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Register a new userSpatio-temporal spread of perturbations in a driven dissipative Duffing chain: an OTOC approach
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Out-of-time-ordered correlators (OTOC) have been extensively used as a major tool for exploring quantum chaos and also recently, there has been a classical analogue. Studies have been limited to closed systems. In this work, we probe an open classical many-body system, more specifically, a spatially extended driven dissipative chain of coupled Duffing oscillators using the classical OTOC to investigate the spread and growth (decay) of an initially localized perturbation in the chain. Correspondingly, we find three distinct types of dynamical behavior, namely the sustained chaos, transient chaos and non-chaotic region, as clearly exhibited by different geometrical shapes in the heat map of OTOC. To quantify such differences, we look at instantaneous speed (IS), finite time Lyapunov exponents (FTLE) and velocity dependent Lyapunov exponents (VDLE) extracted from OTOC. Introduction of these quantities turn out to be instrumental in diagnosing and demarcating different regimes of dynamical behavior. To gain control over open nonlinear systems, it is important to look at the variation of these quantities with respect to parameters. As we tune drive, dissipation and coupling, FTLE and IS exhibit transition between sustained chaos and non-chaotic regimeswith intermediate transient chaos regimes and highly intermittent sustained chaos points. In the limit of zero nonlinearity, we present exact analytical results for the driven dissipative harmonic system and we find that our analytical results can very well describe the non-chaotic regime as well as the late time behavior in the transient regime of the Duffing chain. We believe, this analysis is an important step forward towards understanding nonlinear dynamics, chaos and spatio-temporal spread of perturbations in many-particle open systems.
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We present exact results for the classical version of the Out-of-Time-Order Commutator (OTOC) for a family of power-law models consisting of $N$ particles in one dimension and confined by an external harmonic potential. These particles are interacting via power-law interaction of the form $propto sum_{substack{i, j=1 (i eq j)}}^N|x_i-x_j|^{-k}$ $forall$ $k>1$ where $x_i$ is the position of the $i^text{th}$ particle. We present numerical results for the OTOC for finite $N$ at low temperatures and short enough times so that the system is well approximated by the linearized dynamics around the many body ground state. In the large-$N$ limit, we compute the ground-state dispersion relation in the absence of external harmonic potential exactly and use it to arrive at analytical results for OTOC. We find excellent agreement between our analytical results and the numerics. We further obtain analytical results in the limit where only linear and leading nonlinear (in momentum) terms in the dispersion relation are included. The resulting OTOC is in agreement with numerics in the vicinity of the edge of the light cone. We find remarkably distinct features in OTOC below and above $k=3$ in terms of going from non-Airy behaviour ($1<k<3$) to an Airy universality class ($k>3$). We present certain additional rich features for the case $k=2$ that stem from the underlying integrability of the Calogero-Moser model. We present a field theory approach that also assists in understanding certain aspects of OTOC such as the sound speed. Our findings are a step forward towards a more general understanding of the spatio-temporal spread of perturbations in long-range interacting systems.
The synchronization transition between two coupled replicas of spatio-temporal chaotic systems in 2+1 dimensions is studied as a phase transition into an absorbing state - the synchronized state. Confirming the scenario drawn in 1+1 dimensional systems, the transition is found to belong to two different universality classes - Multiplicative Noise (MN) and Directed Percolation (DP) - depending on the linear or nonlinear character of damage spreading occurring in the coupled systems. By comparing coupled map lattice with two different stochastic models, accurate numerical estimates for MN in 2+1 dimensions are obtained. Finally, aiming to pave the way for future experimental studies, slightly non-identical replicas have been considered. It is shown that the presence of small differences between the dynamics of the two replicas acts as an external field in the context of absorbing phase transitions, and can be characterized in terms of a suitable critical exponent.
Spontaneous symmetry breaking (SSB) is a key concept in physics that for decades has played a crucial role in the description of many physical phenomena in a large number of different areas, like particle physics, cosmology, and condensed-matter physics. SSB is thus an ubiquitous concept connecting several, both high and low energy, areas of physics and many textbooks describe its basic features in great detail. However, to study the dynamics of symmetry breaking in the laboratory is extremely difficult. In condensed-matter physics, for example, tiny external disturbances cause a preference for the breaking of the symmetry in a particular configuration and typically those disturbances cannot be avoided in experiments. Notwithstanding these complications, here we describe an experiment, in which we directly observe the spontaneous breaking of the temporal phase of a driven system with respect to the drive into two distinct values differing by $pi$.
The attribution method provides a direction for interpreting opaque neural networks in a visual way by identifying and visualizing the input regions/pixels that dominate the output of a network. Regarding the attribution method for visually explaining video understanding networks, it is challenging because of the unique spatiotemporal dependencies existing in video inputs and the special 3D convolutional or recurrent structures of video understanding networks. However, most existing attribution methods focus on explaining networks taking a single image as input and a few works specifically devised for video attribution come short of dealing with diversified structures of video understanding networks. In this paper, we investigate a generic perturbation-based attribution method that is compatible with diversified video understanding networks. Besides, we propose a novel regularization term to enhance the method by constraining the smoothness of its attribution results in both spatial and temporal dimensions. In order to assess the effectiveness of different video attribution methods without relying on manual judgement, we introduce reliable objective metrics which are checked by a newly proposed reliability measurement. We verified the effectiveness of our method by both subjective and objective evaluation and comparison with multiple significant attribution methods.
To make progress in understanding the issue of memory loss and history dependence in evolving complex systems, we consider the mixing rate that specifies how fast the future states become independent of the initial condition. We propose a simple measure for assessing the mixing rate that can be directly applied to experimental data observed in any metric space $X$. For a compact phase space $X subset R^M$, we prove the following statement. If the underlying dynamical system has a unique physical measure and its dynamics is strongly mixing with respect to this measure, then our method provides an upper bound of the mixing rate. We employ our method to analyze memory loss for the system of slowly sheared granular particles with a small inertial number $I$. The shear is induced by the moving walls as well as by the linear motion of the support surface that ensures approximately linear shear throughout the sample. We show that even if $I$ is kept fixed, the rate of memory loss (considered at the time scale given by the inverse shear rate) depends erratically on the shear rate. Our study suggests a presence of bifurcations at which the rate of memory loss increases with the shear rate while it decreases away from these points. We also find that the memory loss is not a smooth process. Its rate is closely related to frequency of the sudden transitions of the force network. The loss of memory, quantified by observing evolution of force networks, is found to be correlated with the loss of correlation of shear stress measured on the system scale. Thus, we have established a direct link between the evolution of force networks and macroscopic properties of the considered system.
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