ${\lambda }_D/L_K$ of the Debye length to the Ekman depth, while $\eta$(TP) measures the ratio $L/L_K$ of the channel width to the Ekman depth. The effect of rotation is always to reduce the axial flow rate along the direction of the applied electric field, accompanied by a (secondary) transverse flow. In the SP case, the plot on the velocity plane for each $\omega$ shows an interesting closed EO Ekman spiral. The size of the spiral shrinks with increasing $\omega$. The transverse flow is so significant that the volume transport associated with the EO Ekman spiral turns clockwise 45° to the applied field near $\omega =0$ and gradually turns at a right angle to the applied field as $\omega$ is increased. In contrast, in the TP case, the transverse flow rate is smaller than the axial flow rate when $\omega$ is small. The transverse flow rates at all $K$ are observed to reach their maxima at $\omega$ of order 1. The volume transport is nearly at a zero angle to the applied field near $\omega =0$ and gradually turns to 45° to the applied field as $\omega$ is increased. In the limit of $\omega \to \infty ,$ for both SP and TP cases, the entire system forms a rigid body rotation—there is neither axial nor transverse flow.

Carnot's theorem poses a fundamental limit on the maximum efficiency achievable from an engine that works between two reservoirs at thermal equilibrium. We extend this result to the case of arbitrary nonthermal stationary reservoirs, even with quantum coherence. In order to do this we prove that a single nonthermal reservoir is formally equivalent to multiple equilibrium ones. Finally, we discuss the possibility of realizing an engine that works at unit efficiency by exploiting quantum coherence present in the reservoir.

The viscosity and diffusivities of silicate melts under high-pressure, high-temperature conditions are difficult to obtain experimentally. Estimation and extrapolation of transport coefficients are further complicated by their extreme sensitivity to melt composition. Our molecular-dynamics simulations show that, over a broad range of melt composition, temperature, and pressure, the diffusivities correlate with the excess entropy; approximations to the latter can be obtained from the knowledge of the radial distribution function. Using this structure-thermodynamics-dynamics relationship, we show that transport properties of silicate melts can be estimated quantitatively using static structure factor data from experiments.

A molecular statistical theory of the smectic $A$ phase is developed taking into account specific interactions between different molecular fragments which enables one to describe different microscopic scenario of the transition into the smectic phase. The effects of nanoscale segregation are described using molecular models with different combinations of attractive and repulsive sites. These models have been used to calculate numerically coefficients in the mean filed potential as functions of molecular model parameters and the period of the smectic structure. The same coefficients are calculated also for a conventional smectic with standard Gay-Berne interaction potential which does not promote the segregation. The free energy is minimized numerically to calculate the order parameters of the smectic $A$ phases and to study the nature of the smectic transition in both systems. It has been found that in conventional materials the smectic order can be stabilized only when the orientational order is sufficiently high, In contrast, in materials with nanosegregation the smectic order develops mainly in the form of the orientational-translational wave while the nematic order parameter remains relatively small. Microscopic mechanisms of smectic ordering in both systems are discussed in detail, and the results for smectic order parameters are compared with experimental data for materials of various molecular structure.

We use analytical considerations and computer simulations to show that the membrane spectrum of thermal fluctuations is governed by the mechanical and not the intrinsic tension. Our study highlights the fact that the commonly used quadratic approximation of the Helfrich effective Hamiltonian is not rotationally invariant. We demonstrate that this nonphysical feature leads to a calculated mechanical tension that differs dramatically from the correct mechanical tension. Specifically, our results suggest that the mechanical and intrinsic tensions vanish simultaneously, which contradicts recent theoretical predictions derived for the approximated Hamiltonian.

We study the evolution of networks when the creation and decay of links are based on the position of nodes in the network measured by their centrality. We show that the same network dynamics arise under various centrality measures, and solve analytically the network evolution. During the complete evolution, the network is characterized by nestedness: the neighborhood of a node is contained in the neighborhood of the nodes with larger degree. We find a discontinuous transition in the network density between hierarchical and homogeneous networks, depending on the rate of link decay. We also show that this evolution mechanism leads to double power-law degree distributions, with interrelated exponents.