Time-scale Separation Research Articles

A two-timescale model of plankton–oxygen dynamics predicts formation of oxygen minimum zones and global anoxia

Decline of the dissolved oxygen in the ocean is a growing concern, as it may eventually lead to global anoxia, an elevated mortality of marine fauna and even a mass extinction. Deoxygenation of the ocean often results in the formation of oxygen minimum zones (OMZ): large domains where the abundance of oxygen is much lower than that in the surrounding ocean environment. Factors and processes resulting in the OMZ formation remain controversial. We consider a conceptual model of coupled plankton–oxygen dynamics that, apart from the plankton growth and the oxygen production by phytoplankton, also accounts for the difference in the timescales for phyto- and zooplankton (making it a “slow-fast system”) and for the implicit effect of upper trophic levels resulting in density dependent (nonlinear) zooplankton mortality. The model is investigated using a combination of analytical techniques and numerical simulations. The slow-fast system is decomposed into its slow and fast subsystems. The critical manifold of the slow-fast system and its stability is then studied by analyzing the bifurcation structure of the fast subsystem. We obtain the canard cycles of the slow-fast system for a range of parameter values. However, the system does not allow for persistent relaxation oscillations; instead, the blowup of the canard cycle results in plankton extinction and oxygen depletion. For the spatially explicit model, the earlier works in this direction did not take into account the density dependent mortality rate of the zooplankton, and thus could exhibit Turing pattern. However, the inclusion of the density dependent mortality into the system can lead to stationary Turing patterns. The dynamics of the system is then studied near the Turing bifurcation threshold. We further consider the effect of the self-movement of the zooplankton along with the turbulent mixing. We show that an initial non-uniform perturbation can lead to the formation of an OMZ, which then grows in size and spreads over space. For a sufficiently large timescale separation, the spread of the OMZ can result in global anoxia.

Unsupervised detection of large-scale weather patterns in the northern hemisphere via Markov State Modelling: from blockings to teleconnections

Detecting recurrent weather patterns and understanding the transitions between such regimes are key to advancing our knowledge of the low-frequency variability of the atmosphere and have important implications in terms of weather and climate-related risks. We adopt an analysis pipeline inspired by Markov State Modelling and detect in an unsupervised manner the dominant winter mid-latitude Northern Hemisphere weather patterns in the Atlantic and Pacific sectors. The daily 500 hPa geopotential height fields are first classified in about 200 microstates. The weather dynamics are then represented on the basis of these microstates and the slowest decaying modes are identified from the spectral properties of the transition probability matrix. These modes are defined on the basis of the nonlinear dynamical processes of the system and not as tentative metastable states, as often done in Markov state analysis. When focusing on a shifting longitudinal window of 60∘, we find that the longitude-dependent estimate of the longest relaxation time is smaller where stronger baroclinic activity is found. In the Atlantic and Pacific sectors slow relaxation processes are mainly related to transitions between blocked regimes and zonal flow. We also find strong evidence of a dynamical regime associated with the simultaneous Atlantic-Pacific blocking. When the analysis is performed on a broader geographical region of the Atlantic sector, we discover that the slowest relaxation modes of the system are associated with transitions between dynamical regimes that resemble teleconnection patterns like the North Atlantic Oscillation and weather regimes like the Scandinavian and Greenland blocking, yet have a much stronger dynamical foundation than classical methods based e.g. on EOF analysis. Our method clarifies that, as a result of the lack of a time-scale separation in the atmospheric variability of the mid-latitudes, there is no clear-cut way to represent the atmospheric dynamics in terms of few, well-defined modes of variability. The approach proposed here can be seamlessly applied across different regions of the globe for detecting regional modes of variability, and has a great potential for intercomparing climate models and for assessing the impact of climate change on the low-frequency variability of the atmosphere.

The Michaelis–Menten Reaction at Low Substrate Concentrations: Pseudo-First-Order Kinetics and Conditions for Timescale Separation

We demonstrate that the Michaelis–Menten reaction mechanism can be accurately approximated by a linear system when the initial substrate concentration is low. This leads to pseudo-first-order kinetics, simplifying mathematical calculations and experimental analysis. Our proof utilizes a monotonicity property of the system and Kamke’s comparison theorem. This linear approximation yields a closed-form solution, enabling accurate modeling and estimation of reaction rate constants even without timescale separation. Building on prior work, we establish that the sufficient condition for the validity of this approximation is s0≪K\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$s_0 \\ll K$$\\end{document}, where K=k2/k1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$K=k_2/k_1$$\\end{document} is the Van Slyke–Cullen constant. This condition is independent of the initial enzyme concentration. Further, we investigate timescale separation within the linear system, identifying necessary and sufficient conditions and deriving the corresponding reduced one-dimensional equations.

A Bis-Porphyrin Cavitand Breathing-In to Constrict Bucky Balls.

Bis-porphyrin cages have long been exploited to bind fullerenes selectively for various applications. The major consideration for an effective binding here had been the cavity size. Herein, we structurally demonstrate that a bis-Ni-porphyrin cavitand having even a smaller cavity can host a larger fullerene by a breathing and ruffling mechanism. It has also been shown that both the electronic and steric influence at the meso- positions of the porphyrin in fact dictate the binding character. The smaller cavity of 2NiD exhibits preferential binding for C70 over C60; however, surprisingly, the larger cavities in 2HD and 2NiTD display stronger affinities for C60 over the larger fullerene. We show here that the structural elasticity infused both by the metalloporphyrins and the connecting bridges play a major role in directing the binding. These conclusions have adequately been supported by structural and spectroscopic investigations. Additionally, the suitability of one of the conjugates for photoinduced charge-separation has been investigated using ultrafast transient absorption measurements. 2NiD has a charge separation timescale of ~0.8 ps, while charge recombination occurs at a longer timescale of ~920 ps.

Impulse generated from detonation waves in non-premixed and partially premixed reactants

Rotating detonation engines (RDEs) are the subject of research in the combustion community due to the prospects of enhanced thermal efficiency and power density when compared to current deflagrative-based aerospace combustors. Many current simulations presume premixed reactants and therefore miss the important characteristics of transient mixing, wave-induced mixing, and injector design/spacing that are known to play a pivotal role in the system performance. Very ambitious large eddy simulations are being conducted, but necessarily on a limited number of realistic and complex cases, thus limiting their utility in deriving fundamental understanding. For these reasons, a two dimensional parametric study was conducted to assess propagation of a detonation across an idealized array of mixing/injection sites, parametrically characterized by the width and axial mixing profile.Under such non-premixed conditions, discrete energy release and interdependence are observed. The discrete energy release sites frequently create pressures that exceed idealized Chapman-Jouguet (CJ) predictions based on perfect and uniform mixtures. The local higher pressure is shown to be caused by delayed heat release behind the shock, near constant pressure combustion, and additional compression due to the non-uniformities present. The resulting compression and the near constant pressure combustion are accompanied by time scale separation of exothermic and endothermic reactions due to the mixing efficiencies in the non/poorly-premixed cases. In contrast, the better mixed cases show that the detonation wave is sustained but unburnt fuel and oxidizer exist behind the main combustion wave and impulse performance suffers. Results show that for the conditions modeled there exists an optimal injector spacing to maximize the impulse produced and that discrete injection impulses can exceed that of the premixed systems. These somewhat counter-intuitive results imply that the detailed mixing evolution, provoked by the passage of the wave can lead to an extended heat release zone that elevates the pressure over a longer distance along the wavefront. These revelations provide potential for optimizing injector configurations for real non-premixed systems in order to exploit these physics.

Computational efficiency study of a micro-macro Markov chain Monte Carlo method for molecular dynamics

We study the numerical properties of a recently introduced micro-macro Markov chain Monte Carlo (mM-MCMC) scheme that accelerates sampling of Gibbs distributions when there is a time-scale separation between the complete (microscopic) molecular dynamics and the slow dynamics of a low dimensional reaction coordinate. The micro-macro Markov chain works in three steps: 1) compute the reaction coordinate value associated to the current molecular state; 2) generate a new macroscopic proposal using a proposal distribution; 3) reconstruct a molecular configuration that is consistent with the newly sampled macroscopic value. Here, we systematically study the impact of method parameters on the efficiency of the micro-macro Markov chain Monte Carlo method, namely the chosen macroscopic dynamics and the microscopic reconstruction distribution. We specifically investigate the impact of the macroscopic and reconstruction proposal distributions and their parameters on this efficiency on two prototypical molecular systems, a three-atom molecule and butane. We find, through detailed computational experiments, that the efficiency is largest when these proposal distributions are close to their time-invariant counterparts and that it is robust with respect to parameter changes.

Emergence of Memory in Equilibrium versus Nonequilibrium Systems.

Experiments often probe observables that correspond to low-dimensional projections of high-dimensional dynamics. In such situations distinct microscopic configurations become lumped into the same observable state. It is well known that correlations between the observable and the hidden degrees of freedom give rise to memory effects. However, how and under which conditions these correlations emerge remain poorly understood. Here we shed light on two fundamentally different scenarios of the emergence of memory in minimal stationary systems, where observed and hidden degrees of freedom either evolve cooperatively or are coupled by a hidden nonequilibrium current. In the reversible setting the strongest memory manifests when the timescales of hidden and observed dynamics overlap, whereas, strikingly, in the driven setting maximal memory emerges under a clear timescale separation. Our results hint at the possibility of fundamental differences in the way memory emerges in equilibrium versus driven systems that may be utilized as a "diagnostic" of the underlying hidden transport mechanism.

Transition from time-variant to static networks: Timescale separation in N-intertwined mean-field approximation of susceptible-infectious-susceptible epidemics.

We extend the N-intertwined mean-field approximation (NIMFA) for the susceptible-infectious-susceptible (SIS) epidemiological process to time-varying networks. Processes on time-varying networks are often analyzed under the assumption that the process and network evolution happen on different timescales. This approximation is called timescale separation. We investigate timescale separation between disease spreading and topology updates of the network. We introduce the transition times [under T]̲(r) and T[over ¯](r) as the boundaries between the intermediate regime and the annealed (fast changing network) and quenched (static network) regimes, respectively, for a fixed accuracy tolerance r. By analyzing the convergence of static NIMFA processes, we analytically derive upper and lower bounds for T[over ¯](r). Our results provide insights and bounds on the time of convergence to the steady state of the static NIMFA SIS process. We show that, under our assumptions, the upper-transition time T[over ¯](r) is almost entirely determined by the basic reproduction number R_{0} of the network. The value of the upper-transition time T[over ¯](r) around the epidemic threshold is large, which agrees with the current understanding that some real-world epidemics cannot be approximated with the aforementioned timescale separation.

Tuning limit cycles with a noise: Survival and collapse.

We consider a general class of limit cycle oscillators driven by an additive Gaussian white noise. Based on the separation of timescales, we construct the equationof motion for slow dynamics after appropriate averaging over the fast motion. The equationfor slow motion whose coefficients are modified by noise characteristics is solved to obtain the analytic solution in the long time limit. We show that with increase of noise strength, the loop area of the limit cycle decreases until a critical value is reached, beyond which the limit cycle collapses. We determine the noise threshold from the condition for removal of secular divergence of the perturbation series and work out two explicit examples of Van der Pol and Duffing-Van der Pol oscillators for corroboration between the theory and numerics.

Shortcuts of Freely Relaxing Systems Using Equilibrium Physical Observables.

Many systems, when initially placed far from equilibrium, exhibit surprising behavior in their attempt toequilibrate. Striking examples are the Mpemba effect and the cooling-heating asymmetry. These anomalous behaviors can be exploited to shorten the time needed to cool down (or heat up) a system. Though, a strategy to design these effects in mesoscopic systems is missing. We bring forward a description that allows us to formulate such strategies, and, along the way, makes natural these paradoxical behaviors. In particular, we study the evolution of macroscopic physical observables of systems freely relaxing under the influence of one or two instantaneous thermal quenches. The two crucial ingredients in our approach are timescale separation and a nonmonotonic temperature evolution of an important state function. We argue that both are generic features near a first-order transition. Our theory is exemplified with the one-dimensional Ising model in a magnetic field using analytic results and numerical experiments.

Pseudo-marginal approximation to the free energy in a micro-macro Markov chain Monte Carlo method.

We introduce a generalized micro-macro Markov chain Monte Carlo (mM-MCMC) method with pseudo-marginal approximation to the free energy that is able to accelerate sampling of the microscopic Gibbs distributions when there is a time-scale separation between the macroscopic dynamics of a reaction coordinate and the remaining microscopic degrees of freedom. The mM-MCMC method attains this efficiency by iterating four steps: (i) propose a new value of the reaction coordinate, (ii) accept or reject the macroscopic sample, (iii) run a biased simulation that creates a microscopic molecular instance that lies close to the newly sampled macroscopic reaction coordinate value, and (iv) microscopic accept/reject step for the new microscopic sample. In the present paper, we eliminate the main computational bottleneck of earlier versions of this method: the necessity to have an accurate approximation of free energy. We show that the introduction of a pseudo-marginal approximation significantly reduces the computational cost of the microscopic accept/reject step while still providing unbiased samples. We illustrate the method's behavior on several molecular systems with low-dimensional reaction coordinates.

The impact of architecture on the behavior of siloxane-grafted polymethacrylate

In this paper, we studied for the first time the impact of the molecular weight, Mn, and the topology on the molecular dynamics of a series of grafted poly(poly(dimethylsiloxane methacrylate)) (polyPDMSMA). The behavior of grafted (molecular brushes) polyPDMSMA and their three-armed counterparts of tailor-made properties prepared by Atom Transfer Radical Polymerization (ATRP) was monitored by means of broadband dielectric spectroscopy (BDS), differential scanning calorimetry (DSC) and comprehensive rheology analysis. Surprisingly, the dielectric loss spectra of all studied polyPDMSMA exhibit the presence of two dielectric relaxation processes above their glass transition temperature, Tg. A comparison of dielectric, calorimetric and mechanical data indicated that the faster and slower relaxation processes identified in loss spectra can be related to the segmental and chain dynamics, respectively. Interestingly, we observed that the “local” segmental dynamics of studied polyPDMSMA, involving the mobility of few repeatable units creating a segment, is almost unaffected by neither the length of polymer chain, number of grafts nor its topology. This is also reflected in a weak molecular weight-dependences of Tg of studied polyPDMSMA independently to their topology. On the other hand, the time-scale separation between segmental and the chain modes as well as the amplitude of the latter process increase with growing Mn of all examined polyPDMSMA. Additionally, it should be mentioned that the mechanical data collected for samples characterized by high molecular weight (of Mn > 280 kg/mol) reveal the emergence of additional process (most likely related to the grafts and/or the star arm relaxation) at the intermediate regime, independently to their topology. Interestingly, this mode was observed to be much slower for the star-shaped macromolecules when compared to the grafted polymers, despite their lower Mn. This implies that the observed differences are not a result of different molecular weights, but rather originate from a various molecular mechanism due to the different topology of the studied samples. In this context, one can assume that in the case of high-molecular-weight macromolecules, the backbone architecture has a significant influence on the behavior of polymers, including their molecular dynamics.

Separation of evolutionary timescales in coevolving species

Many coevolutionary processes, including host-parasite and host-symbiont interactions, involve one species or trait which evolves much faster than the other. Whether or not a coevolutionary trajectory converges depends on the relative rates of evolutionary change in the two species, and so current adaptive dynamics approaches generally either determine convergence stability by considering arbitrary (often comparable) rates of evolutionary change or else rely on necessary or sufficient conditions for convergence stability. We propose a method for determining convergence stability in the case where one species is expected to evolve much faster than the other. This requires a second separation of timescales, which assumes that the faster evolving species will reach its evolutionary equilibrium (if one exists) before a new mutation arises in the more slowly evolving species. This method, which is likely to be a reasonable approximation for many coevolving species, both provides straightforward conditions for convergence stability and is less computationally expensive than traditional analysis of coevolution models, as it reduces the trait space from a two-dimensional plane to a one-dimensional manifold. In this paper, we present the theory underlying this new separation of timescales and provide examples of how it could be used to determine coevolutionary outcomes from models.

СИНТЕЗ ДВОКОНТУРНИХ СИСТЕМ КЕРУВАННЯ НАПРУГОЮ РЕВЕРСИВНИХ ПІДВИЩУВАЛЬНИХ DC-DC ПЕРЕТВОРЮВАЧІВ

The paper deals with the design and analysis of cascaded DC-link voltage control systems for bidirectional buck-boost DC-DC converters. Its model is significantly nonlinear and non-minimum-phase, which makes it impossible to achieve high dynamic quality indicators using standard methods of linear control theory. A new method for design and analysis of DC-DC converters control systems based on partial feedback linearization and the subsequent application of linear PI voltage and current controllers is proposed. It provides that the resulting model is composed of the feedback inter-connected linear asymptotically stable subsystems with bilinear properties. The resulting system is linearized in the vicinity of the trajectories corresponding to the power balance equation. Such form allows to apply the theory of cas-caded systems with two time-scale separation of the control loops dynamics. It not only ensures system stability but also allows to specify the process quality indicators using methods available to control engineers. References 14, figures 9.

In-Cell Association of a Bioorthogonal Tubulin.

Studies of proteins from one organism in another organism's cells have shown that such exogenous proteins stick more, pointing toward coevolution of the cytoplasm and protein surface to minimize stickiness. Here we flip this question around by asking whether exogenous proteins can assemble efficiently into their target complexes in a non-native cytoplasm. We use as our model system the assembly of BtubA and BtubB from Prosthecobacter hosted in human U-2 OS cells. BtubA and B evolved from eukaryotic tubulins after horizontal gene transfer, but they have low surface sequence identity with the homologous human tubulins and do not respond to tubulin drugs such as nocodazole. In U-2 OS cells, BtubA and B assemble efficiently into dimers compared to in vitro, and the wild-type BtubA and B proteins subsequently are able to form microtubules as well. We find that generic crowding effects (Ficoll 70 in vitro) contribute significantly to efficient dimer assembly when compared to sticking interactions (U-2 OS cell lysate in vitro), consistent with the notion that a generic mechanism such as crowding can be effective at driving assembly of exogenous proteins, even when protein-cytoplasm quinary structure and sticking have been modified in a non-native cytoplasm. A simple Monte Carlo model of in vitro and in-cell interactions, treating BtubA and B as sticky dipoles in a matrix of sticky or nonsticky crowders, rationalizes all the experimental trends with two adjustable parameters and reveals nucleation as the likely mechanism for the time-scale separation between dimer- and tubule formation in-cell and in vitro.

On the Convergence to the Non-equilibrium Steady State of a Langevin Dynamics with Widely Separated Time Scales and Different Temperatures

On the Convergence to the Non-equilibrium Steady State of a Langevin Dynamics with Widely Separated Time Scales and Different Temperatures

Multiplicative Renormalization of Stochastic Differential Equations for the Abelian Sandpile Model

The long-term, large-scale behavior in a problem of stochastic nonlinear dynamics corresponding to the Abelian sandpile model is studied with the use of the quantum-field theory renormalization group approach. We prove the multiplicative renormalization of the model including an infinite number of coupling parameters, calculate an infinite number of renormalization constants, identify a plane of fixed points in the infinite dimensional space of coupling parameters, discuss their stability and critical scaling in the model, and formulate a simple law relating the asymptotic size of an avalanche to a model exponent quantifying the time-scale separation between the slow energy injection and fast avalanche relaxation processes.

Metastable dynamics of Rydberg atomic system under electromagnetically induced transparency

Open quantum system, coupled with an external bath is a key research field of quantum physics. Steady state is the state in which any initial state converges after a long time and is usually of the most interest. However, relatively speaking, the nonequilibrium dynamical processes of quantum many-body systems have been rarely studied. This is mainly due to the fact that quantum many-body systems generally have interactions, and the Hilbert space required for a complete description of their dynamical processes will grow exponentially with the number of particles increasing, and the computational difficulty will increase dramatically as well. Hence, it is a difficult problem to completely describe their dynamical processes completely. With the development of quantum technologies, the interest in the nonequilibrium dynamics of open quantum many-body systems is aroused. A common phenomenon is the metastable state, where the system initially relaxes into a long-lived state and then converges to the final stationary state for a longer time. In this paper, we establish a low-dimensional approximation to describe the metastability dynamics in Markovian open quantum system, based on the spectra of the Liouvillian super-operator. The separation of time scales implies a splitting in the spectrum, and this spectral division allows us to eliminate the fast decay modes by using the perturbation method, and then we establish the effective description in the low-lying eigenmodes subspace. Furthermore, we study the dynamics process of the Rydberg atomic system under electromagnetically induced transparency (EIT) conditions and find that the system can handle metastable dynamics if the atomic interactions are considered. We compare the effective dynamics in the subspace with the actual dynamics in the full space, and the results show that the effective dynamics works well on condition that the perturbation approximation holds. Our work provides a feasible idea and method for establishing an effective and simplified description of the dynamical process of open quantum many-body systems.

Steady-State Supersaturation Distributions for Clouds under Turbulent Forcing

Abstract The supersaturation equation for a vertically moving adiabatic cloud parcel is analyzed. The effects of turbulent updrafts are incorporated in the shape of a stochastic Lagrangian model, with spatial and time correlations expressed in terms of turbulent kinetic energy. Using the Fokker–Planck equation, the steady-state probability distributions of supersaturation are analytically computed for a number of approximations involving the time-scale separation between updraft fluctuations and phase relaxation, and droplet or ice particle size fluctuations. While the analytical results are presented in general for single-phase clouds, the calculated distributions are used to compute mixed-phase cloud properties—mixed fraction and mean liquid water content in an initially icy cloud—and are argued to be useful for generalizing and constructing new parameterization schemes. Significance Statement Supersaturation is the fuel for the development of clouds in the atmosphere. In this paper, our goal is to better understand the supersaturation budget of clouds embedded in a turbulent environment by analyzing the basic equations of cloud microphysics. It is found that the turbulent characteristics of an air parcel substantially affect the cloud’s supersaturation budget and hence its life cycle. This is also shown in the context of mixed-phase clouds where, depending on the turbulent regime, different liquid-to-ice ratios are found. Consequently, the theoretical approach of this paper is crucial to develop tools to parameterize small-scale atmospheric features, like clouds, into global circulation models to improve climate projections for the future.

Hierarchical Sliding Mode Control for Underactuated Multicopters With Time-Scale Separation Enforcement

Hierarchical Sliding Mode Control for Underactuated Multicopters With Time-Scale Separation Enforcement