Exact Steady-state Solution Research Articles

High-resolution particle-in-cell simulations of two-dimensional Bernstein–Greene–Kruskal modes

We present two-dimensional (2D) particle-in-cell (PIC) simulations of 2D Bernstein–Greene–Kruskal modes, which are exact nonlinear steady-state solutions of the Vlasov–Poisson equations, on a 2D plane perpendicular to a background magnetic field, with a cylindrically symmetric electric potential localized on the plane. PIC simulations are initialized using analytic electron distributions and electric potentials from the theory. We confirm the validity of such solutions using high-resolutions up to a 20482 grid. We show that the solutions are dynamically stable for a stronger background magnetic field, while keeping other parameters of the model fixed, but become unstable when the field strength is weaker than a certain value. When a mode becomes unstable, we observe that the instability begins with the excitation of azimuthal electrostatic waves that ends with a spiral pattern.

Modelling solar coronal magnetic fields with physics-informed neural networks

ABSTRACT We present a novel numerical approach aiming at computing equilibria and dynamics structures of magnetized plasmas in coronal environments. A technique based on the use of neural networks that integrates the partial differential equations of the model, and called physics-informed neural networks (PINNs), is introduced. The functionality of PINNs is explored via calculation of different magnetohydrodynamic (MHD) equilibrium configurations, and also obtention of exact two-dimensional steady-state magnetic reconnection solutions. Advantages and drawbacks of PINNs compared to traditional numerical codes are discussed in order to propose future improvements. Interestingly, PINNs is a meshfree method in which the obtained solution and associated different order derivatives are quasi-instantaneously generated at any point of the spatial domain. We believe that our results can help to pave the way for future developments of time dependent MHD codes based on PINNs.

An integrated modelling approach for targeted degradation: insights on optimization, data requirements and PKPD predictions from semi- or fully-mechanistic models and exact steady state solutions

The value of an integrated mathematical modelling approach for protein degraders which combines the benefits of traditional turnover models and fully mechanistic models is presented. Firstly, we show how exact solutions of the mechanistic models of monovalent and bivalent degraders can provide insight on the role of each system parameter in driving the pharmacological response. We show how on/off binding rates and degradation rates are related to potency and maximal effect of monovalent degraders, and how such relationship can be used to suggest a compound optimization strategy. Even convoluted exact steady state solutions for bivalent degraders provide insight on the type of observations required to ensure the predictive capacity of a mechanistic approach. Specifically for PROTACs, the structure of the exact steady state solution suggests that the total remaining target at steady state, which is easily accessible experimentally, is insufficient to reconstruct the state of the whole system at equilibrium and observations on different species (such as binary/ternary complexes) are necessary. Secondly, global sensitivity analysis of fully mechanistic models for PROTACs suggests that both target and ligase baselines (actually, their ratio) are the major sources of variability in the response of non-cooperative systems, which speaks to the importance of characterizing their distribution in the target patient population. Finally, we propose a pragmatic modelling approach which incorporates the insights generated with fully mechanistic models into simpler turnover models to improve their predictive ability, hence enabling acceleration of drug discovery programs and increased probability of success in the clinic.

Noise induced bistability in a fluctuating environment

Bistability in some systems does not occur at the mean-field level but is induced by noise. In this work, we consider a system which exhibits bistability induced by noise but in the presence of environmental fluctuations. The system can transition between two environmental states at a specified rate. It has absorbing boundaries in one state and can switch between two metastable states in the other. At small values of a system parameter (ϵ), the model exhibits noise-induced switching behaviour and the probability distribution is bimodal which becomes unimodal at large values of ϵ. We calculate the exact eigenvalues and steady-state solutions of the system. We further extend the investigation by calculating the mean switching time between the two metastable states in the small parameter limit and compare its accuracy with simulation results.

ON AN IMPORTANT OBSERVATION REGARDING SOME HYDROMAGNETIC MOTIONS OF MAXWELL FLUIDS AND ITS APPLICATIONS*

The problem of exact solutions for isothermal motions of non­Newtonian fluids is of interest yet and a new way to get them is wel­come. In this work an important observation regarding the governing equations corresponding to some isothermal hydromagnetic unidirec­tional motions of incompressible Maxwell fluids is brought to light. It allows us to easily determine exact solutions for motions with shear stress or velocity on the boundary when similar solutions for motions with velocity, respectively shear stress on the boundary are know. To exemplify, the solutions of some hydromagnetic motion problems of Maxwell fluids with velocity on the boundary are used to generate exact steady state solutions for similar motions of same fluids with shear stress on the boundary. These solutions are very important for the experimental researchers who want to know the required time to reach the steady state.

On an Important Remark Concerning Some MHD Motions of Second-Grade Fluids through Porous Media and Its Applications

In this work it is proven that the governing equations for the fluid velocity and non-trivial shear stress corresponding to some isothermal MHD unidirectional motions of incompressible second-grade fluids through a porous medium have identical forms. This important remark is used to provide exact steady-state solutions for motions with shear stress on the boundary when similar solutions of some motions with velocity on the boundary are known. Closed-form expressions are provided both for the fluid velocity and the corresponding shear stress and Darcy’s resistance. As a check of the results that are obtained here, the solutions corresponding to motions over an infinite flat plate are presented in different forms whose equivalence is graphically proven. In the case of the motions between infinite parallel plates, the fluid behavior is symmetric with respect to the median plane due to the boundary conditions.

Exact steady-state solution of fractals damped, and forced systems

In this article, the methodology for deriving the exact steady-state solutions of forced oscillatory systems has been extended to obtain the solution of fractal damped and forced differential equations using the two-scale fractal dimension transform. Finally, a numerical example is given to illustrate the applicability of the proposed approach in obtaining the frequency–amplitude response curves for a forced and damped fractal oscillator. Numerical results show that the fractal parameter values shift the frequency–amplitude curves to the left and right from the resonance curve. The present method sheds new light on solving damped and forced fractal oscillators.

Collective stochastic resonance behaviors of two coupled harmonic oscillators driven by dichotomous fluctuating frequency* *Project supported by the National Natural Science Foundation of China for the Youth (Grant Nos. 11501385 and 11801385).

The collective behaviors of two coupled harmonic oscillators with dichotomous fluctuating frequency are investigated, including stability, synchronization, and stochastic resonance (SR). First, the synchronization condition of the system is obtained. When this condition is satisfied, the mean-field behavior is consistent with any single particle behavior in the system. On this basis, the stability condition and the exact steady-state solution of the system are derived. Comparative analysis shows that, the stability condition is stronger than the synchronization condition, that is to say, when the stability condition is satisfied, the system is both synchronous and stable. Simulation analysis indicates that increasing the coupling strength will reduce the synchronization time. In weak coupling region, there is an optimal coupling strength that maximizes the output amplitude gain (OAG), thus the coupling-induced SR behavior occurs. In strong coupling region, the two particles are bounded as a whole, so that the coupling effect gradually disappears.

Dissipative generation of pure steady states and a gambler's ruin problem

We consider an open quantum system, with dissipation applied only to a part of its degrees of freedom, evolving via a quantum Markov dynamics. We demonstrate that, in the Zeno regime of large dissipation, the relaxation of the quantum system towards a pure quantum state is linked to the evolution of a classical Markov process towards a single absorbing state. The rates of the associated classical Markov process are determined by the original quantum dynamics. Extension of this correspondence to absorbing states with internal structure allows us to establish a general criterion for having a Zeno-limit nonequilibrium stationary state of arbitrary finite rank. An application of this criterion is illustrated in the case of an open XXZ spin-1/2 chain dissipatively coupled at its edges to baths with fixed and different polarizations. For this system, we find exact nonequilibrium steady-state solutions of ranks 1 and 2.

Quantum transport with electronic relaxation in electrodes: Landauer-type formulas derived from the driven Liouville–von Neumann approach

We derive the exact steady-state solutions for the simplest model systems of resonant tunneling and tunneling with destructive quantum interference from the driven Liouville-von Neumann (DLvN) approach. Under the finite-state lead condition (the two electrodes have finite states), we analyze the asymptotic behavior of the steady-state current in the two limits of electronic relaxation. Under the infinite-state lead condition, the steady-state solutions of the two model systems can be cast as Landauer-type current formulas. According to the formulas, we show that the transmission functions near the resonant peak and the antiresonant dip can be significantly influenced by electronic relaxation in the electrodes. Moreover, under intermediate and strong electronic relaxation conditions, we analytically show that the steady-state current of the DLvN approach dramatically deviates from the Landauer current when destructive quantum interference occurs. In the regime of zero electronic relaxation, our results are reduced to the Landauer formula, indicating that the DLvN approach is equivalent to the Landauer approach when the leads have infinite states without any electronic relaxation.

Response of a multi-degree-of-freedom system with a pounding vibration neutralizer to harmonic and random excitation

Exact steady-state solutions are obtained for the motion of a multi-degree-of-freedom system (MDOF) that is provided, at an arbitrary location within the primary structure, with a highly-nonlinear auxiliary mass damper which resembles a conventional linear dynamic vibration neutralizer (DVN) whose relative motion with respect to the primary system is constrained to remain within a specified gap, thus operating as a “pounding DVN.” This configuration of a conventional DVN with motion-limiting stops could be quite useful when a primary structure with a linear DVN is subjected to transient loads (e.g., earthquakes) that may cause excessive relative motion between the auxiliary and primary systems. Under the assumption that the motion of the composite nonlinear MDOF system under harmonic excitation is undergoing steady-state motion with two symmetric impacts per period of the excitation, an exact, closed-form solution is obtained for the motion of the MDOF as well as the nonlinear damper. This solution is subsequently used to develop an approximate analytical procedure to estimate the stationary response of the pounding DVN when subjected to random excitation with white spectral density and Gaussian probability distribution. Since the vibration control effectiveness of auxiliary mass dampers, whether linear or not, deteriorates considerably when dealing with transient dynamic loads, an extensive investigation is provided for assessing the influence and interaction effects of the major nonlinear system parameters, so as to provide useful guidelines for quantifying the tradeoffs in selecting a suitable strategy to enhance the mitigation capabilities of the subject class of dampers, under stationary as well as nonstationary deterministic and random dynamic environments.

Binary Expression Enhances Reliability of Messaging in Gene Networks.

The promoter state of a gene and its expression levels are modulated by the amounts of transcription factors interacting with its regulatory regions. Hence, one may interpret a gene network as a communicating system in which the state of the promoter of a gene (the source) is communicated by the amounts of transcription factors that it expresses (the message) to modulate the state of the promoter and expression levels of another gene (the receptor). The reliability of the gene network dynamics can be quantified by Shannon’s entropy of the message and the mutual information between the message and the promoter state. Here we consider a stochastic model for a binary gene and use its exact steady state solutions to calculate the entropy and mutual information. We show that a slow switching promoter with long and equally standing ON and OFF states maximizes the mutual information and reduces entropy. That is a binary gene expression regime generating a high variance message governed by a bimodal probability distribution with peaks of the same height. Our results indicate that Shannon’s theory can be a powerful framework for understanding how bursty gene expression conciliates with the striking spatio-temporal precision exhibited in pattern formation of developing organisms.

Kinetic flux ropes: Bernstein–Greene–Kruskal modes for the Vlasov–Poisson–Ampère system

Electrostatic structures have been observed in many regions of space plasmas, including the solar wind, the magnetosphere, and the auroral acceleration region. One possible theoretical description of some of these structures is the concept of Bernstein–Greene–Kruskal (BGK) modes, which are exact nonlinear steady-state solutions of the Vlasov–Poisson system of equations in collisionless kinetic theory. We generalize the exact solutions of two-dimensional BGK modes in a magnetized plasma with a finite magnetic field strength [Ng et al., Phys. Plasmas 13, 055903 (2006)] to cases with azimuthal magnetic fields so that these structures carry the electric current as well as the steady electric and magnetic fields. Such nonlinear solutions now exactly satisfy the Vlasov–Poisson–Ampère system of equations. Explicit examples with either positive or negative electric potential structure are provided.

Phase transformations in metastable liquids combined with polymerization.

This paper is concerned with the theory of nucleation and growth of crystals in a metastable polymer melt with allowance for the polymerization of a monomer. A mathematical model consisting of the heat balance equation, equations governing the particle-radius distribution function and the polymerization degree is formulated. The exact steady-state analytical solutions are found. In the case of unsteady-state crystallization with polymerization, the particle-size distribution function is determined analytically for different space-time regions by means of the Laplace transform. Two functional integro-differential equations governing the dimensionless temperature and polymerization degree are deduced. These equations are solved by means of the saddle-point technique for the evaluation of a Laplace-type integral. The time-dependent distribution function, temperature and polymerization degree at different polymerization rates and nucleation kinetics are derived with allowance for the main contribution to the Laplace-type integral. In addition, the general analytical solution by means of the saddle-point technique and an example showing how to construct the analytical solutions in particular cases are given in the appendices. The analytical method developed in the present paper can be used to describe the similar phase transition phenomena in the presence of chemical reactions. This article is part of the theme issue 'Heterogeneous materials: metastable and non-ergodic internal structures'.

Exact steady-state solution to air pumping from underground partly covered by an impermeable tarp

A pipe is located in a semi-infinite two-dimensional porous medium partially covered by a tarp. A constant pressure gradient may be imposed at infinity on the medium while air is pumped into the pipe. The governing Laplace equation is solved by extending the famous Keldysh–Sedov formulae to doubly connected domains by reduction of the mixed problem to a Riemann–Hilbert problem with discontinuous coefficients; the solution is obtained by combining factorization and functional equations. Then, the influence of the various geometrical and flow parameters is presented and discussed.

Response of Pounding Dynamic Vibration Neutralizer Under Harmonic and Random Excitation

Exact steady-state solutions are obtained for the motion of an single-degree-of-freedom (SDOF) system that is provided with a highly nonlinear auxiliary mass damper (AMD), which resembles a conventional dynamic vibration neutralizer (DVN), whose relative motion with respect to the primary system is constrained to remain within a specified gap, thus operating as a “pounding DVN.” This configuration of a conventional DVN with motion-limiting stops could be quite useful when a primary structure with a linear DVN is subjected to transient loads (e.g., earthquakes) that may cause excessive relative motion between the auxiliary and primary systems. Under the assumption that the motion of the nonlinear system under harmonic excitation is undergoing steady-state motion with two impacts per period of the excitation, an exact, closed-form solution is obtained for the system motion. This solution is subsequently used to develop an approximate analytical solution for the stationary response of the pounding DVN when subjected to random excitation with white spectral density and Gaussian probability distribution. Comparison between the analytically estimated rms response of the primary system and its corresponding response obtained via numerical simulation shows that the analytical estimates are quite accurate when the coupling (tuning parameters) between the primary system and the damper are weak, but only moderately accurate when the linear components of the tuning parameters are optimized. It is also shown that under nonstationary, the pounding DVN provides slightly degraded performance compared to the linear one but simultaneously limits the damper-free motion to specified design constraints.

Synthesis of oscillators for exact desired periodic solutions having a finite number of harmonics

The synthesis of autonomous oscillators with exact desired periodic steady-state solution is described in this contribution. The vector field of the oscillator differential equation is built up with a conservative and a dissipative part. Both parts are synthesized using an algebraic function describing the desired limit cycle. The desired periodic motion is restricted by a finite numbers of harmonics, whereby the amplitude and the phase shift of every harmonic can be freely chosen, depending on the specific application. Afterwards the synthesis of a periodically driven oscillator with an exact desired periodic response is described. For this purpose, the differential equation of the autonomous oscillator is extended by a state-dependent compensation term that equals the excitation at the steady-state solution. Here the freely definable amplitudes and phase angles of the oscillator motion are restricted by the existence and stability conditions for synchronization.

Exact steady-state solutions of 3D toroidal flows and their stability

The purpose of the article is three-fold. The first is to get the Navier–Stokes equations in the toroidal coordinates by the methods of Riemannian geometry, which govern the motion of flows confined in a nested tori with inner torus rotating at a constant horizontal angular velocity. The second is to obtain the corresponding steady-state solutions by using the Implicit Function Theorem and iterative techniques. The last objective is to show the linear stability of the derived steady-states.

Exact steady states of the periodically forced and damped Duffing oscillator

We present an analytical methodology to compute the exact steady states of the strongly nonlinear Duffing oscillator with arbitrary viscous damping and under the action of a time-periodic excitation. In contrast to current analytical approaches that consider harmonic excitations to produce approximate steady-state solutions, the excitations considered in this work are multi-harmonic and are computed in closed form. We consider the exact fundamental resonance plot of this strongly nonlinear system and present an exact depiction of the dynamic balance between the internal and external forces leading to it in terms of rotating vectors in the complex plane with modulated amplitudes and modulated frequencies of rotation. Moreover, an infinity of exact steady-state solutions seems to be possible, each corresponding to a different time-periodic excitation. Generalization of the presented methodology to systems with other type of nonlinearities or with many degrees of freedom is presumably straightforward.

High Order Well-Balanced Weighted Compact Nonlinear Schemes for the Gas Dynamic Equations under Gravitational Fields

AbstractIn this study, we propose a high order well-balanced weighted compact nonlinear (WCN) scheme for the gas dynamic equations under gravitational fields. The proposed scheme is an extension of the high order WCN schemes developed in (S. Zhang, S. Jiang, C.-W Shu, J. Comput. Phys. 227 (2008) 7294-7321). For the purpose of maintaining the exact steady state solution, the well-balanced technique in (Y. Xing, C.-W Shu, J. Sci. Comput. 54 (2013) 645-662) is employed to split the source term into two terms. The proposed scheme can maintain the isothermal equilibrium solution exactly, genuine high order accuracy and resolve small perturbations of the hydrostatic balance state on the coarse meshes. Furthermore, in order to capture the strong discontinuities and large gradients, the fifth-order upwind weighted nonlinear interpolations together with the fourth/sixth order cell-centered compact schemes with local characteristic projections are used to construct different WCN schemes. Several representative one- and two-dimensional examples are simulated to demonstrate the good performance of the proposed schemes.