The kinetics of nucleation of a core-shell composite nucleus that consists of a core of stable final phase surrounded by a wetting layer of intermediate metastable phase is studied using the kinetic theory of binary nucleation not only in the size and composition space but also in the component space. The steady-state solution of the Fokker-Planck equation is considered. Various formulas for the critical nucleus at the saddle point as well as for the postcritical nucleus are derived. The kinetics of nucleation at the saddle point is more appropriately characterized in the size and composition space, while the kinetics of the postcritical nucleus is more appropriately described in the component space. Although both the free-energy landscape and the reaction rates play decisive role to determine the kinetics of nucleation at the saddle point, the details of the free-energy landscape are irrelevant to the kinetics of the postcritical nucleus.

A theoretical analysis is presented of a nematic liquid crystal confined between substrates patterned with squares that promote vertical and planar alignment. Two approaches are used to elucidate the behavior across a wide range of length scales: Monte Carlo simulation of hard particles and Frank-Oseen continuum theory. Both approaches predict bistable degenerate azimuthal alignment in the bulk along the edges of the squares; the continuum calculation additionally reveals the possibility of an anchoring transition to diagonal alignment if the polar anchoring energy associated with the pattern is sufficiently weak. Unlike the striped systems previously analyzed, the Monte Carlo simulations suggest that there is no “bridging” transition for sufficiently thin cells. The extent to which these geometrically patterned systems resemble topographically patterned substrates, such as square wells, is also discussed.

We investigate the microtubule polymerization dynamics with catastrophe and rescue events for three different confinement scenarios, which mimic typical cellular environments: (i) The microtubule is confined by rigid and fixed walls, (ii) it grows under constant force, and (iii) it grows against an elastic obstacle with a linearly increasing force. We use realistic catastrophe models and analyze the microtubule dynamics, the resulting microtubule length distributions, and force generation by stochastic and mean field calculations; in addition, we perform stochastic simulations. Freely growing microtubules exhibit a phase of bounded growth with finite microtubule length and a phase of unbounded growth. The main results for the three confinement scenarios are as follows: (i) In confinement by fixed rigid walls, we find exponentially decreasing or increasing stationary microtubule length distributions instead of bounded or unbounded phases, respectively. We introduce a realistic model for wall-induced catastrophes and investigate the behavior of the average length as a function of microtubule growth parameters. (ii) Under a constant force, the boundary between bounded and unbounded growth is shifted to higher tubulin concentrations and rescue rates. The critical force $f_c$ for the transition from unbounded to bounded growth increases logarithmically with tubulin concentration and the rescue rate, and it is smaller than the stall force. (iii) For microtubule growth against an elastic obstacle, the microtubule length and polymerization force can be regulated by microtubule growth parameters. For zero rescue rate, we find that the average polymerization force depends logarithmically on the tubulin concentration and is always smaller than the stall force in the absence of catastrophes and rescues. For a nonzero rescue rate, we find a sharply peaked steady-state length distribution, which is tightly controlled by microtubule growth parameters. The corresponding average microtubule length self-organizes such that the average polymerization force equals the critical force $f_c$ for the transition from unbounded to bounded growth. We also investigate the force dynamics if growth parameters are perturbed in dilution experiments. Finally, we show the robustness of our results against changes of catastrophe models and load distribution factors.

Proteins participating in a protein-protein interaction network can be grouped into homology classes following their common ancestry. Proteins added to the network correspond to genes added to the classes, so the dynamics of the two objects are intrinsically linked. Here we first introduce a statistical model describing the joint growth of the network and the partitioning of nodes into classes, which is studied through a combined mean-field and simulation approach. We then employ this unified framework to address the specific issue of the age dependence of protein interactions through the definition of three different node wiring or divergence schemes. A comparison with empirical data indicates that an age-dependent divergence move is necessary in order to reproduce the basic topological observables together with the age correlation between interacting nodes visible in empirical data. We also discuss the possibility of nontrivial joint partition and topology observables.

Many transcription factors bind to DNA with a remarkable lack of specificity, so that regulatory binding sites compete with an enormous number of nonregulatory “decoy” sites. For an autoregulated gene, we show decoy sites decrease noise in the number of unbound proteins to a Poisson limit that results from binding and unbinding. This noise buffering is optimized for a given protein concentration when decoys have a 1/2 probability of being occupied. Decoys linearly increase the time to approach steady state and exponentially increase the time to switch epigenetically between bistable states.

Tracers in a turbulent flow separate according to the celebrated $t^{3/2}$ Richardson-Obukhov law, which is usually explained by a scale-dependent effective diffusivity. Here, supported by state-of-the-art numerics, we revisit this argument. The Lagrangian correlation time of velocity differences increases too quickly for validating this approach, but acceleration differences decorrelate on dissipative time scales. Phenomenological arguments are used to relate the behavior of separations to that of a “local energy dissipation,” defined as the average ratio between the cube of the longitudinal velocity difference and the distance between the two tracers. This quantity is shown to stabilize on short time scales and this results in an asymptotic diffusion $\propto$$t^{1/2}$ of velocity differences. The time of convergence to this regime is shown to be that of deviations from Batchelor's initial ballistic regime, given by a scale-dependent energy dissipation time rather than the usual turnover time. It is finally demonstrated that the fluid flow intermittency should not affect this long-time behavior of the relative motion.

In human crowds, interactions among individuals give rise to a variety of self-organized collective motions that help the group to effectively solve the problem of coordination. However, it is still not known exactly how humans adjust their behavior locally, nor what are the direct consequences on the emergent organization. One of the underlying mechanisms of adjusting individual motions is the stepping dynamics. In this paper, we present first quantitative analysis on the stepping behavior in a one-dimensional pedestrian flow studied under controlled laboratory conditions. We find that the step length is proportional to the velocity of the pedestrian, and is directly related to the space available in front of him, while the variations of the step duration are much smaller. This is in contrast with locomotion studies performed on isolated pedestrians and shows that the local density has a direct influence on the stepping characteristics. Furthermore, we study the phenomena of synchronization—walking in lock step—and show its dependence on flow densities. We show that the synchronization of steps is particularly important at high densities, which has direct impact on the studies of optimizing pedestrians’ flow in congested situations. However, small synchronization and antisynchronization effects are found also at very low densities, for which no steric constraints exist between successive pedestrians, showing the natural tendency to synchronize according to perceived visual signals.

A spectral element approach is introduced to determine the Floquet exponents (FEs) of unstable periodic orbits (UPOs) stabilized by extended delayed feedback control (EDFC). The spectral approach does not require solving time-dependent eigenproblems that existing methods require. Instead, the spectral approach determines the stability of the delay differential equations of the system by numerical approximation. The method is capable of analyzing systems whose UPOs arise from bifurcations other than period-doubling. Results are presented for stabilizing UPOs in Duffing systems. The FEs calculated by the spectral approach are compared to published results for two examples. In both cases, the spectral method results agree well with those determined by previous methods. In addition, the spectral method was used to analyze a high-dimensional, asymmetrical system with a UPO in chaos arising from tori doubling following a Hopf bifurcation.

In order to understand the onset of convective instability and multiple stable convection patterns of buoyancy-driven convection of cold water near its density maximum in a vertical cylindrical container heated from below, a series of three-dimensional numerical simulations were performed. The aspect ratio of the container was 2 and Prandtl number of cold water was 11.57. The sidewall was considered to be perfectly adiabatic, and the density inversion parameter was fixed at 0.3. The result shows that the density inversion phenomenon in cold water has an important effect on the critical Rayleigh number for the onset of convection and the pattern formation at higher Rayleigh numbers. When the Rayleigh number varies from 3×10${}^3$ to 1.2×10${}^5$, eight stable, steady convection patterns are obtained under different initial conditions. The coexistence of multiple stable steady flow patterns is also observed within some specific ranges of the Rayleigh number.

Theoretical consideration of the optical response of nascent free electrons in the process of laser filamentation reveals that the initial microscopically inhomogeneous charge distribution causes a transient electromagnetic response of the medium that differs drastically from that of a homogeneous plasma with the same degree of ionization. An analytical model, describing the forced oscillations of virtually isolated and expanding electron clouds, predicts considerable enhancement of these oscillations caused by transient resonance with the laser field. The transient resonance processes should play a role in the currently accepted picture of filament formation dynamics.

The truncation of a pair potential at a distance $r_{\mathrm{c}}$ is well known to imply, in general, an impulsive correction to the pressure and other moments of the first derivatives of the potential. That, depending on $r_{\mathrm{c}}$, the truncation may also be of relevance to higher derivatives is shown theoretically for the Born contributions to the elastic moduli obtained using the stress-fluctuation formalism in $d$ dimensions. Focusing on isotropic liquids for which the shear modulus $G$ must vanish by construction, the predicted corrections are tested numerically for binary mixtures and polydisperse Lennard-Jones beads in, respectively, $d=3$ and 2 dimensions. Both models being glass formers, we comment briefly on the temperature ($T$) dependence of the (corrected) shear modulus $G\left(T\right)$ around the glass transition temperature $T_{\mathrm{g}}$.

A deeper understanding of nonequilibrium phenomena is needed to reveal the principles governing natural and synthetic molecular machines. Recent work has shown that when a thermodynamic system is driven from equilibrium then, in the linear response regime, the space of controllable parameters has a Riemannian geometry induced by a generalized friction tensor. We exploit this geometric insight to construct closed-form expressions for minimal-dissipation protocols for a particle diffusing in a one-dimensional harmonic potential, where the spring constant, inverse temperature, and trap location are adjusted simultaneously. These optimal protocols are geodesics on the Riemannian manifold and reveal that this simple model has a surprisingly rich geometry. We test these optimal protocols via a numerical implementation of the Fokker-Planck equation and demonstrate that the friction tensor arises naturally from a first-order expansion in temporal derivatives of the control parameters, without appealing directly to linear response theory.

Using the bond-propagation algorithm, we study the Ising model on a rectangle of size $M\times N$ with free boundaries. For five aspect ratios, $\rho =M/N=1$, 2, 4, 8, and 16, the critical free energy, internal energy and specific heat are calculated. The largest size reached is $M\times N=64\times {10}^6$. The accuracy of the free energy reaches ${10}^{-26}$. Based on these accurate data, we determine exact expansions of the critical free energy, internal energy, and specific heat. With these expansions, we extract the bulk, surface, and corner parts of the free energy, internal energy, and specific heat. The fitted bulk free energy density is given by $f_{\infty }=0.92969539834161021499\left(1\right)$, compared with Onsager's exact result $f_{\infty }=0.929695398341610214985\cdots .$ We confirm the conformal field theory (CFT) prediction of the corner free energy, in which the central charge of the Ising model is found to be $c=0.5\pm 1\times {10}^{-10}$, compared with the CFT result $c=0.5$. We find that not only the corner free energy but also the corner internal energy and specific heat are geometry independent, i.e., independent of aspect ratio. The implication of this finding for finite-size scaling is discussed. In the second-order correction of the free energy, we confirm the geometry dependence predicted by CFT and determine a geometry-independent constant beyond CFT. High-order corrections are also obtained.

We consider the directed percolation process as a prototype of systems displaying a nonequilibrium phase transition into an absorbing state. The model is in a critical state when the activation probability is adjusted at some precise value $p_c$. Criticality is lost as soon as the probability to activate sites at the first attempt, $p_1$, is changed. We show here that criticality can be restored by “compensating” the change in $p_1$ by an appropriate change of the second time activation probability $p_2$ in the opposite direction. At compensation, we observe that the bulk exponents of the process coincide with those of the normal directed percolation process. However, the spreading exponents are changed and take values that depend continuously on the pair $(p_1,p_2)$. We interpret this situation by acknowledging that the model with modified initial probabilities has an infinite number of absorbing states.

In this paper we present a generalization of Berreman's model for the elastic contribution to the surface free-energy density of a nematic liquid crystal in presence of a sawtooth substrate which favors homeotropic anchoring as a function of the wave number of the surface structure $q$, the tilt angle $\alpha$, and the surface anchoring strength $w$. In addition to the previously reported nonanalytic contribution proportional to $-q\ln q$, due to the nucleation of disclination lines at the wedge bottoms and apexes of the substrate, the next-to-leading contribution is proportional to $q$ for a given substrate roughness, in agreement with Berreman's predictions. We characterize this term, finding that it has two contributions: the deviations of the nematic director field with respect to a reference field corresponding to the isolated disclination lines and their associated core free energies. Comparison with the results obtained from the Landau-de Gennes model shows that our model is quite accurate in the limit $wL>1$, when strong anchoring conditions are effectively achieved.

We discovered a striped pattern of gene expression with circadian rhythms in growing plant roots using bioluminescent imaging of gene expression. Our experimental analysis revealed that the stripe wave in the bioluminescent image originated at the root tip and was caused by a continuous phase resetting of circadian oscillations. Some complex stripe waves containing arrhythmic regions were also observed. We succeeded in describing the formation mechanisms of these patterns using a growing phase oscillator network with a phase-resetting boundary condition.

Whether the evolution of natural river networks pursues a certain optimal state has been a most intriguing and fundamental question. There have been many optimality hypotheses proposed but it has yet to be proved which of these best serves as a quantitative signature of river network development. Here, this fundamental question is investigated for the five hypotheses of “minimum total energy expenditure,” “minimum total energy dissipation rate,” “minimum total stream power,” “minimum global energy expenditure rate,” and “minimum topological energy.” Using simple example landscapes, I examined whether any of these hypotheses pursues both the treelike river network formation and the concave stream longitudinal profile, the two characteristic patterns of natural landscapes. It is found that none of these hypotheses captures both patterns under the steady-state condition where the balance between tectonic uplift and sediment loss is satisfied. These findings are further verified through simulations of landscapes that satisfy given optimality criteria using an optimization method.