We systematically test and compare different reinforcement learning schemes in a complementarity game [J. Jost and W. Li, Physica A 345, 245 (2005)] played between members of two populations. More precisely, we study the Roth-Erev, Bush-Mosteller, and SoftMax reinforcement learning schemes. A modified version of Roth-Erev with a power exponent of 1.5, as opposed to 1 in the standard version, performs best. We also compare these reinforcement learning strategies with evolutionary schemes. This gives insight into aspects like the issue of quick adaptation as opposed to systematic exploration or the role of learning rates.

Bistable regulatory elements are important for nongenetic inheritance, increase of cell-to-cell heterogeneity allowing adaptation, and robust responses at the population level. Here, we study computationally the bistable genetic toggle switch—a small regulatory network consisting of a pair of mutual repressors—in growing and dividing bacteria. We show that as cells with an inhibited growth exhibit high stability of toggle states, cell growth and divisions lead to a dramatic increase of toggling rates. The toggling rates were found to increase with rate of cell growth, and can be up to six orders of magnitude larger for fast growing cells than for cells with the inhibited growth. The effect is caused mainly by the increase of protein and mRNA burst sizes associated with the faster growth. The observation that fast growth dramatically destabilizes toggle states implies that rapidly growing cells may vigorously explore the epigenetic landscape enabling nongenetic evolution, while cells with inhibited growth adhere to the local optima. This can be a clever population strategy that allows the slow growing (but stress resistant) cells to survive long periods of unfavorable conditions. Simultaneously, at favorable conditions, this stress resistant (but slowly growing—or not growing) subpopulation may be replenished due to a high switching rate from the fast growing population.

We propose an online method of detecting combustion instability based on the concept of dynamical system theory, including the characterization of the dynamic behavior of combustion instability. As an important case study relevant to combustion instability encountered in fundamental and practical combustion systems, we deal with the combustion dynamics close to lean blowout (LBO) in a premixed gas-turbine model combustor. The relatively regular pressure fluctuations generated by thermoacoustic oscillations transit to low-dimensional intermittent chaos owing to the intermittent appearance of burst with decreasing equivalence ratio. The translation error, which is characterized by quantifying the degree of parallelism of trajectories in the phase space, can be used as a control variable to prevent LBO.

Many applications in computational science require computing the elements of a function of a large matrix. A commonly used approach is based on the the evaluation of the eigenvalue decomposition, a task that, in general, involves a computing time that scales with the cube of the size of the matrix. We present here a method that can be used to evaluate the elements of a function of a positive-definite matrix with a scaling that is linear for sparse matrices and quadratic in the general case. This methodology is based on the properties of the dynamics of a multidimensional harmonic potential coupled with colored-noise, generalized Langevin equation thermostats. This “$f$-thermostat” approach allows us to calculate directly elements of functions of a positive-definite matrix by carefully tailoring the properties of the stochastic dynamics. We demonstrate the scaling and the accuracy of this approach for both dense and sparse problems and compare the results with other established methodologies.

A linear stability analysis of the Navier-Stokes (NS) granular hydrodynamic equations is performed to determine the critical length scale for the onset of vortices and clusters instabilities in granular dense binary mixtures. In contrast to previous attempts, our results (which are based on the solution to the inelastic Enskog equation to NS order) are not restricted to nearly elastic systems since they take into account the complete nonlinear dependence of the NS transport coefficients on the coefficients of restitution ${\alpha }_{ij}$. The theoretical predictions for the critical length scales are compared to molecular dynamics (MD) simulations in flows of strong dissipation (${\alpha }_{ij}\ge 0.7$) and moderate solid volume fractions ($\varphi \le 0.2$). We find excellent agreement between MD and kinetic theory for the onset of velocity vortices, indicating the applicability of NS hydrodynamics to polydisperse flows even for strong inelasticity, finite density, and particle dissimilarity.

The dynamics of an electron in a strong laser field can be significantly altered by radiation reaction. This usually results in a strongly damped motion, with the electron losing a large fraction of its initial energy. Here we show that the electron dynamics in a bichromatic laser pulse can be indirectly controlled by a comparatively small radiation reaction force through its interplay with the Lorentz force. By changing the relative phase between the two frequency components of the bichromatic laser field, an ultrarelativistic electron bunch colliding head-on with the laser pulse can be deflected in a controlled way, with the deflection angle being independent of the initial electron energy. The effect is predicted to be observable with laser powers and intensities close to those of current state-of-the-art petawatt laser systems.

We study the combined effects of noise and detector sensitivity on a dynamical process that generates intermittent events mimicking the behavior of complex systems. By varying the sensitivity level of the detector we move between two forms of complexity, from inverse power law to Mittag-Leffler interevent time survival probabilities. Here fluctuations fight against complexity, causing an exponential truncation to the survival probability. We show that fluctuations of relatively weak intensity have a strong effect on the generation of Mittag-Leffler complexity, providing a reason why stretched exponentials are frequently found in nature. Our results afford a more unified picture of complexity resting on the Mittag-Leffler function and encompassing the standard inverse power law definition.

We investigate the scaling regimes of the Kardar-Parisi-Zhang (KPZ) equation in the presence of spatially correlated noise with power-law decay $D\left(p\right)\sim p^{-2\rho }$ in Fourier space, using a nonperturbative renormalization group approach. We determine the full phase diagram of the system as a function of $\rho$ and the dimension $d$. In addition to the weak-coupling part of the diagram, which agrees with the results from Europhys. Lett. 47, 14 (1999) and Eur. Phys. J. B 9, 491 (1999), we find the two fixed points describing the short-range- (SR) and long-range- (LR) dominated strong-coupling phases. In contrast with a suggestion in the references cited above, we show that, for all values of $\rho$, there exists a unique strong-coupling SR fixed point that can be continuously followed as a function of $d$. We show in particular that the existence and the behavior of the LR fixed point do not provide any hint for 4 being the upper critical dimension of the KPZ equation with SR noise.

The use of statistical complexity metrics has yielded a number of successful methodologies to differentiate and identify signals from complex systems where the underlying dynamics cannot be calculated. The Mori-Zwanzig framework from statistical mechanics forms the basis for the generalized non-Markov parameter (NMP). The NMP has been used to successfully analyze signals in a diverse set of complex systems. In this paper we show that the Mori-Zwanzig framework masks an elegantly simple closed form of the first NMP, which, for $C^1$ smooth autocorrelation functions, is solely a function of the second moment (spread) and amplitude envelope of the measured power spectrum. We then show that the higher-order NMPs can be constructed in closed form in a modular fashion from the lower-order NMPs. These results provide an alternative, signal processing-based perspective to analyze the NMP, which does not require an understanding of the Mori-Zwanzig generating equations. We analyze the parametric sensitivity of the zero-frequency value of the first NMP, which has been used as a metric to discriminate between states in complex systems. Specifically, we develop closed-form expressions for three instructive systems: band-limited white noise, the output of white noise input to an idealized all-pole filter,$f$ and a simple harmonic oscillator driven by white noise. Analysis of these systems shows a primary sensitivity to the decay rate of the tail of the power spectrum.

Dense simple fluids appear to have a long-time, very slowly evolving state which—for high-enough density—approaches a glassy, nonergodic phase. In this high-density regime, the equilibrating system decays via a three-step process as identified in mode-coupling theory (MCT). MCT, however, is an inherently phenomenological theory without prospects for improvement, and so a systematic theory has been recently developed which naturally allows one to calculate cumulants between the fundamental particle density and response fields self-consistently in a perturbation expansion in a pseudopotential. This paper details the extension of this theory and specifically addresses the universal nature of this erogodic-nonergodic transition. We find that the nonergodic state can be characterized by the associated static equilibrium state, and—nontrivially—that these states are identical regardless of whether a system is driven by Smoluchowski or Newtonian dynamics. In addition, though the specific response fields differ between Smoluchowski and Newtonian dynamics, the two share identical linear fluctuation-dissipation relations connecting the $\rho$-$B$ cumulants. While we show this universality of nonergodic states within perturbation theory, we expect it to be true more generally. At leading order in the perturbation expansion, one obtains a kinetic kernel quadratic in the density analogous to the “one-loop” theory of MCT. At this one-loop level one finds vertex corrections which depend on the three-point equilibrium cumulants. Here we assume these vertex-corrections can be ignored and instead focus on computing the contributions of higher-order loops. We show that it is possible to sum up all of the loop contributions and find that these higher-order loops do not change the nonergodic state parameters substantially.

We study why the calculation of current correlation functions (CCFs) still suffers from finite-size effects even when the periodic boundary condition is taken. Two important one-dimensional, momentum-conserving systems are investigated as examples. Intriguingly, it is found that the state of a system recurs in the sense of microcanonical ensemble average, and such recurrence may result in oscillations in CCFs. Meanwhile, we find that the sound mode collisions induce an extra time decay in a current so that its correlation function decays faster (slower) in a smaller (larger) system. Based on these two unveiled mechanisms, a procedure for correctly evaluating the decay rate of a CCF is proposed, with which our analysis suggests that the global energy CCF decays as $\sim t^{-\frac{2}{3}}$ in the diatomic hard-core gas model and in a manner close to $\sim t^{-\frac{1}{2}}$ in the Fermi-Pasta-Ulam-$\beta$ model.

An approach to obtain the mobility of nonspherical particles is proposed by averaging the drag force orientationally, and two other widely used approaches in the literature, the averaged-collision-integral and averaged-drift-velocity methods, are summarized and extended. The concept of orientationally averaged collision integrals based on Chapman-Enskog theory for small gas-phase ions is re-examined for macromolecular ions whose surface cannot be treated as specular, but with inelastic interactions. A well accepted collision model considering inelastic collisions is Epstein's theory, which has been extended to include long-range potential forces by Li and Wang [Phys. Rev. E 68, 061206 (2003)] for spherical particles. This work extends Li and Wang's spherical particle theory to convex nonspherical particles considering long-range potential, and simplifies this collision integral to a product of the averaged projection area and an enhancement factor for short-range interactions (hard collisions), which is independent of convex particle shape and is identical to the value for a sphere that people are using. We also show that the averaged projection area of a convex particle in free molecular regime for hard collisions is equal to its mobility diameter. The second approach is the averaged-drift-velocity approach using the friction coefficient in a tensor form, which is often employed in aerosol science. We extend this approach in our previous work for axisymmetric particles to develop an expression for the mobility of nonspherical particles in a general form. Furthermore, it is pointed out that this approach is only valid when the particle Brownian rotation is slow compared with the particle translational relaxation time. If the particle Brownian rotation is fast, usually so in the case of very small ions and particles, we propose an “averaged-drag-force” approach. The three approaches are then compared for a randomly oriented rod and the protein GroEL. We show that for a cylinder rod in the free molecular regime at random orientation, the averaged-drag-force approach is identical to the averaged-collision-integral approach for short-range interactions (hard collisions). We then summarize the relationship between collision-integral based approach and tensor based approaches. For readers only interested in implementation of the theory, we provide useful expressions in Tables I and II.

Nucleation of a new phase almost always starts at a surface. This surface is almost always assumed not to change with time. However, surfaces can roughen, partially dissolve, and change chemically with time. Each of these irreversible changes will change the nucleation rate at the surface, resulting in a time-dependent nucleation rate. Here we use a simple model to show that partial surface dissolution can qualitatively change the nucleation process in a way that is testable in experiment. The changing surface means that the nucleation rate is increasing with time. There is an initial period during which no nucleation occurs, followed by relatively rapid nucleation.

Long polymer chains are ubiquitous in biological systems and their mechanical properties have significant impact upon biological processes. Of particular interest is the situation in which polymer chains are wound around each other or around other objects. We have analyzed the parameters of a long Gaussian polymer chain wound around a cylinder as a function of the torque applied to the ends of the chain. We have shown that for sufficiently long polymer chains, an average winding angle and a characteristic radius of the chain can be determined from a modified Bessel function of purely imaginary order, in which the value of the order is equivalent to the applied torque, normalized to the product of the absolute temperature and the Boltzmann constant. The obtained results are consistent with a simplified interpretation in terms of “torsional blobs,” and this could be extended to nonideal chains with excluded volumes. We have also extended our results to the case of a polymer chain rotating in viscous medium. Our results could be used to estimate the mechanical strains that appear in DNA and RNA during transcription, as these might initiate formation of unusual DNA structures, invasion of RNA into the DNA duplex ($R$-loop formation), and modulation of the interactions of DNA and RNA with proteins.

In this paper, we will investigate critical phenomena by considering a model spin glass on scale-free networks. For this purpose, we consider the Ghatak-Sherrington (GS) model, a spin-1 spin-glass model with a crystal field, instead of the usual Ising-type model. Scale-free networks on which the GS model is placed are constructed from the static model, in which the number of vertices is fixed from the beginning. On the basis of the replica-symmetric solution, we obtain the analytical solutions, i.e., free energy and order parameters, and we derive the various phase diagrams consisting of the paramagnetic, ferromagnetic, and spin-glass phases as functions of temperature $T$, the degree exponent $\lambda$, the mean degree $K$, and the fraction of the ferromagnetic interactions $\rho$. Since the present model is based on the GS model, which considers the three states ($S=0,\pm 1$), the $S=0$ state plays a crucial role in the $\lambda$-dependent critical behavior: glass transition temperature $T_g$ has a finite value, even when $2<\lambda <3$. In addition, when the crystal field becomes nonzero, the present model clearly exhibits three types of inverse transitions, which occur when an ordered phase is more entropic than a disordered one.

We examine a system consisting of two reservoirs of particles connected by a channel. In the channel are two oscillating repulsive potential-energy barriers. It is known that such a system can transport particles from one reservoir to the other, even when the chemical potentials in the reservoirs are equal. We use computations and the theory of chaotic transport to study this system. Chaotic transport is described by passage around or through a heteroclinic tangle. Topological properties of the tangle are described using a generalization of homotopic lobe dynamics, which is a theory that gives some properties of intermediate-time behavior from properties of short-time behavior. We compare these predicted properties with direct computation of trajectories.

Macroscopic magnetic systems subjected to external forcing exhibit complex spatiotemporal behaviors as result of dissipative self-organization. Pattern formation from a uniform magnetization state, induced by the combination of a spin-polarized current and an external magnetic field, is studied for spin-transfer nano-oscillator devices. The system is described in the continuous limit by the Landau-Lifshitz-Gilbert equation. The bifurcation diagram of the quintessence parallel state, as a function of the external field and current, is elucidated. We have shown analytically that this state exhibits a spatial supercritical quintic bifurcation, which generates in two spatial dimensions a family of stationary stripes, squares, and superlattice states. Analytically, we have characterized their respective stabilities and bifurcations, which are controlled by a single dimensionless parameter. This scenario is confirmed numerically.

The distribution of zeros of the Husimi function of energy eigenstates has proven to be a very useful tool for the characterization of the quantum transition from order to chaos in systems without degeneracy. In this paper, we show, by means of calculations on the Hénon-Heiles Hamiltonian system, that the quantum order-chaos transition in systems with degeneracy can also be characterized through the distribution of the Husimi function zeros, providing that the appropriate linear combination of degenerate eigenstates is used. If using an arbitrary linear combination, spurious results can be achieved, suggesting a wrong premature quantum transition to chaos.

The Navier boundary condition for velocity slip on flat surfaces, when expressed in tensor form, is readily extended to surfaces of any shape. We test this assertion using molecular dynamics simulations of flow in channels with flat and curved walls and for rotating cylinders and spheres, all for a wide range of solid-liquid interaction strengths. We find that the slip length as conventionally measured at a flat wall in Couette flow is the same as that for all other cases with curved and rotating boundaries, provided the atomic interactions are the same and boundary shape is properly taken into account. These results support the idea that the slip length is a material property, transferable between different flow configurations.

Under the limit of infinite Prandtl number, we derive analytical expressions for the large-scale quantities, e.g., Péclet number Pe, Nusselt number Nu, and rms value of the temperature fluctuations ${\theta }_{\mathrm{rms}}$. We complement the analytical work with direct numerical simulations, and show that $\mathrm{Nu}\sim {\mathrm{Ra}}^{\gamma }$ with $\gamma \approx (0.30–0.32)$, $\mathrm{Pe}\sim {\mathrm{Ra}}^{\eta }$ with $\eta \approx (0.57–0.61)$, and ${\theta }_{\mathrm{rms}}\sim \mathrm{const}$. The Nusselt number is observed to be an intricate function of $\mathrm{Pe}$, ${\theta }_{\mathrm{rms}}$, and a correlation function between the vertical velocity and temperature. Using the scaling of large-scale fields, we show that the energy spectrum $E_u\left(k\right)\sim k^{-13/3}$, which is in a very good agreement with our numerical results. The entropy spectrum $E_{\theta }\left(k\right)$, however, exhibits dual branches consisting of $k^{-2}$ and $k^0$ spectra; the $k^{-2}$ branch corresponds to the Fourier modes $\widehat{\theta }(0,0,2n)$, which are approximately $-1/\left(2n\pi \right)$. The scaling relations for Prandtl number beyond ${10}^2$ match with those for infinite Prandtl number.