Traveling Density Waves Research Articles

Scalar Active Mixtures: The Nonreciprocal Cahn-Hilliard Model

Pair interactions between active particles need not follow Newton's third law. In this work we propose a continuum model of pattern formation due to non-reciprocal interaction between multiple species of scalar active matter. The classical Cahn-Hilliard model is minimally modified by supplementing the equilibrium Ginzburg-Landau dynamics with particle number conserving currents which cannot be derived from a free energy, reflecting the microscopic departure from action-reaction symmetry. The strength of the asymmetry in the interaction determines whether the steady state exhibits a macroscopic phase separation or a traveling density wave displaying global polar order. The latter structure, which is equivalent to an active self-propelled smectic phase, coarsens via annihilation of defects, whereas the former structure undergoes Ostwald ripening. The emergence of traveling density waves, which is a clear signature of broken time-reversal symmetry in this active system, is a generic feature of any multi-component mixture with microscopic non-reciprocal interactions.

Emergence of traveling density waves in cyclic multiparticle transport.

Multiparticle flow through a cyclic array of K connected compartments with a preferential direction is found to be able to organize itself in traveling waves. This behavior is connected with the transition between uniform flow and cluster formation. When the bias in the system is large, the particles flow freely in the preferred direction, with all compartments being equally filled at all times. Conversely, when the bias is small the particles cluster together in one compartment. The transition between these two regimes is found to involve an intermediate state in which the flow exhibits a density peak traveling periodically around the system. We relate the emergence of this traveling wave to a Hopf bifurcation and analytically derive the critical value of the "symmetry parameter" at which this bifurcation occurs. This critical value proves to be independent of the number of compartments, but the width of the intermediate regime (and thus the chance of observing traveling wave solutions) decreases sharply with growing K. The reverse transition follows a different course and takes place at a significantly lower value of the symmetry parameter; it is an abrupt transition from a clustered state to a uniform flow without an intermediate regime of stable traveling waves.

Interacting particles in a periodically moving potential: traveling wave and transport.

We study a system of interacting particles in a periodically moving external potential, within the simplest possible description of paradigmatic symmetric exclusion process on a ring. The model describes diffusion of hardcore particles where the diffusion dynamics is locally modified at a uniformly moving defect site, mimicking the effect of the periodically moving external potential. The model, though simple, exhibits remarkably rich features in particle transport, such as polarity reversal and double peaks in particle current upon variation of defect velocity and particle density. By tuning these variables, the most efficient transport can be achieved in either direction along the ring. These features can be understood in terms of a traveling density wave propagating in the system. Our results could be experimentally tested, e.g., in a system of colloidal particles driven by a moving optical tweezer.

Classical Time Crystals

We consider the possibility that classical dynamical systems display motion in their lowest-energy state, forming a time analogue of crystalline spatial order. Challenges facing that idea are identified and overcome. We display arbitrary orbits of an angular variable as lowest-energy trajectories for nonsingular Lagrangian systems. Dynamics within orbits of broken symmetry provide a natural arena for formation of time crystals. We exhibit models of that kind, including a model with traveling density waves.

Traffic Optimization at Junctions to Improve Vehicular Flows

The aim of this work is to improve urban traffic viability through an appropriate choice of yielding and stop signs or red and green phases for traffic lights in junctions with two entering and one exiting roads (junctions of 2×1 type). We consider a macroscopic fluid-dynamic model able to capture the traffic evolution. We analyze different functionals measuring networks performance in terms of average velocity, average traveling time, total flux, density, stop and go waves, average traveling time, weighted with the number of cars moving on roads, and kinetic energy. Right of way parameters which optimize the latter two functionals are obtained. Simulations of simple junctions of 2×1type have been used to test the correctness of the analytical results. Then, global performance of optimization procedures has been investigated on Re di Roma Square, in Italy. In particular, we discuss cases in which the functionals are optimized locally at each junction for different values of right of way parameters. We show that for the chosen initial data the only algorithm for the maximization of velocity assures globally the best performance for the network, also in terms of average traveling times and kinetic energy.

Quantum Quench in One Dimension: Coherent Inhomogeneity Amplification and “Supersolitons”

We study a quantum quench in a 1D system possessing Luttinger liquid (LL) and Mott insulating ground states before and after the quench, respectively. We show that the quench induces power law amplification in time of any particle density inhomogeneity in the initial LL ground state. The scaling exponent is set by the fractionalization of the LL quasiparticle number relative to the insulator. As an illustration, we consider the traveling density waves launched from an initial localized density bump. While these waves exhibit a particular rigid shape, their amplitudes grow without bound.

Quantitative analysis of highly transient fuel sprays by time-resolved x-radiography

Microsecond time-resolved synchrotron x-radiography has been used to elucidate the structure and dynamics of optically turbid, multiphase, direct-injection gasoline fuel sprays. The combination of an ultrafast x-ray framing detector and tomographic analysis allowed three-dimensional reconstruction of the dynamics of the entire 1-ms-long injection cycle. Striking, detailed features were observed, including complex traveling density waves, and unexpected axially asymmetric flows. These results will facilitate realistic computational fluid dynamic simulations of high-pressure sprays and combustion.

The maser mechanism for solar millisecond spike generation in inhomogeneous plasma

The generation of solar millisecond spikes on the maser cyclotron resonance is studied by taking into account the large-scale monotonic magnetic eld inhomogeneity and the small-scale plasma inhomogeneity. It is shown that the presence of these inhomogeneities can have a signicant eect on the generation of electromagnetic waves. In this case the structure of the inhomogeneities determines the spectral and temporal parameters of emission. A modeling of the spike dynamic spectrum in the presence of a small-scale inhomogeneity of coronal plasma is performed. It is shown that a simple model of the plasma inhomogeneity (traveling density wave) can account for many of the observed features of spikes.

Probabilistic Description of Traffic Flow

The study of traffic flow is investigated by different means. Well established theories are (i) kinematic models based on partial differential equations to describe traveling density waves, and (ii) deterministic models using nonlinear car‐following equations to determine trajectories of moving cars, as well as (iii) large-scale simulation hopping models like cellular automata. An important intermediate approach is (iv) the stochastic or probabilistic attempt to understand phenomena like “Stau aus dem Nichts” (phantom jam) on long crowded roads. Initiated by the old argument that road traffic is a stochastic process, we develop a new probabilistic theory based on Markov processes to improve our understanding of traffic flow and its three phases (free flow, synchronized motion, wide moving jams) discovered by Kerner. As an introductory example, first we consider a dissolution of a car queue described by the stochastic master equation as a one-step decay process. Furtheron more realistic models are developed to investigate the nucleation, growth and condensation as well as dissolution of car clusters on a circular one-lane freeway. In analogy to usual aggregation phenomena such as the formation of liquid droplets in supersaturated vapour the clustering behavior in traffic flow is described by the master equation. At overcritical densities the transition from the initial free particle situation (free flow of vehicles) to the final congested state, where one or several big aggregates of cars have been formed, is shown. In dependence on the concentration of cars on the road the stationary solution of the master equation is derived analytically. The obtained fundamental diagram as flow-density-relation indicates clearly the different regimes of traffic flow (free jet of cars, coexisting phase of jams and isolated cars, highly viscous heavy traffic). In the (thermodynamic) limit of infinite number of vehicles on an infinite long road the analytical solution for the fundamental diagram is in agreement with experimental traffic flow data. As a particular example we take into account measurements from German highways presented by Kerner and Rehborn.

Traveling density wave models for earthquakes and driven threshold systems

Driven threshold systems are now used to model sandpiles, earthquakes, magnetic depinning transitions, integrate-and-fire neural networks, and driven foams. We analyze a physically motivated model which has many of the same properties as the hard threshold models, but in which all of the nonequilibrium physics is obtained from a Lyapunov functional. The ideas apply to mean-field systems, and lead to a number of predictions, including scaling exponents and metastable lifetimes for nucleating droplets. The former predictions are supported, for example, by data observed for earthquake fault systems. An interesting consequence of the model is that time appears as a scaling field, leading to temporal scaling laws similar to those observed in nature.

Dynamics of a traveling density wave model for earthquakes.

We propose a unifying new framework for friction and earthquakes. Predictions from theory include (1) scaling exponents, and (2) metastable lifetimes for nucleating slip droplets. Earthquake simulations produce Gutenberg-Richter distributions of events with b values similar to observed declustered values, and populations of characteristic earthquakes, with associated aftershocks. Time is a relevant scaling field, justifying statistical time-to-failure analyses. When applied to integrate-and-fire neurons, the theory yields an explicit equation for the cell potential.

On density-wave oscillations in two-phase flows

Density-wave oscillations in two-phase boiling flow systems have been studied numerically using a robust model based on two non-linear, functional, integro-differential equations. Results of several numerical simulations are used to gain insight into the physical mechanism behind density-wave oscillations. For a wide range of parameter values we find that: (1) traveling density-waves do not play an important role during the oscillations, and that oscillations may persist with very weak traveling density waves; (2) the oscillation period is between three and four times the channel transit time rather than twice as commonly reported; and (3) the variation in mixture velocity, in general, plays a more important role than the variation in mixture density in determining the channel pressure drop characteristics. A physical mechanism for these non-linear oscillations applicable to a large region of parameter space, as suggested by—and consistent with—the results of the numerical experiments, is proposed.

Observation of Plasma Density Waves in n-InSb in Transverse Magnetic Fields

Electron-hole plasma density oscillations in n-InSb have been investigated in transverse magnetic fields using a microwave probe technique. Experiments were done at 77°K and 210°K, using single pulses. The damped oscillations observed at 210°K have been identified as traveling density waves with frequencies ranging from 2 Mc–13 Mc, depending on the magnetic field intensity. The oscillations observed at 77°K were growing waves. The oscillations were identified as standing waves both in the longitudinal and transverse directions. Their frequency ranged from 20 Mc to 50 Mc depending on the magnetic field and the current strength. Both types of oscillations were seen in breakdown region of the current strength. Both types of oscillations were seen in breakdown region of the current-voltage curve. Incoherent waves, similar to noise, were observed above certain critical magnetic fields.

Traveling Density Waves in Positive Columns

The continuity equations for positive ions and for electrons coupled to each other by the Coulomb force and the effect of ionizing collisions have a solution representing traveling density waves whose frequencies are widely different from the usual plasma oscillations.