Bistability and time crystals in long-ranged directed percolation

AbstractStochastic processes govern the time evolution of a huge variety of realistic systems throughout the sciences. A minimal description of noisy many-particle systems within a Markovian picture and with a notion of spatial dimension is given by probabilistic cellular automata, which typically feature time-independent and short-ranged update rules. Here, we propose a simple cellular automaton with power-law interactions that gives rise to a bistable phase of long-ranged directed percolation whose long-time behaviour is not only dictated by the system dynamics, but also by the initial conditions. In the presence of a periodic modulation of the update rules, we find that the system responds with a period larger than that of the modulation for an exponentially (in system size) long time. This breaking of discrete time translation symmetry of the underlying dynamics is enabled by a self-correcting mechanism of the long-ranged interactions which compensates noise-induced imperfections. Our work thus provides a firm example of a classical discrete time crystal phase of matter and paves the way for the study of novel non-equilibrium phases in the unexplored field of driven probabilistic cellular automata.

Download Full-text

EXTENDED APPLICATION OF CONSTRUCTIVE CRITERIA FOR ERGODICITY OF INTERACTING PARTICLE SYSTEMS

International Journal of Modern Physics C ◽

10.1142/s0129183196000028 ◽

1996 ◽

Vol 07 (01) ◽

pp. 1-18 ◽

Cited By ~ 2

Author(s):

HANS DE JONG ◽

CHRISTIAN MAES

Keyword(s):

Cellular Automata ◽

Discrete Time ◽

Continuous Time ◽

Interacting Particle Systems ◽

Particle Systems ◽

Probabilistic Cellular Automata ◽

Interacting Particle ◽

Spin Flip ◽

Extended Application ◽

Computational Aspects

We discuss computational aspects of verifying constructive criteria for ergodicity of interacting particle systems. Both discrete time (probabilistic cellular automata) and continuous time spin flip dynamics are considered. We also investigate how the criteria have to be adapted if stirring is added to the dynamics.

Download Full-text

An in-depth numerical study of the two-dimensional Kuramoto–Sivashinsky equation

Proceedings of The Royal Society A Mathematical Physical and Engineering Sciences ◽

10.1098/rspa.2014.0932 ◽

2015 ◽

Vol 471 (2179) ◽

pp. 20140932 ◽

Cited By ~ 6

Author(s):

A. Kalogirou ◽

E. E. Keaveny ◽

D. T. Papageorgiou

Keyword(s):

Numerical Study ◽

Period Doubling ◽

System Size ◽

Spatial Dimension ◽

Two Dimensional ◽

Time Dynamics ◽

Long Time ◽

Spatio Temporal ◽

Carry Over ◽

Sivashinsky Equation

The Kuramoto–Sivashinsky equation in one spatial dimension (1D KSE) is one of the most well-known and well-studied partial differential equations. It exhibits spatio-temporal chaos that emerges through various bifurcations as the domain length increases. There have been several notable analytical studies aimed at understanding how this property extends to the case of two spatial dimensions. In this study, we perform an extensive numerical study of the Kuramoto–Sivashinsky equation (2D KSE) to complement this analytical work. We explore in detail the statistics of chaotic solutions and classify the solutions that arise for domain sizes where the trivial solution is unstable and the long-time dynamics are completely two-dimensional. While we find that many of the features of the 1D KSE, including how the energy scales with system size, carry over to the 2D case, we also note several differences including the various paths to chaos that are not through period doubling.

Download Full-text

Linear Time Invariant Homogeneous Matrix Descriptor (Singular) Discrete-Time Systems with Non Consistent Initial Conditions

2010 Sixth International Conference on Networking and Services ◽

10.1109/icns.2010.60 ◽

2010 ◽

Cited By ~ 1

Author(s):

Athanasios A. Pantelous ◽

Athanasios D. Karageorgos ◽

Grigoris I. Kalogeropoulos

Keyword(s):

Discrete Time ◽

Initial Conditions ◽

Linear Time ◽

Linear Time Invariant ◽

Discrete Time Systems ◽

Homogeneous Matrix ◽

Time Invariant ◽

Time Systems

Download Full-text

The VIT Transform Approach to Discrete-Time Signals and Linear Time-Varying Systems

Eng ◽

10.3390/eng2010008 ◽

2021 ◽

Vol 2 (1) ◽

pp. 99-125

Author(s):

Edward W. Kamen

Keyword(s):

Discrete Time ◽

Initial Conditions ◽

Linear Time ◽

Order Term ◽

Initial Time ◽

Time Varying ◽

Varying Coefficients ◽

Time Signals ◽

Time Varying Systems ◽

Time Varying Coefficients

A transform approach based on a variable initial time (VIT) formulation is developed for discrete-time signals and linear time-varying discrete-time systems or digital filters. The VIT transform is a formal power series in z−1, which converts functions given by linear time-varying difference equations into left polynomial fractions with variable coefficients, and with initial conditions incorporated into the framework. It is shown that the transform satisfies a number of properties that are analogous to those of the ordinary z-transform, and that it is possible to do scaling of z−i by time functions, which results in left-fraction forms for the transform of a large class of functions including sinusoids with general time-varying amplitudes and frequencies. Using the extended right Euclidean algorithm in a skew polynomial ring with time-varying coefficients, it is shown that a sum of left polynomial fractions can be written as a single fraction, which results in linear time-varying recursions for the inverse transform of the combined fraction. The extraction of a first-order term from a given polynomial fraction is carried out in terms of the evaluation of zi at time functions. In the application to linear time-varying systems, it is proved that the VIT transform of the system output is equal to the product of the VIT transform of the input and the VIT transform of the unit-pulse response function. For systems given by a time-varying moving average or an autoregressive model, the transform framework is used to determine the steady-state output response resulting from various signal inputs such as the step and cosine functions.

Download Full-text

A Discontinuity of the Energy of Quantum Walk in Impurities

Symmetry ◽

10.3390/sym13071134 ◽

2021 ◽

Vol 13 (7) ◽

pp. 1134

Author(s):

Kenta Higuchi ◽

Takashi Komatsu ◽

Norio Konno ◽

Hisashi Morioka ◽

Etsuo Segawa

Keyword(s):

Unit Circle ◽

Stationary State ◽

Discrete Time ◽

Quantum Walk ◽

Time Limit ◽

The Other ◽

Unitary Matrix ◽

Initial State ◽

Time Quantum ◽

Long Time

We consider the discrete-time quantum walk whose local dynamics is denoted by a common unitary matrix C at the perturbed region {0,1,⋯,M−1} and free at the other positions. We obtain the stationary state with a bounded initial state. The initial state is set so that the perturbed region receives the inflow ωn at time n(|ω|=1). From this expression, we compute the scattering on the surface of −1 and M and also compute the quantity how quantum walker accumulates in the perturbed region; namely, the energy of the quantum walk, in the long time limit. The frequency of the initial state of the influence to the energy is symmetric on the unit circle in the complex plain. We find a discontinuity of the energy with respect to the frequency of the inflow.

Download Full-text

Evolution of solitary waves in a two-pycnocline system

Journal of Fluid Mechanics ◽

10.1017/s0022112009991819 ◽

2009 ◽

Vol 642 ◽

pp. 235-277 ◽

Cited By ~ 11

Author(s):

M. NITSCHE ◽

P. D. WEIDMAN ◽

R. GRIMSHAW ◽

M. GHRIST ◽

B. FORNBERG

Keyword(s):

Solitary Waves ◽

Initial Conditions ◽

Oscillation Period ◽

Resonance Phenomenon ◽

Numerical Code ◽

Asymptotic Model ◽

Narrow Region ◽

Near Resonance ◽

Long Time ◽

The Waves

Over two decades ago, some numerical studies and laboratory experiments identified the phenomenon of leapfrogging internal solitary waves located on separated pycnoclines. We revisit this problem to explore the behaviour of the near resonance phenomenon. We have developed a numerical code to follow the long-time inviscid evolution of isolated mode-two disturbances on two separated pycnoclines in a three-layer stratified fluid bounded by rigid horizontal top and bottom walls. We study the dependence of the solution on input system parameters, namely the three fluid densities and the two interface thicknesses, for fixed initial conditions describing isolated mode-two disturbances on each pycnocline. For most parameter values, the initial disturbances separate immediately and evolve into solitary waves, each with a distinct speed. However, in a narrow region of parameter space, the waves pair up and oscillate for some time in leapfrog fashion with a nearly equal average speed. The motion is only quasi-periodic, as each wave loses energy into its respective dispersive tail, which causes the spatial oscillation magnitude and period to increase until the waves eventually separate. We record the separation time, oscillation period and magnitude, and the final amplitudes and celerity of the separated waves as a function of the input parameters, and give evidence that no perfect periodic solutions occur. A simple asymptotic model is developed to aid in interpretation of the numerical results.

Download Full-text