Degenerate Steady State Research Articles

Coupled activity-current fluctuations in open quantum systems under strong symmetries

Strong symmetries in open quantum systems lead to broken ergodicity and the emergence of multiple degenerate steady states. From a quantum jump (trajectory) perspective, the appearance of multiple steady states is related to underlying dynamical phase transitions (DPTs) at the fluctuating level, leading to a dynamical coexistence of different transport channels classified by symmetry. In this paper we investigate how strong symmetries affect both the transport properties and the activity patterns of a particular class of Markovian open quantum system, a three-qubit model under the action of a magnetic field and in contact with a thermal bath. We find a pair of twin DPTs in exciton current statistics, induced by the strong symmetry and related by time reversibility, where a zero-current exchange-antisymmetric phase coexists with a symmetric phase of negative exciton current. On the other hand, the activity statistics exhibits a single DPT where the symmetric and antisymmetric phases of different but nonzero activities dynamically coexists. Interestingly, the maximum current and maximum activity phases do not coincide for this three-qubits system. We also investigate how symmetries are reflected in the joint large deviation statistics of the activity and the current, a central issue in the characterization of the complex quantum jump dynamics. The presence of a strong symmetry under nonequilibrium conditions implies non-analyticities in the dynamical free energy in the dual activity-current plane (or equivalently in the joint activity-current large deviation function), including an activity-driven current lockdown phase for activities below some critical threshold. Remarkably, the DPT predicted around the steady state and its Gallavotti–Cohen twin dual are extended into lines of first-order DPTs in the current-activity plane, with a nontrivial structure which depends on the transport and activity properties of each of the symmetry phases. Finally, we also study the effect of a symmetry-breaking, ergodicity-restoring dephasing channel on the coupled activity-current statistics for this model. Interestingly, we observe that while this dephasing noise destroys the symmetry-induced DPTs, the underlying topological symmetry leaves a dynamical fingerprint in the form of an intermittent, bursty on/off dynamics between the different symmetry sectors.

The Multiplicity and Types of Steady States for a Continuous Stirred-Tank Reactor: a Liquid–Liquid Heterogeneous System

The types of the steady states for a continuous stirred-tank reactor are studied depending on the process parameters, in particular, a parameter not affecting the position of the steady states, by way of example of a liquid–liquid heterogeneous system, in which an exothermic reaction proceeds. It is shown that the change in the coarse steady states occurs through degenerate steady states.

Steady state types for CSTR. Heterogeneous system liquid - liquid

For a heterogeneous liquid - liquid system in which an exothermal reaction takes place, the topology and stability of the steady states of CSTR are investigated. It is shown that the thermal effect of the reaction, the magnitude of the energy barrier and the intensity of mass transfer strongly influence the type of steady state that arises. In this case, a change in the topological picture occurs through degenerate steady states.

Types of Steady States of a Continuous Stirred-Tank Reactor: Liquid–Liquid Heterogeneous System

The topology and stability of steady states of a continuous stirred-tank reactor were studied for a liquid–liquid heterogeneous system in which an exothermic reaction occurs. It was shown that the heat of the reaction, its energy barrier, and the mass transfer rate strongly affect the type of the structure of the emerging steady state, and the topological pattern changes through a series of degenerate steady states.

Transitions of Spherical Thermohaline Circulation to Multiple Equilibria

The main aim of the paper is to investigate the transitions of the thermohaline circulation in a spherical shell in a parameter regime which only allows transitions to multiple equilibria. We find that the first transition is either continuous (Type-I) or drastic (Type-II) depending on the sign of the transition number. The transition number depends on the system parameters and $l_c$, which is the common degree of spherical harmonics of the first critical eigenmodes, and it can be written as a sum of terms describing the nonlinear interactions of various modes with the critical modes. We obtain the exact formulas of this transition number for $l_c=1$ and $l_c=2$ cases. Numerically, we find that the main contribution to the transition number is due to nonlinear interactions with modes having zero wave number and the contribution from the nonlinear interactions with higher frequency modes is negligible. In our numerical experiments we encountered both types of transition for $Le<1$ but only continuous transition for $Le>1$. In the continuous transition scenario, we rigorously prove that an attractor in the phase space bifurcates which is homeomorphic to the 2$l_c$ dimensional sphere and consists entirely of degenerate steady state solutions.

Preclusion of switch behavior in networks with mass-action kinetics

We study networks taken with mass-action kinetics and provide a Jacobian criterion that applies to an arbitrary network to preclude the existence of multiple positive steady states within any stoichiometric class for any choice of rate constants. We are concerned with the characterization of injective networks, that is, networks for which the species formation rate function is injective in the interior of the positive orthant within each stoichiometric class. We show that a network is injective if and only if the determinant of the Jacobian of a certain function does not vanish. The function consists of components of the species formation rate function and a maximal set of independent conservation laws. The determinant of the function is a polynomial in the species concentrations and the rate constants (linear in the latter) and its coefficients are fully determined. The criterion also precludes the existence of degenerate steady states. Further, we relate injectivity of a network to that of the network obtained by adding outflow, or degradation, reactions for all species.

On bifurcation of Prion disease: A Chemical Reaction Network approach

Fatal neurodegenerative diseases such as bovine spongiform encephalopathy in cattle, scrapie in sheep, and Creutzfeldt-Jakob disease in humans are caused by prions. A prion is a protein encoded by a normal cellular gene. The cellular form of the prion, namely PrPC, is benign but can be reconfigured into a disease-causing form (named scrapie), PrPSc, by a conformational change from α—helix to β—sheets. Prions replicate by this conformational change; that is, PrPSc interacts with PrPC producing a new molecule of PrPSc. This kind of replication is modeled in this contribution as an autocatalytic Chemical Reaction Network (CRN). The Mass Action Kinetic (MAK) ODEs support three distinct dynamical scenarios: multiple steady states, sustained oscillations and a degenerate steady state. We perform equilibria analysis of each dynamical scenario.

A General Model for Binary Cell Fate Decision Gene Circuits with Degeneracy: Indeterminacy and Switch Behavior in the Absence of Cooperativity

BackgroundThe gene regulatory circuit motif in which two opposing fate-determining transcription factors inhibit each other but activate themselves has been used in mathematical models of binary cell fate decisions in multipotent stem or progenitor cells. This simple circuit can generate multistability and explains the symmetric “poised” precursor state in which both factors are present in the cell at equal amounts as well as the resolution of this indeterminate state as the cell commits to either cell fate characterized by an asymmetric expression pattern of the two factors. This establishes the two alternative stable attractors that represent the two fate options. It has been debated whether cooperativity of molecular interactions is necessary to produce such multistability.Principal FindingsHere we take a general modeling approach and argue that this question is not relevant. We show that non-linearity can arise in two distinct models in which no explicit interaction between the two factors is assumed and that distinct chemical reaction kinetic formalisms can lead to the same (generic) dynamical system form. Moreover, we describe a novel type of bifurcation that produces a degenerate steady state that can explain the metastable state of indeterminacy prior to cell fate decision-making and is consistent with biological observations.ConclusionThe general model presented here thus offers a novel principle for linking regulatory circuits with the state of indeterminacy characteristic of multipotent (stem) cells.

On the bifurcation analysis of a food web of four species

This paper is concerned with bifurcations of equilibria and the chaotic dynamics of a food web containing a bottom prey X, two competing predators Y and Z on X, and a super-predator W only on Y. Conditions for the existence of all equilibria and the stability properties of most equilibria are derived. A two-dimensional bifurcation diagram with the aid of a numerical method for identifying bifurcation curves is constructed to show the bifurcations of equilibria. We prove that the dynamical system possesses a line segment of degenerate steady states for the parameter values on a bifurcation line in the bifurcation diagram. Numerical simulations show that these degenerate steady states can help to switch the stabilities between two far away equilibria when the system crosses this bifurcation line. Some observations concerned with chaotic dynamics are also made via numerical simulations. Different routes to chaos are found in the system. Relevant calculations of Lyapunov exponents and power spectra are included to support the chaotic properties.

Bifurcation analysis for spherically symmetric systems using invariant theory

We study a degenerate steady state bifurcation problem with spherical symmetry. This singularity, with the five dimensional irreducible action ofO(3), has been studied by several authors for codimensions up to 2. We look at the case where the topological codimension is 3, theC∞-codimension is 5. We find a tertiary Hopf bifurcation and a heteroclinic orbit. Our analysis does not use any specific properties of the five dimensional representation and can in principle be used for higher representations as well. The computations are based on invariant theory and orbit space reduction.

Reaction-Diffusion Dissipative Systems—Detailed Stability Analysis-Pattern of Growth and Effect of Inhomogenity

A detailed stability analysis of the one dimensional steady state solutions for the Brusselator model under the conditions of diffusion of initial (non-autocatalytic) components has been performed both for zero flux as well as fixed boundary conditions. In addition to subcritical as well as supercritical bifurcations, situations have been observed where all solution branches at a bifurcation point are unstable. A case of degenerate steady state bifurcation (2 solutions emanating from the same bifurcation point) has also been noticed. A transient simulation of the system in growth reveals the importance of growth rate on the pattern selection process and suggests that the selection of branches at a bifurcation point may be influenced by perturbations/ fluctuations. It also indicates that a stability analysis of the bifurcation diagram alone cannot decide the state of the system in a transient process, and under certain situations complex behavior may be observed at limit points. Numerical calculations on coupled cells indicate that a heterogenity in the system can introduce multiple (two) time scales in the system. As the ratio of time scales increases, aperiodic or irregular oscillations are observed for the 'fast' variable. A combination of cells with one cell in a steady-state mode and the other in a periodic motion results in a combined motion of the entire system. For a distributed parameter system, a heterogenity can cause development of sharp local concentration gradients, alter the stability properties of steady state as well as periodic solutions and can cause partitioning of the system.

Transverse Instabilities in Solid State Plasmas under Steady Electric and Magnetic Fields

Effects of a dc electric field on the behavior of circularly polarized, low frequency waves in infinite plasmas under a dc magnetic field in which densities of mobile carriers of different sign of charge are not equal are investigated. The wave shows spatial growth and instability. Collisions are beneficial to these. According to Sturrock's criterion, the spatially growing waves can not be used for amplification and the instability leads to spontaneous oscillation. The dielectric constant tensor is calculated for a quite general case of arbitrary directions of wave propagation and dc electric field by the hydrodynamic approximation and for the case of parallel electric and magnetic fields but arbitrary direction of propagation by the kinetic theory using a completely degenerate steady state distribution and a simple collision term. By using the dielectric constant tensor thus obtained, it is found that in Bi-like material at lower temperatures, the inclined field enhances instability especially of inclined waves and the hydrodynamic approximation is a fairly good approximation.