Time-periodic Steady State Research Articles

Chemotaxis driven formation of time periodic pattern and steady state pattern

This paper is concerned with a chemotaxis-fluid model with realistic boundary conditions matching the experiments of Kessler et al. Different from the initial and boundary value problem, the existence of time periodic solution is proved under no any smallness assumptions on chemotactic coefficient. Subsequently, we investigate the reason of pattern formation. It is generally known that there is no pattern formation for the homogeneous boundary value problem. Therefore, the pattern formation is partly caused by non-homogeneous boundary conditions, but chemotaxis is also an indispensable condition. In fact, it is proved that the density of bacteria will tend to be evenly distributed when the chemotactic intensity tends to 0. At last, the idea of finding time periodic pattern is also applied to prove the steady-state pattern formation of chemotaxis-Navier-Stokes model.

Using tensor network states for multi-particle Brownian ratchets.

The study of Brownian ratchets has taught how time-periodic driving supports a time-periodic steady state that generates nonequilibrium transport. When a single particle is transported in one dimension, it is possible to rationalize the current in terms of the potential, but experimental efforts have ventured beyond that single-body case to systems with many interacting carriers. Working with a lattice model of volume-excluding particles in one dimension, we analyze the impact of interactions on a flashing ratchet's current. To surmount the many-body problem, we employ the time-dependent variational principle applied to binary tree tensor networks. Rather than propagating individual trajectories, the tensor network approach propagates a distribution over many-body configurations via a controllable variational approximation. The calculations, which reproduce Gillespie trajectory sampling, identify and explain a shift in the frequency of maximum current to higher driving frequency as the lattice occupancy increases.

Nonequilibrium phase transition in transport through a driven quantum point contact

We study transport of noninteracting fermions through a periodically driven quantum point contact (QPC) connecting two tight-binding chains. Initially, each chain is prepared in its own equilibrium state, generally with a bias in chemical potentials and temperatures. We examine the heating rate (or, alternatively, energy increase per cycle) in the nonequilibrium time-periodic steady state established after initial transient dynamics. We find that the heating rate vanishes identically when the driving frequency exceeds the bandwidth of the chain. We first establish this fact for a particular type of QPC where the heating rate can be calculated analytically. Then we verify numerically that this nonequilibrium phase transition is present for a generic QPC. Finally, we derive this effect perturbatively in leading order for cases when the QPC Hamiltonian can be considered as a small perturbation. Strikingly, we discover that for certain QPCs the current averaged over the driving cycle also vanishes above the critical frequency, despite a persistent bias. This shows that a driven QPC can act as a frequency-controlled quantum switch.

Charging a quantum battery via nonequilibrium heat current.

When a quantum system is subject to a thermal gradient it may sustain a steady nonequilibrium heat current by entering into a so-called nonequilibrium steady state (NESS). Here we show that NESS constitute a thermodynamic resource that can be exploited to charge a quantum battery. This adds to the list of recently reported sources available at the nanoscale, such as coherence, entanglement, and quantum measurements. We elucidate this concept by showing analytic and numerical studies of a two-qubit quantum battery that is alternatively charged by an incoherent heat flow and discharged by application of a properly chosen unitary gate. The presence of a NESS for the charging step guarantees steady operation with positive power output. Decreasing the duration of the charging step results in a time-periodic steady state accompanied by increased efficiency and output power. The device is amenable to implementation with different nanotechnology platforms.

Propagation phenomena for partially degenerate nonlocal dispersal models in time and space periodic habitats

This paper is concerned with propagation phenomena of a general class of partially degenerate nonlocal dispersal cooperative systems in time and space periodic habitats. We first show that such system has a finite spreading speed interval in any direction and there is a spreading speed for the partially degenerate system under certain conditions. Next, we prove that if the wave speed is greater than the spreading speed, there exists a time and space periodic traveling wave solution connecting the stable positive time and space periodic steady state and 0. Finally, we apply these results to two species partially degenerate competition systems and a partially degenerate epidemic model with nonlocal dispersal in time and space periodic habitats.

High-Harmonic Generation in Mott Insulators.

Using Floquet dynamical mean-field theory, we study the high-harmonic generation in the time-periodic steady states of wide-gap Mott insulators under ac driving. In the strong-field regime, the harmonic intensity exhibits multiple plateaus, whose cutoff energies ε_{cut}=U+mE_{0} scale with the Coulomb interaction U and the maximum field strength E_{0}. In this regime, the created doublons and holons are localized because of the strong field and the mth plateau originates from the recombination of mth nearest-neighbor doublon-holon pairs. In the weak-field regime, there is only a single plateau in the intensity, which originates from the recombination of itinerant doublons and holons. Here, ε_{cut}=Δ_{gap}+αE_{0}, with Δ_{gap} the band gap and α>1. We demonstrate that the Mott insulator shows a stronger high-harmonic intensity than a semiconductor model with the same dispersion as the Mott insulator, even if the semiconductor bands are broadened by impurity scattering to mimic the incoherent scattering in the Mott insulator.

Boundary Time Crystals

In this work we introduce boundary time crystals. Here continuous time-translation symmetry breaking occurs only in a macroscopic fraction of a many-body quantum system. After introducing their definition and properties, we analyze in detail a solvable model where an accurate scaling analysis can be performed. The existence of the boundary time crystals is intimately connected to the emergence of a time-periodic steady state in the thermodynamic limit of a many-body open quantum system. We also discuss connections to quantum synchronization.

Energy dissipation and fluctuations in a driven liquid

Minimal models of active and driven particles have recently been used to elucidate many properties of nonequilibrium systems. However, the relation between energy consumption and changes in the structure and transport properties of these nonequilibrium materials remains to be explored. We explore this relation in a minimal model of a driven liquid that settles into a time periodic steady state. Using concepts from stochastic thermodynamics and liquid state theories, we show how the work performed on the system by various nonconservative, time-dependent forces-this quantifies a violation of time reversal symmetry-modifies the structural, transport, and phase transition properties of the driven liquid.

A Fast Solution Method for the Boundary Value Problem of Capacitively Coupled Plasmas

Solution methods of capacitively coupled plasmas (CCPs) have been facing a monumental challenge because the solution for the overwhelming number of radio frequency (RF) cycles is in demand before the plasma evolves to a time periodic steady state. It is known that the conventional solution methods require an extremely long calculation time, hence these methods would be limited in industrial applications that need a fast development period. In this paper, a new numerical simulation method for CCPs by COMSOL Multiphysics® is presented. The solution of CCPs in the time domain is converted to a boundary value problem in the additional dimension space to the computational space dimension of the CCP reactor. The fast solutions of the axisymmetric two-dimensional CCPs in Ar and Ar/N2 are performed and the computational efficiency is examined.

A parallelization method for time periodic steady state in simulation of radio frequency sheath dynamics

In order to reduce the computing time in simulation of radio frequency (rf) plasma sources, various numerical schemes were developed. It is well known that the upwind, exponential, and power-law schemes can efficiently overcome the limitation on the grid size for fluid transport simulations of high density plasma discharges. Also, the semi-implicit method is a well-known numerical scheme to overcome on the simulation time step. However, despite remarkable advances in numerical techniques and computing power over the last few decades, efficient multi-dimensional modeling of low temperature plasma discharges has remained a considerable challenge. In particular, there was a difficulty on parallelization in time for the time periodic steady state problems such as capacitively coupled plasma discharges and rf sheath dynamics because values of plasma parameters in previous time step are used to calculate new values each time step. Therefore, we present a parallelization method for the time periodic steady state problems by using period-slices. In order to evaluate the efficiency of the developed method, one-dimensional fluid simulations are conducted for describing rf sheath dynamics. The result shows that speedup can be achieved by using a multithreading method.

Bifurcation analysis of a diffusive plant-wrack model with tide effect on the wrack

This paper deals with the spatial, temporal and spatiotemporal dynamics of a spatial plant-wrack model. The parameter regions for the stability and instability of the unique positive constant steady state solution are derived, and the existence of time-periodic orbits and non-constant steady state solutions are proved by bifurcation method. The nonexistence of positive nonconstant steady state solutions are studied by energy method and Implicit Function Theorem. Numerical simulations are presented to verify and illustrate the theoretical results.

Spatiotemporal Patterns of a Homogeneous Diffusive Predator–Prey System with Holling Type III Functional Response

The dynamics of a diffusive predator–prey system with Holling type-III functional response subject to Neumann boundary conditions is investigated. The parameter region for the stability and instability of the unique constant steady state solution is derived, and the existence of time-periodic orbits and non-constant steady state solutions are proved by bifurcation method and Leray–Schauder degree theory. The effect of various parameters on the existence and nonexistence of spatiotemporal patterns is analyzed. These results show that the impact of Holling type-III response essentially increases the system spatiotemporal complexity.

Single-particle stochastic heat engine.

We have performed an extensive analysis of a single-particle stochastic heat engine constructed by manipulating a Brownian particle in a time-dependent harmonic potential. The cycle consists of two isothermal steps at different temperatures and two adiabatic steps similar to that of a Carnot engine. The engine shows qualitative differences in inertial and overdamped regimes. All the thermodynamic quantities, including efficiency, exhibit strong fluctuations in a time periodic steady state. The fluctuations of stochastic efficiency dominate over the mean values even in the quasistatic regime. Interestingly, our system acts as an engine provided the temperature difference between the two reservoirs is greater than a finite critical value which in turn depends on the cycle time and other system parameters. This is supported by our analytical results carried out in the quasistatic regime. Our system works more reliably as an engine for large cycle times. By studying various model systems, we observe that the operational characteristics are model dependent. Our results clearly rule out any universal relation between efficiency at maximum power and temperature of the baths. We have also verified fluctuation relations for heat engines in time periodic steady state.

Fluctuation relations for heat engines in time-periodic steady states

A fluctuation relation for heat engines has been derived recently. In the beginning, the system is in contact with the cooler bath. The system is then coupled to the hotter bath and external parameters are changed cyclically, eventually bringing the system back to its initial state, once the coupling with the hot bath is switched off. In this work, we lift the condition of initial thermal equilibrium and derive a new fluctuation relation for the central system (heat engine) being in a time-periodic steady state (TPSS). Carnot’s inequality for classical thermodynamics follows as a direct consequence of this fluctuation theorem even in the TPSS. For the special cases of the absence of hot bath and no extraction of work, we obtain the integral fluctuation theorem for total entropy and the generalized exchange fluctuation theorem, respectively. Recently, microsized heat engines have been realized experimentally in the TPSS. We numerically simulate the same model and verify our proposed theorems.

Classical and thermodynamic work fluctuations

We have studied the nature of classical work (W c ) and thermodynamic work (W) fluctuations in systems driven out of equilibrium both in transient and time periodic steady state. As the observation time of a large number of trajectories increases, the fraction of trajectories which exhibit excursions away from the typical behaviour, namely, W c < 0, W < Δ F and dissipated heat Q < 0 decreases as anticipated for macroscopic time scales. An analytical expression for such trajectories is obtained. Trajectory for which W c < 0 may not correspond to W < Δ F or Q < 0. The applicability of steady state fluctuation theorems is discussed in our linear as well as nonlinear models.

Total entropy production fluctuation theorems in a nonequilibrium time-periodic steady state

We investigate the total entropy production of a Brownian particle in a driven bistable system. This system exhibits the phenomenon of stochastic resonance. We show that in the time-periodic steady state, the probability density function for the total entropy production satisfies Seifert’s integral and detailed fluctuation theorems over finite time trajectories.

On Rapid Computation of Time Periodic Steady State in Simulation of Capacitively Coupled RF Plasma

This paper proposes a methodology reducing the number of RF cycles necessary to reach time periodic steady-state solutions for efficient simulations of a three-moment plasma model. Our methodology employs a feedback control approach in conjunction with an implicit time integration scheme. Feedback gains are rigorously estimated by taking into account plasma responses during simulation. Through the one-dimensional simulation of a parallel plate capacitively coupled RF argon plasma, the proposed feedback control approach demonstrates its capability to dramatically reduce the number of RF cycles required to reach the time periodic steady state, resulting in several factors of speedup in simulation time. The simulation result showed that electrons emitted in a temporary cathodic sheath could acquire supersonic velocities and shocks could form near the corresponding sheath boundary as they enter a bulk plasma region.

Automatic detection of Hopf bifurcations on the solution path of a parametrized nonlinear circuit

A numerical technique for the automatic detection of Hopf bifurcations on the solution path of a continuously parametrized nonlinear circuit operating in time-periodic steady state is discussed. The algorithm is based on the piecewise harmonic-balance technique, is truly general-purpose, and can be applied to any kind of nonlinear microwave subsystem without restrictions on circuit topology and device models. This method can lead to software tools for the solution of complex CAD problems, such as the detection of spurious tones in any nonlinear circuit and the analysis of the injection locking of oscillators. >