Biphasic amplitude dynamics (BAD) of oscillation have been observed in many biological systems. However, the specific topology structure and regulatory mechanisms underlying these biphasic amplitude dynamics remain elusive. Here, we searched all possible two-node circuit topologies and identified the core oscillator that enables robust oscillation. This core oscillator consists of a negative feedback loop between two nodes and a self-positive feedback loop of the input node, which result in the fast and slow dynamics of the two nodes, thereby achieving relaxation oscillation. Landscape theory was employed to study the stochastic dynamics and global stability of the system, allowing us to quantitatively describe the diverse positions and sizes of the Mexican hat. With increasing input strength, the size of the Mexican hat exhibits a gradual increase followed by a subsequent decrease. The self-activation of input node and the negative feedback on input node, which dominate the fast dynamics of the input node, were observed to regulate BAD in a bell-shaped manner. Both deterministic and statistical analysis results reveal that BAD is characterized by the linear and nonlinear dependence of the oscillation trough and crest on the input strength. In addition, combining with computational and theoretical analysis, we addressed that the linear response of trough to input is predominantly governed by the negative feedback, while the nonlinear response of crest is jointly regulated by the negative feedback loop and the self-positive feedback loop within the oscillator. Overall, this study provides a natural and physical basis for comprehending the occurrence of BAD in oscillatory systems, yielding guidance for the design of BAD in synthetic biology applications.

Single-molecule stretching experiments are widely utilized within the fields of physics and chemistry to characterize the mechanics of individual bonds or molecules, as well as chemical reactions. Analytic relations describing these experiments are valuable, and these relations can be obtained through the statistical thermodynamics of idealized model systems representing the experiments. Since the specific thermodynamic ensembles manifested by the experiments affect the outcome, primarily for small molecules, the stretching device must be included in the idealized model system. Though the model for the stretched molecule might be exactly solvable, including the device in the model often prevents analytic solutions. In the limit of large or small device stiffness, the isometric or isotensional ensembles can provide effective approximations, but the device effects are missing. Here a dual set of asymptotically correct statistical thermodynamic theories are applied to develop accurate approximations for the full model system that includes both the molecule and the device. The asymptotic theories are first demonstrated to be accurate using the freely jointed chain model and then using molecular dynamics calculations of a single polyethylene chain.

The population annealing algorithm is a population-based equilibrium version of simulated annealing. It can sample thermodynamic systems with rough free-energy landscapes more efficiently than standard Markov chain Monte Carlo alone. A number of parameters can be fine-tuned to improve the performance of the population annealing algorithm. While there is some numerical and theoretical work on most of these parameters, there appears to be a gap in the literature concerning the role of resampling in population annealing which this work attempts to close. The two-dimensional Ising model is used as a benchmarking system for this study. At first various resampling methods are implemented and numerically compared. In a second part the exact solution of the Ising model is utilized to create an artificial population annealing setting with effectively infinite Monte Carlo updates at each temperature. This limit is first performed on finite population sizes and subsequently extended to infinite populations. This allows us to look at resampling isolated from other parameters. Many results are expected to generalize to other systems.

We introduce a mathematical approach for the description of driver overacceleration in a microscopic traffic flow model. The model, in which no driver overreaction occurs, explains the empirical nucleation nature of traffic breakdown.

We examine motility-induced phase separation (MIPS) in two-dimensional run-and-tumble disk systems using both machine learning and noise fluctuation analysis. Our measures suggest that within the MIPS state there are several distinct regimes as a function of density and run time, so that systems with MIPS transitions exhibit an active fluid, an active crystal, and a critical regime. The different regimes can be detected by combining an order parameter extracted from principal component analysis with a cluster stability measurement. The principal component-derived order parameter is maximized in the critical regime, remains low in the active fluid, and has an intermediate value in the active crystal regime. We demonstrate that machine learning can better capture dynamical properties of the MIPS regimes compared to more standard structural measures such as the maximum cluster size. The different regimes can also be characterized via changes in the noise power of the fluctuations in the average speed. In the critical regime, the noise power passes through a maximum and has a broad spectrum with a $1/f^{1.6}$ signature, similar to the noise observed near depinning transitions or for solids undergoing plastic deformation.

In this work we consider an efficient algorithm for variational calculations of quantum few-particle systems in $S, P$, and $D$ states of the even parity using all-particle explicitly correlated Gaussian (ECG) basis sets. We primarily focus on the description of states where the dominant configuration contains either two particles in $p$ states or a single particle in a $d$ state (all other particles are in $s$ states). The basis functions we consider are products of spherically symmetric ECGs and bipolar harmonics. We introduced a scheme for deriving expressions for matrix elements of the overlap, kinetic and potential energy, as well as their derivatives with respect to the nonlinear parameters of the Gaussians. This allowed us to improve the efficiency of numerical calculations of the matrix elements (which is the most critical part of any code that uses ECGs) by one to two orders of magnitude compared to previous implementations. We provide a complete set of formulas for all basic matrix elements using the formalism of matrix differential calculus and discuss some technical details relevant to their efficient implementation. Lastly, we report a few example calculations of the ${}^2{D}^e$ states of the Li atom, ${}^4{P}^e$ and ${}^2{D}^e$ states of the B atom, and ${}^3{P}^e$ states of the C atom.

The corner transfer matrix renormalization group (CTMRG) algorithm has been extensively used to investigate both classical and quantum two-dimensional (2D) lattice models. The convergence of the algorithm can strongly vary from model to model depending on the underlying geometry and symmetries, and the presence of algebraic correlations. An important factor in the convergence of the algorithm is the lattice symmetry, which can be broken due to the necessity of mapping the problem onto the square lattice. We propose a variant of the CTMRG algorithm, designed for models with $C_3$-symmetry, which we apply to the conceptually simple yet numerically challenging problem of the triangular lattice Ising antiferromagnet in a field, at zero and low temperatures. We study how the finite-temperature three-state Potts critical line in this model approaches the ground-state Kosterlitz-Thouless transition driven by a reduced field ($h/T$). In this particular instance, we show that the $C_3$-symmetric CTMRG leads to much more precise results than both existing results from exact diagonalization of transfer matrices and Monte Carlo.

We study the time evolution of a single qubit in contact with a bath, within the framework of projection operator methods. Employing the so-called modified Redfield theory, which also treats energy conserving interactions nonperturbatively, we are able to study the regime beyond the scope of the ordinary approach. Reduced equations of motion for the qubit are derived in an idealistic system where both the bath and system-bath interactions are modeled by Gaussian distributed random matrices. In the strong decoherence regime, a simple relation between the bath correlation function and the decoherence process induced by the energy conserving interaction is found. It implies that energy conserving interactions slow down the relaxation process, which leads to a Zeno freezing if they are sufficiently strong. Furthermore, our results are also confirmed in numerical simulations.

The strong enhancement of tunneling couplings typically observed in tunneling splittings in the quantum map is investigated. We show that the transition from instanton to noninstanton tunneling, which is known to occur in tunneling splittings in the space of the inverse Planck constant, takes place in a parameter space as well. By applying the absorbing perturbation technique, we find that the enhancement invoked as a result of local avoided crossings and that originating from globally spread interactions over many states should be distinguished and that the latter is responsible for the strong and persistent enhancement. We also provide evidence showing that the coupling across the separatrix in phase space is crucial in explaining the behavior of tunneling splittings by performing the wave-function-based observation. In the light of these findings, we examine the validity of the resonance-assisted tunneling theory.

Active stimuli-responsive materials, intrinsically powered by chemical reactions, have immense capabilities that can be harnessed for designing synthetic systems for a variety of biomimetic applications. It goes without saying that the key aspect involved in the designing of such systems is to accurately estimate the amount of energy and power available for transduction through various mechanisms. Belousov-Zhabotinsky (BZ) reactions are dynamical systems, which exhibit self-sustained nonlinear chemical oscillations, as their catalyst undergoes periodic redox cycles in the presence of reagents. The unique feature of BZ reaction based active systems is that they can continuously perform mechanical work by transducing energy from sustained chemical oscillations. The objective of our work is to use bifurcation analyses to identify oscillatory regimes and quantify energy-power characteristics of the BZ reaction based on nanocatalyst activity and BZ reaction formulations. Our approach involves not only the computation of higher order Lyapunov and frequency coefficients but also Hamiltonian functions, through normal form reduction of the kinetic model of the BZ reaction. Ultimately, using these calculations, we determine amplitude, frequency, and energy-power densities, as a function of the nanocatalysts’ activity and BZ formulations. As normal form representations are applicable to any dynamical system, we believe that our framework can be extended to other self-sustained active systems, including systems based on stimuli-responsive materials.

We study hypersensitivity to initial-state perturbation in the unitary dynamics of a multiqubit system. We use the quantum state metric, introduced by Girolami and Anza [Phys. Rev. Lett. 126, 170502 (2021)], which can be interpreted as a quantum Hamming distance. To provide a proof of principle, we take the multiqubit implementation of the quantum kicked top, a paradigmatic system known to exhibit quantum chaotic behavior. Our findings confirm that the observed hypersensitivity corresponds to commonly used signatures of quantum chaos. Furthermore, we demonstrate that the proposed metric can detect quantum chaos in the same regime and under analogous initial conditions as in the corresponding classical case.

The reservoir computing approach utilizes a time series of measurements as input to a high-dimensional dynamical system known as a reservoir. However, the approach relies on sampling a random matrix to define its underlying reservoir layer, which leads to numerous hyperparameters that need to be optimized. Here, we propose a nonlocally coupled pendulum model with higher-order interactions as a novel reservoir, which requires no random underlying matrices and fewer hyperparameters. We use Bayesian optimization to explore the hyperparameter space within a minimal number of iterations and train the coupled pendulums model to reproduce the chaotic attractors, which simplifies complicated hyperparameter optimization. We illustrate the effectiveness of our technique with the Lorenz system and the Hindmarsh-Rose neuronal model, and we calculate the Pearson correlation coefficients between time series and the Hausdorff metrics in the phase space. We demonstrate the contribution of higher-order interactions by analyzing the interaction between different reservoir configurations and prediction performance, as well as computations of the largest Lyapunov exponents. The chimera state is found as the most effective dynamical regime for prediction. The findings, where we present a new reservoir structure, offer potential applications in the design of high-performance modeling of dynamics in physical systems.

Traditional models for molecular (Brownian) motors predominantly depend on nonequilibrium driving, while particle interactions rigorously adhere to Newton's third law. However, numerous living and natural systems at various scales seem to defy this well-established law. In this study, we investigated the transport of mixed Brownian particles in a two-dimensional ratchet potential with nonreciprocal interactions. Our findings reveal that these nonreciprocal interactions can introduce a zero-mean nonequilibrium driving force. This force is capable of disrupting the thermodynamic equilibrium and inducing directed motion. The direction of this motion is determined by the asymmetry of the potential. Interestingly, the average velocity is a peaked function of the degree of nonreciprocity, while the effective diffusion consistently increases with the increase of nonreciprocity. There exists an optimal temperature or packing fraction at which the average velocity reaches its maximum value. We share a mechanism for particle rectification, devoid of particle-autonomous nonequilibrium drive, with potential usage in systems characterized by nonreciprocal interactions.

Cyclic loading on granular packings and amorphous media exhibits a transition from reversible elastic behavior to irreversible plasticity. The present study compares the irreversibility transition and microscopic details of colloidal polycrystals under oscillatory tensile-compressive and shear strain. Under both modes, the systems exhibit a reversible to irreversible transition. However, the strain amplitude at which the transition is observed is larger in the shear strain than in the tensile-compressive mode. The threshold strain amplitude is confirmed by analyzing the dynamical properties, such as mobility and atomic strain (von Mises shear strain and the volumetric strain). The structural changes are quantified using a hexatic order parameter. Under both modes of deformation, dislocations and grain boundaries in polycrystals disappear, and monocrystals are formed. We also recognize the dislocation motion through grains. The key difference is that strain accumulates diagonally in oscillatory tensile-compressive deformation, whereas in shear deformation, strain accumulation is along the $x$ or $y$ axis.

This paper proposes a transient three-dimensional model to simulate microwave-induced discharges in a waveguide-based plasma source under intermediate pressures. A plane-symmetric simplification method is applied to simplify half of the microwave plasma source in the calculation domain, dramatically reducing the demand for computational resources and calculation time. Meanwhile, the numerical simulations remain in three dimensions without dimensionality reduction, which allows us to directly calculate the efficiency of power coupling from the incident microwave to the plasma. Besides, the computation decrease improves the convergence performance of the mathematical model, making it possible to model the entire discharge process from $1\times {10}^{-9}$ to $1\times {10}^4\mathrm{s}$. This period covers the instantaneous microwave breakdown to the formation of a stable plasma column near steady state. The results have revealed the electromagnetic waveguide structure change of the microwave plasma source during the discharge process. Several microwave power-coupling efficiencies of the waveguide-based plasma source with different thicknesses and permittivities of the glass tube are calculated and compared with the experimentally measured data. Furthermore, the effects of the glass tube on the electromagnetic modes of the traveling microwave propagating along the plasma column and the discharge properties are also investigated. The numerically obtained results generally agree with the theoretical analysis and the experimental data in our previous studies, demonstrating the validity of the proposed mathematical model and the plane-symmetric simplification method.

Using schematic model potentials, we calculate exactly the virial coefficients of a classical gas up to sixth order and use them to calculate the virial expansion of basic thermodynamic quantities such as pressure, density, and compressibility. At sufficiently strong couplings, as expected, the virial expansion fails to converge. However, at least for the interactions and parameter ranges we explored, we find that Padé-Borel resummation methods are very effective in improving the convergence of the expansion.

Exact mechanisms of thermal conductivity in liquids are not well understood, despite a rich research history. A vibrational model of energy transfer in dense simple liquids with soft pairwise interactions seems adequate to partially fill this gap. The purpose of the present paper is to define its applicability domain and to demonstrate how well it works within the identified applicability domain in the important case of the Lennard-Jones model system. The existing results from molecular dynamics simulations are used for this purpose. Additionally, we show that a freezing density scaling approach represents a very powerful tool to estimate the thermal conductivity coefficient across essentially the entire gas-liquid region of the phase diagram, including metastable regions. A simple practical expression serving this purpose is proposed.

We investigate ergodicity, chaos, and thermalization for a one-dimensional classical gas of hard rods confined to an external quadratic or quartic trap, which breaks microscopic integrability. To quantify the strength of chaos in this system, we compute its maximal Lyapunov exponent numerically. The approach to thermal equilibrium is studied by considering the time evolution of particle position and velocity distributions and comparing the late-time profiles with the Gibbs state. Remarkably, we find that quadratically trapped hard rods are highly nonergodic and do not resemble a Gibbs state even at extremely long times, despite compelling evidence of chaos for four or more rods. On the other hand, our numerical results reveal that hard rods in a quartic trap exhibit both chaos and thermalization, and equilibrate to a Gibbs state as expected for a nonintegrable many-body system.

Study of dynamical systems using partial state observation is an important problem due to its applicability to many real-world systems. We address the problem by studying an echo state network (ESN) framework with partial state input with partial or full state output. Application to the Lorenz system and Chua's oscillator (both numerically simulated and experimental systems) demonstrate the effectiveness of our method. We show that the ESN, as an autonomous dynamical system, is capable of making short-term predictions up to a few Lyapunov times. However, the prediction horizon has high variability depending on the initial condition—an aspect that we explore in detail using the distribution of the prediction horizon. Further, using a variety of statistical metrics to compare the long-term dynamics of the ESN predictions with numerically simulated or experimental dynamics and observed similar results, we show that the ESN can effectively learn the system's dynamics even when trained with noisy numerical or experimental data sets. Thus, we demonstrate the potential of ESNs to serve as cheap surrogate models for simulating the dynamics of systems where complete observations are unavailable.