Multiple Stationary States Research Articles

Block approximations for probabilistic mixtures of elementary cellular automata

Probabilistic Cellular Automata are a generalization of Cellular Automata. Despite their simple definition, they exhibit fascinating and complex behaviours. The stationary behaviour of these models changes when model parameters are varied, making the study of their phase diagrams particularly interesting. The block approximation method, also known in this context as the local structure approach, is a powerful tool for studying the main features of these diagrams, improving upon Mean Field results. This work considers systems with multiple stationary states, aiming to understand how their interactions give rise to the structure of the phase diagram. Additionally, it shows how a simple algorithmic implementation of the block approximation allows for the effective study of the phase diagram even in the presence of several absorbing states.

Symmetry breaking and non-ergodicity in a driven-dissipative ensemble of multilevel atoms in a cavity

Dissipative light-matter systems can display emergent collective behavior. Here, we report a Z2-symmetry-breaking phase transition in a system of multilevel Rb87 atoms strongly coupled to a weakly driven two-mode optical cavity. In the symmetry-broken phase, nonergodic dynamics manifests in the emergence of multiple stationary states with disjoint basins of attraction. This feature enables the amplification of a small atomic population imbalance into a characteristic macroscopic cavity transmission signal. Our experiment does not only showcase strongly dissipative atom-cavity systems as platforms for probing nontrivial collective many-body phenomena, but also highlights their potential for hosting technological applications in the context of sensing, density classification, and pattern retrieval dynamics within associative memories. Published by the American Physical Society 2024

Multiple Stationary States of Chemical Reactions in a Closed System

Multiple Stationary States of Chemical Reactions in a Closed System

A computational model of self-organized shape dynamics of active surfaces in fluids

Mechanochemical processes on surfaces such as the cellular cortex or epithelial sheets, play a key role in determining patterns and shape changes of biological systems. To understand the complex interplay of hydrodynamics and material flows on such active surfaces requires novel numerical tools. Here, we present a finite-element method for an active deformable surface interacting with the surrounding fluids. The underlying model couples surface and bulk hydrodynamics to surface flow of a diffusible species which generates active contractile forces. The method is validated with previous results based on linear stability analysis and shows almost perfect agreement regarding predicted patterning. Away from the linear regime we find rich non-linear behavior, such as the presence of multiple stationary states. We study the formation of a contractile ring on the surface and the corresponding shape changes. Finally, we explore mechanochemical pattern formation on various surface geometries and find that patterning strongly adapts to local surface curvature. The developed method provides a basis to analyze a variety of systems that involve mechanochemical pattern formation on active surfaces interacting with surrounding fluids.

Well-posedness and stationary solutions of McKean-Vlasov (S)PDEs

This paper is composed of two parts. In the first part we consider McKean-Vlasov Partial Differential Equations (PDEs), obtained as thermodynamic limits of interacting particle systems (i.e. in the limit N→∞, where N is the number of particles). It is well-known that, even when the particle system has a unique invariant measure (stationary solution), the limiting PDE very often displays a phase transition: for certain choices of (coefficients and) parameter values, the PDE has a unique stationary solution, but as the value of the parameter varies multiple stationary states appear. In the first part of this paper, we add to this stream of literature and consider a specific instance of a McKean-Vlasov type equation, namely the Kuramoto model on the torus perturbed by a symmetric double-well potential, and show that this PDE undergoes the type of phase transition just described, as the diffusion coefficient is varied. In the second part of the paper, we consider a rather general class of McKean-Vlasov PDEs on the torus (which includes both the original Kuramoto model and the Kuramoto model in double well potential of part one) perturbed by (strong enough) infinite-dimensional additive noise. To the best of our knowledge, the resulting Stochastic PDE, which we refer to as the Stochastic McKean-Vlasov equation, has not been studied before, so we first study its well-posedness. We then show that the addition of noise to the PDE has the effect of restoring uniqueness of the stationary state in the sense that, irrespective of the choice of coefficients and parameter values in the McKean-Vlasov PDE, the Stochastic McKean-Vlasov PDE always admits at most one invariant measure.

Non-mean-field Vicsek-type models for collective behavior

We consider interacting particle dynamics with Vicsek-type interactions, and their macroscopic Partial Differential Equation (PDE) limit, in the non-mean-field regime; that is, we consider the case in which each particle/agent in the system interacts only with a prescribed subset of the particles in the system (for example, those within a certain distance). In this non-mean-field regime the influence between agents (i.e. the interaction term) can be normalized either by the total number of agents in the system (global scaling) or by the number of agents with which the particle is effectively interacting (local scaling). We compare the behavior of the globally scaled and the locally scaled systems in many respects, considering for each scaling both the PDE and the corresponding particle model. In particular, we observe that both the locally and globally scaled particle system exhibit pattern formation (i.e. formation of traveling-wave-like solutions) within certain parameter regimes, and generally display similar dynamics. The same is not true of the corresponding PDE models. Indeed, while both PDE models have multiple stationary states, for the globally scaled PDE such (space-homogeneous) equilibria are unstable for certain parameter regimes, with the instability leading to traveling wave solutions, while they are always stable for the locally scaled one, which never produces traveling waves. This observation is based on a careful numerical study of the model, supported by further analysis.

Analysis of divergent bifurcations in the dynamics of wheeled vehicles

Abstract. This paper presents the bifurcation approach to analyze divergent loss of stability of the symmetric solution of a nonlinear dynamic model in Lyapunov's critical case of a single zero root. Under such a condition, material birth-annihilation bifurcations of multiple stationary states take place. Moreover, the equilibrium surface of stationary states in a small neighborhood of the symmetric solution is a generalized Whitney fold. In the simplest case of a fold peculiarity, the corresponding bifurcation manifold locally coincides with the discriminant manifold of a third-degree polynomial that determines the manifold of stationary states in a small neighborhood of the symmetric solution. An algorithm to construct the corresponding polynomial is introduced. Through the algorithm, the bifurcation manifold is determined, and the conditions for safe/unsafe loss of stability of the symmetric solution are derived analytically. The proposed approach to analyze divergent loss of stability is implemented for a nonlinear bicycle model of a two-axle wheeled vehicle. It represents a further development of Pevzner–Pacejka's well-known graph-analytical method. The paper determines the critical values of constructive parameters that are responsible for safe/unsafe loss of stability of the vehicle's straight-line motion, and it discusses strategies for the bifurcation approach to analyze divergent loss of stability.

First-principles prediction of multiple stationary states in glass-forming liquids.

In the present work, the Non-Equilibrium Self-Consistent Generalized Langevin Equation (NESCGLE) theory is used to predict the final state of glass-forming liquids subjected to different cooling processes. We show that the NESCGLE theory correctly describes two essential features of the glass transition. Such features are the structural recovery and the dependence of the final state with the cooling rate. We demonstrate that below a particular temperature Tc, the system is unable to equilibrate, independently of the cooling rate. We show that the equilibrium state is only reached for the quasistatic process. Additionally, we show how, from the NESCGLE theory, it is possible to deduce a relaxation model of structural recovery, for which we obtain molecular expressions of the parameters.

On Oscillations and Noise in Multicomponent Adsorption: The Nature of Multiple Stationary States

Starting from the fact that monocomponent adsorption, whether modeled by Lagergren or nonlinear Riccati equation, does not sustain oscillations, we speculate about the nature of multiple steady state states in multicomponent adsorption with second-order kinetics and about the possibility that multicomponent adsorption might exhibit oscillating behavior, in order to provide a tool for better discerning possible oscillations from inevitable fluctuations in experimental results or a tool for a better control of adsorption process far from equilibrium. We perform an analysis of stability of binary adsorption with second-order kinetics in multiple ways. We address perturbations around the steady state analytically, first in a classical way, then by introducing Langevin forces and analyzing the reaction flux and cross-correlations, then by applying the stochastic chemical master equation approach, and finally, numerically, by using stochastic simulation algorithms. Our results show that stationary states in this model are stable nodes. Hence, experimental results with purported oscillations in response should be addressed from the point of view of fluctuations and noise analysis.

A non-linear kinetic model of self-propelled particles with multiple equilibria

We introduce and analyse a continuum model for an interacting particle system of Vicsek type. The model is given by a non-linear kinetic partial differential equation (PDE) describing the time-evolution of the density $f_t$, in the single particle phase-space, of a collection of interacting particles confined to move on the one-dimensional torus. The corresponding stochastic differential equation for the position and velocity of the particles is a conditional McKean-Vlasov type of evolution (conditional in the sense that the process depends on its own law through its own conditional expectation). In this paper, we study existence and uniqueness of the solution of the PDE in consideration. Challenges arise from the fact that the PDE is neither elliptic (the linear part is only {\em hypoelliptic}) nor in gradient form. Moreover, for some specific choices of the interaction function and for the simplified case in which the density profile does not depend on the spatial variable, we show that the model exhibits multiple stationary states (corresponding to the particles forming a coordinated clockwise/anticlockwise rotational motion) and we study convergence to such states as well. Finally, we prove mean-field convergence of an appropriate $N$-particles system to the solution of our PDE: more precisely, we show that the empirical measures of such a particle system converge weakly, as $N \rightarrow \infty$, to the solution of the PDE.

Zeroth law of thermodynamics for thermalized open quantum systems having constants of motion

We study the evolution of an open quantum system described by a dynamical semigroup having the Lindblad superoperator as a generator. This generator may have an eigenfunction with a zero eigenvalue referred to as a constant of motion (COM). An open quantum system has a unique stationary state if and only if it has no COMs. A system with multiple stationary states has a basis of COMs, any COM of the system is a linear combination of the basis COMs. The basis divides the space of system states into subspaces. In each subspace, its own stationary state is formed, and any stationary state of the system is a linear combination of these states. Usually, neither the basis of COMs nor even their number is known. We demonstrate that finding the stationary state of the system does not require looking for COMs. Instead, one can construct a set of "invariant" subspaces. If the system evolution begins from one of these subspaces, the system will remain in it, arriving at a stationary state independent of evolution in other subspaces. We suggest a direct way of finding the invariant subspaces by studying the evolution of the system. We show that the sets of invariant subspaces and subspaces generated by the basis of COMs are equivalent. A stationary state of the system is a weighted sum of stationary states in each invariant subspace; weighting factors are determined by the initial state of the system.

Time delays, population, and economic development.

This research develops an augmented Solow model with population dynamics and time delays. The model produces either a single stationary state or multiple stationary states (able to characterise different development regimes). The existence of time delays may cause persistent fluctuations in both economic and demographic variables. In addition, the work identifies in a simple way the reasons why economics affects demographics and vice versa.

The epidemiological models of Karl-Peter Hadeler.

The most frequently cited articles out of KP Hadeler's 45 papers with epidemiological applications are summarized. Parasitic diseases which increase the death rate of the hosts proportional to the integer number of parasites present were described by integral equations for the generating function of the age- and time-dependent number of parasites. A model was derived for a population structured by the continuous level of parasitic infection.Stimulated by the spread of AIDS a new class of epidemic models was developed which take into account explicitly the formation and separation of pairs. For predator-prey populations with parasitic infections threshold conditions for the persistence of the predator were derived. The interaction of epidemics and demography was analysed. Several epidemiological conditions led to backward bifurcations associated with multiple infective stationary states.

Multiple stationary filamentary states in a planar dc-driven gas discharge-semiconductor system

Formation of stationary current filaments is studied in a planar layered structure of gas discharge coupled to a high-Ohmic semiconductor and driven by a stationary voltage (the gas discharge-semiconductor system, GDSS). The simulations are carried out in 2D. The model is based on the diffusion-drift theory of gas discharges. Multiple steady-state solutions in GDSS are found for the first time. The computed discharge parameters are found to be in reasonable agreement with the experiment. Results support the evidence that the thermal mechanism is responsible for the instability in GDSS, under considered conditions. This is also supported by the linear stability analysis of the homogeneous stationary state: with heating of the gas excluded, the instability does not form a spatial structure, it is homogeneous and purely growing without oscillations. Including the gas heating into the model changes the dispersion relation drastically.

Efficient tracing and stability analysis of multiple stationary and periodic states with exploitation of commercial CFD software

A computational approach is presented that enables commercial Computational Fluid Dynamics (CFD) codes to detect Hopf bifurcations and with necessary adjustments, compute the frequency and amplitude of the periodic orbits that arise from the Hopf point. The proposed computational framework, which combines a homemade Matlab code with Ansys/Fluent, is an extension of a previously presented methodology for the efficient tracing of solution branches that contain turning points. The need for the special attention to periodic orbits springs from recently published results that indicate that time-periodic states occur in industrial-scale Chemical Vapor Deposition (CVD) reactors. Nevertheless the method presented here is not limited to deposition processes; in fact it treats the CFD process model as a “black box” and requires no alteration of the commercial software. To prove the effectiveness of the computational framework, it is implemented here on the benchmark case of laminar flow around a cylinder, where Hopf bifurcations have been identified via eigenvalue analysis. Once the method is validated, it is implemented on a rotating-disk commercial CVD reactor model in the region of parameter space where Hopf points are observed. In both the benchmark and the industrial-scale CVD cases, stable and unstable, stationary and periodic states are computed for the same parameter values.

Gene regulatory networks and epigenetic modifications in cell differentiation

It is becoming increasingly clear that the functionalities of an organism are mostly derived from regulation of its gene repertoire. Specialized cell types are created from pluripotent stem cells by regulating expression of genes. In eukaryotes, genes are primarily regulated by gene regulatory networks consisting of highly sequence-specific transcription factors and epigenetic modifications. The former mode of regulation is more readily reversible and non-heritable across cell generations, whereas the latter mode is less reversible and heritable. In this article, we explore the relationship between cell differentiation and the two modes of regulation of gene expression, focusing primarily on pluripotent and multipotent stem cells. Recent studies suggest that stem cells execute different gene expression programs, probably driven by one or more gene regulatory network(s). It is now also evident that as stem cells differentiate to more specialized progeny cells, rewriting of epigenetic marks occurs in parallel with the change in the pattern of gene expression. A conceptual framework is put forward in which it is proposed that the cell fate determining gene regulatory network in a pluripotent or multipotent cell has the capability to exist in multiple stationary states with each stationary state dictating a particular pattern of gene expression. We also propose that the broad pattern of gene expression in each stationary state, termed the lineage biased state or LIBS, resembles that of a more differentiated progeny cell. The differentiation process leading to a particular progeny cell involves rewriting of epigenetic marks that result in upregulation of genes in a LIBS and silencing of genes involved in alternative LIBS; thus selecting a particular pattern of gene expression and making a lineage commitment.

Nonsmooth Dynamic Behaviors Inherited from an Ecohydrological Model: Mutation, Bifurcation, and Chaos

The existence of nontrivial dynamic behaviors in a hydrological system is intensively discussed in the literature. However, most of the work has been done from the nonlinear data analysis perspective, with only a few exceptions, due to the mathematical difficulties for theoretical analysis. In this study, a simple but comprehensive enough ecohydrological model with the pulsed atmospheric forcing was developed from the process analysis perspective. The model was then utilized to analyze the non-trivial dynamic behaviors in a coupled ecohydrological system qualitatively and numerically. Our results confirm the existence of multiple stationary states discussed by many researchers. Furthermore, parameter bifurcation was studied and the phenomenon of mutation is found to be rather common. Also, the chaotic characteristic of the system state is obtained under some specific parameters. Parts of these behaviors were seldom reported through the deterministic dynamic analysis done previously.

A multi-state single-molecule switch actuated by rotation of an encapsulated cluster within a fullerene cage

We demonstrate a single-molecule switch based on tunneling electron-driven rotation of a triangular Sc3N cluster within an icosahedral C80 fullerene cage among three pairs of enantiomorphic configurations. Scanning tunneling microscopy imaging of switching within single molecules and electronic structure theory identify the conformational isomers and their isomerization pathways. Bias-dependent action spectra and modeling identify the antisymmetric stretch vibration of Sc3N cluster to be the gateway for energy transfer from the tunneling electrons to the cluster rotation. Hierarchical switching of conductivity through the internal cluster motion among multiple stationary states while maintaining a constant shape, is advantageous for the integration of endohedral fullerene-based single-molecule memory and logic devices into parallel molecular computing architectures.

Semi-analytical solutions for cubic autocatalytic reaction-diffusion equations; the effect of a precursor chemical

Semi-analytical solutions for a cubic autocatalytic reaction, with linear decay and a precursor chemical, are considered. The model is coupled with diffusion and considered in a one-dimensional reactor. In this model the reactant is supplied by two mechanisms, diffusion via the cell boundaries and decay of an abundant precursor chemical present in the reactor. The Galerkin method is used to approximate the spatial structure of the reactant and autocatalyst concentrations in the reactor. Ordinary differential equations are then obtained as an approximation to the governing partial differential equations and analyzed to obtain semi-analytical results for the reaction-diffusion cell. Singularity theory and a local stability analysis are used to determine the regions of parameter space in which the different types of bifurcation diagrams and Hopf bifurcations occur. The effect of the precursor chemical concentration is examined in detail and some novel behaviours are identified. References P. Gray and S. K. Scott. Chemical oscillations and instabilities: nonlinear chemical kinetics. Oxford University Press, London, 1990. P. Gray and S. K. Scott. Autocatalytic reactions in the isothermal continuous, stirred tank reactor: isolas and other forms of multistability. Chem. Engng. Sci., 38:29--43, 1983. doi:10.1016/0009-2509(83)80132-8 P. Gray and S. K. Scott. Autocatalytic reactions in the isothermal continuous, stirred tank reactor: oscillations and instabilities in the system $a + 2b \to 3b$; $b \to c$. Chem. Engng. Sci., 39:1087--1097, 1984. doi:10.1016/0009-2509(84)87017-7 S. R. Kay, S. K. Scott, and P. G. Lignola. The application of singularity theory to isothermal autocatalytic reactions: the influence of uncatalysed reactions. Proc. Roy. Soc. Lond A, 409:433--448, 1987. doi:10.1098/rspa.1986.0015 S. K. Scott. Isolas, mushrooms and oscillations in isothermal, autocatalytic reaction-diffusion equations. Chem. Engng. Sci., 42:307--315, 1987. doi:10.1016/0009-2509(87)85060-1 S. R. Kay and S. K. Scott. Multiple stationary states, sustained oscillations and transient behavior in autocatalytic reaction-diffusion equations. Proc. Roy. Soc. Lond A, 418:345--364, 1988. doi:10.1098/rspa.1988.0088 J. H. Merkin, D. J. Needham, and S. K. Scott. On the creation, growth and extinction of oscillatory solutions for a simple pooled chemical reaction scheme. SIAM Journal on Applied Mathematics, 47:1040--1060, 1987. http://www.jstor.org/stable/2101706 T. R. Marchant. Cubic autocatalytic reaction-diffusion equations: semi-analytical solutions. Proc. Roy. Soc. Lond A, 458:873--888, 2002. doi:10.1098/rspa.2001.0899 V. Balakotaiah and D. Luss. Multiplicity features of reacting systems. Chem. Engng. Sci., 38:1709--1721, 1983. doi:10.1016/0009-2509(83)85028-3 J. Guckenheimer and P. Holmes. Nonlinear oscillations, dynamical systems, and bifurcations of vector fields. Springer-Verlag, New York, 1983. M. Golubitsky and D. G. Schaeffer. Singularites and groups in bifurcation theory. Springer-Verlag, New York, 1985.

Space charge effects on externally injected current in planar diodes: Existence of multiple stationary states

We study the effects of space charge on the properties of a system with a specified steady current. The problem is solved exactly for a planar one dimensional geometry, but we expect similar results for more realistic systems. We find that in many cases, the stationary current density cannot exceed some fixed value Jmax and in other cases be lower than a fixed Jmin depending on the method of injection. There are also intervals of intermediate current densities, which cannot exist in the stationary conditions. In general, there are values of J in some range where there are two different stationary regimes, which correspond to two different electric fields at the emitter surface. Connection with results obtained from numerical simulations and possible applications are also considered.