Existence Of Stationary States Research Articles

Resetting induced multimodality.

Properties of stochastic systems are defined by the noise type and deterministic forces acting on the system. In out-of-equilibrium setups, e.g., for motions under action of Lévy noises, the existence of the stationary state is not only determined by the potential but also by the noise. Potential wells need to be steeper than parabolic in order to assure the existence of stationary states. The existence of stationary states, in sub-harmonic potential wells, can be restored by stochastic resetting, which is the protocol of starting over at random times. Herein, we demonstrate that the combined action of Lévy noise and Poissonian stochastic resetting can result in the phase transition between non-equilibrium stationary states of various multimodality in the overdamped system in super-harmonic potentials. Fine-tuned resetting rates can increase the modality of stationary states, while for high resetting rates, the multimodality is destroyed as the stochastic resetting limits the spread of particles.

On quantitative hypocoercivity estimates based on Harris-type theorems

This Review concerns recent results on the quantitative study of convergence toward the stationary state for spatially inhomogeneous kinetic equations. We focus on analytical results obtained by means of certain probabilistic techniques from the ergodic theory of Markov processes. These techniques are sometimes referred to as Harris-type theorems. They provide constructive proofs for convergence results in the L1 (or total variation) setting for a large class of initial data. The convergence rates can be made explicit (for both geometric and sub-geometric rates) by tracking the constants appearing in the hypotheses. Harris-type theorems are particularly well-adapted for equations exhibiting non-explicit and non-equilibrium steady states since they do not require prior information on the existence of stationary states. This allows for significant improvements of some already-existing results by relaxing assumptions and providing explicit convergence rates. We aim to present Harris-type theorems, providing a guideline on how to apply these techniques to kinetic equations at hand. We discuss recent quantitative results obtained for kinetic equations in gas theory and mathematical biology, giving some perspectives on potential extensions to nonlinear equations.

Epidemic dynamics on social interaction networks

<abstract> <p>The present paper aims to apply the mathematical ideas of the contagion networks in a discrete dynamic context to the modeling of two current pandemics, i.e., COVID-19 and obesity, that are identified as major risks by the World Health Organization. After providing a reminder of the main tools necessary to model epidemic propagation in a Boolean framework (Hopfield-type propagation equation, notion of centrality, existence of stationary states), we present two applications derived from the observation of real data and involving mathematical models for their interpretation. After a discussion of the obtained results of model simulations, multidisciplinary work perspectives (both on mathematical and biomedical sides) are proposed in order to increase the efficiency of the models currently used and improve both the comprehension of the contagion mechanism and the prediction of the dynamic behaviors of the pandemics' present and future states.</p> </abstract>

Automatic identification of near-stationary traffic states based on the PELT changepoint detection

The existence of stationary states during peak periods has been an underlying assumption of many studies on analysis, operations, control, and management of transportation networks. In Cassidy (1998), a method was proposed to manually identify near-stationary states by visually inspecting transformed curves of cumulative total vehicle counts and occupancies. Such near-stationary states are important for calibrating fundamental diagrams, identifying active bottlenecks and incidents, and quantifying capacity drop magnitudes. To the best of our knowledge, however, there lacks an automatic method that can be applied to efficiently identify near-stationary states from a large amount of data.In this study, we attempt to fill this gap. We start with definitions of steady, stationary, and equilibrium states and discuss their relations. Then we present a novel four-step method for automatically identifying near-stationary states from raw loop-detector data: first, the raw data are pre-processed to obtain healthy datasets, fill in missing values, and normalize averaged vehicle counts and occupancies to the same scale; second, daily time series are partitioned into multiple candidate intervals based on the pruned exact linear time (PELT) changepoint detection method; third, the characteristics of the candidate intervals are calculated; and finally, near-stationary states are selected based on modified Cassidy’s gap and duration criteria. We further close the loop by presenting an algorithm to automatically determine the penalty threshold in the second step and the gap threshold in fourth step to ensure the quantity as well as the quality of identified near-stationary states. In a case study, we apply the proposed method to identify near-stationary states from a large set of 30-s raw loop-detector data at a freeway mainline station. We verify the validity of identified near-stationary states both directly and indirectly. The results show that the identified near-stationary states are valid with high quality and the calibrated triangular fundamental diagram is well-fitted and physically meaningful. We finally conclude by discussing some future improvements and potential applications.

On the stability of stationary states in general road networks

In [Jin, W.-L., 2015. On the existence of stationary states in general road networks. Transportation Research Part B 81, 917–929.], with a discrete map in critical demand levels, it was proved that there exist stationary states for the kinematic wave model of general road networks with constant origin demands, route choice proportions, and destination supplies. In this study we further examine the stability property of stationary states with the same map, and the results will help us to understand the long-term trend of a network traffic system. We first review a network kinematic wave model and properties of stationary states on a link, define the criticality of junctions in stationary states, and discuss information propagation in stationary states on links and junctions. We then present the map and examine information propagation in the map. We apply the map to analytically study the stability of stationary states on ring roads and diverge-merge networks with circular information propagation and compare them with results obtained from the Poincaré map [Jin, W.-L., 2013. Stability and bifurcation in network traffic flow: A Poincaré map approach. Transportation Research Part B 57, 191–208]. We further study the stability property of general stationary states in a grid network. We find that the stability of fixed points of the map is the same as that of stationary states in a network, and the new approach is more general than the Poincaré map approach. We conclude the study with future directions and implications.

Stationary states in quantum walk search

When classically searching a database, having additional correct answers makes the search easier. For a discrete-time quantum walk searching a graph for a marked vertex, however, additional marked vertices can make the search harder by causing the system to approximately begin in a stationary state, so the system fails to evolve. In this paper, we completely characterize the stationary states, or 1-eigenvectors, of the quantum walk search operator for general graphs and configurations of marked vertices by decomposing their amplitudes into uniform and flip states. This infinitely expands the number of known stationary states and gives an optimization procedure to find the stationary state closest to the initial uniform state of the walk. We further prove theorems on the existence of stationary states, with them conditionally existing if the marked vertices form a bipartite connected component and always existing if non-bipartite. These results utilize the standard oracle in Grover's algorithm, but we show that a different type of oracle prevents stationary states from interfering with the search algorithm.

On the existence of stationary states in general road networks

Our daily driving experience and empirical observations suggest that traffic patterns in a road network are relatively stationary during peak periods. In numerous transportation network studies, there has been an implicit conjecture that stationary states exist in a network when origin demands, route choice proportions, and destination supplies are constant. In this study, we first rigorously formulate the conjecture within the framework of a network kinematic wave theory with an invariant junction model. After defining stationary states, we derive a system of algebraic equations in 3-tuples of stationary link flow-rates, demands, and supplies. We then introduce a new definition of junction critical demand levels based on effective demands and supplies. With a map in critical demand levels, we show that its fixed points and, therefore, stationary states exist with the help of Brouwer’s fixed point theorem. For two simple road networks, we show that the map is well-defined and can be used to solve stationary states with a brute-force method. Finally we summarize the study and present some future extensions and applications.

Non-equilibrium Thermodynamic Theory of Ferroelectric Phase Transitions

In this review, the non-equilibrium (or irreversible) thermodynamic theory of ferroelectric phase transitions was expounded systematically. Firstly, the limitations of the classical thermodynamic theory of ferroelectric phase transitions (known as the Landau-Devonshire theory) were reviewed critically. The unpleasant consequence caused by the metastable states hypothesis, and the evidence for the non-existence of metastable states i.e. the logical conflict, and the evidence for the existence of stationary states were shown. Secondly, the non-equilibrium thermodynamic description of (especially the first-order) ferroelectric phase transitions was given. The evolving, local characteristics of ferroelectric phase transitions were described well by using the principle of minimum entropy production and the generalized Onsager reciprocal relations. The thermo-electric coupling case is a reduced one of the thermo-electro-mechanical coupling case. In the realm of non-equilibrium thermodynamics, the relation between the latent heat and the polarization was given, too. Thirdly, the non-equilibrium thermodynamic explanation of the irreversibility of ferroelectric phase transitions, i.e. the thermal hysteresis and the domain occurrences in ferroelectrics, was given. At last, some concluding remarks were made and we looked forward to some possible developments.

On the Existence of Stationary States in General Road Networks

Our daily driving experience and empirical observations suggest that traffic patterns in a road network are relatively stationary during peak periods. In numerous transportation network studies, there has been an implicit conjecture that stationary states exist in a network when origin demands, route choice proportions, and destination supplies are constant. In this study, we first rigorously formulate the conjecture within the framework of a network kinematic wave theory with an invariant junction model. After defining stationary states, we derive a system of algebraic equations in 3-tuples of stationary link flow-rates, demands, and supplies. We then introduce a new definition of junction critical demand levels based on effective demands and supplies. With a map in critical demand levels, we show that its fixed points and, therefore, stationary states exist with the help of Brouwer's fixed point theorem. For two simple road networks, we show that the map is well-defined and can be used to solve stationary states with a brute-force method. Finally we summarize the study and present some future extensions and applications.

Quantum master equation with balanced gain and loss

We present a quantum master equation describing a Bose-Einstein condensate with particle loss on one lattice site and particle gain on the other lattice site whose mean-field limit is a non-Hermitian PT-symmetric Gross-Pitaevskii equation. It is shown that the characteristic properties of PT-symmetric systems, such as the existence of stationary states and the phase shift of pulses between two lattice sites, are also found in the many-particle system. Visualizing the dynamics on a Bloch sphere allows us to compare the complete dynamics of the master equation with that of the Gross-Pitaevskii equation. We find that even for a relatively small number of particles the dynamics are in excellent agreement and the master equation with balanced gain and loss is indeed an appropriate many-particle description of a PT-symmetric Bose-Einstein condensate.

Is Gravitational Radiation a Radiation of the Same Level as Electromagnetic Radiation?

The notion of gravitational radiation as a radiation of the same level as the electromagnetic radiation is based on the theoretically proven and experimentally confirmed facts of the existence of stationary states of an electron in its gravitational field, characterized by the gravitational constant K = 1042 G (G is Newtonian gravitational constant) and unrecoverable space-time curvature Λ. These experimental facts include, in particular, data on the broadening of the spectra of the characteristic radiation of multielectron atoms. This broadening of the spectra can be only due to the additional broadening mechanism, in particular the presence of excited states of electrons in their gravitational field. Another fact is the new line in the X-ray emission spectrum according to the results of observation with MOS-camera of the XMM-Newton observatory. This line unlike other identified lines of electromagnetic radiation cannot be assigned to any atomic transition.

On the existence of stationary states during granular compaction

When submitted to gentle mechanical taps a granular packing slowly compacts until it reaches a stationary state that depends on the tap characteristics. The properties of such stationary states are experimentally investigated. The influence of the initial state, taps properties and tapping protocol are studied. The compactivity of the packings is determinated. Our results strongly support the idea that the stationary states are genuine thermodynamic states.

Effect of negative-ion flux on ion distribution around a spherical probe in electronegative plasmas

The electric potential and the electron- and ion-density profiles near a spherical probe in electronegative plasmas containing negative as well as positive ions are studied. It is shown that for relatively large probe radius and neutral-gas pressure, and/or small electron-to-ion temperature ratio, the density profile of the negative ions can differ from that of Boltzmann if their flux near the probe is outwards. The existence of stationary states is considered and the corresponding maximum negative-ion flux is obtained.

Dynamics of the four-level Λ system in a two-mode cavity

In this paper we investigate the properties of a four-level atomic system interacting with two modes of the electromagnetic field in a cavity. By linearization of the Hamiltonian we show that the corresponding mathematical model is exactly solvable. To obtain simpler effective Hamiltonians the method of multiple scales is applied. It is shown that under some special values of the detunings and coupling constants, exactly predictable collapses and revivals in the atomic populations appear. Also, due to the specific interactions of the atom with the cavity modes and the ensuing measurement of energy or angular momentum and energy, two-mode Schr\"odinger-cat states arise in the cavity. Finally, the stabilization and trapping properties of the system are demonstrated in a lossy cavity within the Wigner-Weisskopf approximation. The existence of stationary states is shown and it is also shown that interference effects can cause cancellation of one line in both transmitted light and spontaneous emission spectra.

Safe Debt, Risky Capital

The author extends the Diamond (1965) model of national debt in two directions. She first introduces technological uncertainty and then she allows the stock of debt to vary. The author studies the dynamical equilibria of the resulting stochastic system and she establishes conditions for the existence of stationary states that are expressed in terms of invariant distributions. Since conditions that ensure the existence of a stochastic stationary state with positive debt are very restrictive, in this model financial instability is a pervasive phenomenon and allocations are in general dynamically inefficient. Copyright 1994 by The London School of Economics and Political Science.

Search and mobility in a housing market with limited supply

In this paper, a non-linear dynamic model is presented for a housing market in which various types of households and dwellings are distinguished. The model is based on a stock-flow framework, in which households have to cope with incomplete information when searching for dwellings. The model includes life-cycle patterns (social mobility, ageing) as well as stochastic dwelling preferences. Simulations are carried out with the model to investigate its properties (existence of stationary states, etc.) under various conditions of housing supply. Special attention is paid to vacancy rates, duration of residence, and length of vacancy chains.

Stationary-state and oscillatory combustion of hydrogen in a well-stirred flow reactor

The existence of stationary-state and oscillatory ignition phenomena is established in hydrogen oxidation in a jet-stirred, Pyrex glass reactor (0.55 dm 3). The experimental observations and their numerical interpretation from a detailed kinetic mechanism are reported. Compositions containing 2H 2 + O 2, H 2 + O 2, and H 2 + 2O 2 were investigated at total pressures mainly in the range 10–20 Torr, with brief excursions up to 80 Torr. The vessel temperature was varied between 670 and 790K and mean residence times were controlled in the range 0.7–8.0 s. Reactant temperatures were measured using a very fine thermocouple in the vessel and light output was monitored by photomultiplier. The experimental conditions straddle events at the second limit. Thus criticality and ignition occurs as the vessel temperature is raised under continuous flow of reactants. Ignition is an oscillatory phenomenon at low pressures, and adiabatic temperature changes occur momentarily during it. Increases of vessel temperature cause oscillatory amplitudes to fall and their frequencies to rise. Eventually the oscillations dwindle to a stable stationary state. In certain circumstances there can be an abrupt transition from low frequency to high frequency oscillations. Detailed investigations show that there can be reaction multiplicity and even birhythmicity in the vicinity of these transition points. The dependences of oscillatory reaction on pressure, temperature, and composition are investigated. The elementary kinetic scheme adopted for a numerical integration of the mass and energy conservation equations comprises 8 species in 33 reactions and is designed to embrace subcritical, “slow reaction” and high temperature ignition phenomena. The effects of third body efficiencies are taken into account where these are established quantitatively and enthalpy change is computed at each step. Heat loss rates are computed using an appropriate heat transfer coefficient at the vessel walls. Excellent agreement between the numerical and experimental results is obtained over wide ranges of conditions. Even complex phenomena such as oscillations of mixed frequency and the existence of hysteresis including birhythmicity are predicted numerically. The oscillatory phenomena that occur at the transition to ignition at the second ( p- T a) limit are due to chain branching coupled to product inhibition. Temperature change is an accompaniment and augments the effect, but it does not play an essential role. By contrast, the origins of the additional complexities associated with reaction multiplicity are thermokinetic in kind.

The asymptotic behavior of the solutions of degenerate parabolic equations

Existence of stationary states is established by means of the method of upper and lower solutions. The structure of the solution set is discussed and a uniqueness property for certain classes is proved by a generalized maximum principle. It is then shown that all solutions of the parabolic equation converge to a stationary state.

A classification of Fokker-Planck models and the small and large noise asymptotics

Then-variable Fokker-Planck equation can be written in an equivalent form as a system ofn + 1 first-order equations by introducing as auxiliary variables the components of the drift velocityRl. The stationary state defines a stationaryR uniquely, which allows an intrinsic classification of the stationary states in terms of the properties ofR, without reference to detailed balance. This representation is very appropriate for the study of questions such as the existence of stationary states and their small and large noise asymptotics, as well as for the construction of models having some specified behavior.R provides also a classification of the dynamics, which corresponds to the hermiticity properties of the associated eigenvalue problem.

Dynamics of a class of processes with Smoluchowski noises

One-dimensional evolution equations with a linear random parameter, which is a colored-noise stochastic process, are analyzed. Exact analytical expression for the probability distribution of the considered processes is given explicitly. The existence of stationary states and critical properties of the systems are considered. An analytical example is studied. It is shown that there exist a class of one-dimensional non-stationary markovian processes for which the one-dimensional distribution is the same as for the processes of interest.