Dissipative Lattice Research Articles

Random exponential attractor for cocycle and application to non-autonomous stochastic lattice systems with multiplicative white noise

We first establish some sufficient conditions for constructing a random exponential attractor for a continuous cocycle on a separable Banach space and weighted spaces of infinite sequences. Then we apply our abstract result to study the existence of random exponential attractors for non-autonomous first order dissipative lattice dynamical systems with multiplicative white noise.

Emergence of spontaneous symmetry breaking in dissipative lattice systems

A cornerstone of the theory of phase transitions is the observation that many-body systems exhibiting a spontaneous symmetry breaking in the thermodynamic limit generally show extensive fluctuations of an order parameter in large but finite systems. In this work, we introduce the dynamical analog of such a theory. Specifically, we consider local dissipative dynamics preparing an equilibrium steady-state of quantum spins on a lattice exhibiting a discrete or continuous symmetry but with extensive fluctuations in a local order parameter. We show that for all such processes, there exist asymptotically stationary symmetry-breaking states, i.e., states that become stationary in the thermodynamic limit and give a finite value to the order parameter. We give results both for discrete and continuous symmetries and explicitly show how to construct the symmetry-breaking states. Our results show in a simple way that, in large systems, local dissipative dynamics satisfying detailed balance cannot uniquely and efficiently prepare states with extensive fluctuations with respect to local operators. We discuss the implications of our results for quantum simulators and dissipative state preparation.

Micropolar dissipative models for the analysis of 2D dispersive waves in periodic lattices

The computation of the dispersion relations for dissipative periodic lattices having the attributes of metamaterials is an actual research topic raising the interest of researchers in the field of acoustics and wave propagation phenomena. We analyze in this contribution the impact of wave damping on the dispersion features of periodic lattices, which are modeled as beam-lattices. The band diagram structure and damping ratio are computed for different repetitive lattices, based on the homogenized continuum response of the initially discrete lattice architecture, modeled as Kelvin-Voigt viscoelastic beams. Three of these lattices (reentrant hexagonal, chiral diamond, hexachiral lattice) are auxetic metamaterials, since they show negative Poisson's ratio. The effective viscoelastic anisotropic continuum behavior of the lattices is first computed in terms of the homogenized stiffness and viscosity matrices, based on the discrete homogenization technique. The dynamical equations of motion are obtained for an equivalent homogenized micropolar continuum evaluated based on the homogenized properties, and the dispersion relation and damping ratio are obtained by inserting an harmonic plane waves Ansatz into these equations. The comparison of the acoustic properties obtained in the low frequency range for the four considered lattices shows that auxetic lattices attenuate waves at lower frequencies compared to the classical hexagonal lattice. The diamond chiral lattice shows the best attenuation properties of harmonic waves over the entire Brillouin zone, and the hexachiral lattice presents better acoustic properties than the reentrant hexagonal lattice. The range of validity of the effective continuum obtained by the discrete homogenization has been assessed by comparing the frequency band structure of this continuum with that obtained by a Floquet-Bloch analysis.

Photonic currents in driven and dissipative resonator lattices

Arrays of coupled photonic cavities driven by external lasers represent a highly controllable setup to explore photonic transport. In this paper we address (quasi)-steady states of this system that exhibit photonic currents introduced by engineering driving and dissipation. We investigate two approaches: in the first one, photonic currents arise as a consequence of a phase difference of applied lasers and in the second one, photons are injected locally and currents develop as they redistribute over the lattice. Effects of interactions are taken into account within a mean-field framework. In the first approach, we find that the current exhibits a resonant behavior with respect to the driving frequency. Weak interactions shift the resonant frequency toward higher values, while in the strongly interacting regime in our mean-field treatment the effect stems from multiphotonic resonances of a single driven cavity. For the second approach, we show that the overall lattice current can be controlled by incorporating few cavities with stronger dissipation rates into the system. These cavities serve as sinks for photonic currents and their effect is maximal at the onset of quantum Zeno dynamics.

Higher‐order quasicontinuum methods for elastic and dissipative lattice models: uniaxial deformation and pure bending

AbstractThe quasicontinuum (QC) method is a numerical strategy to reduce the computational cost of direct lattice computations ‐ in this study we achieve a speed up of a factor of 40. It has successfully been applied to (conservative) atomistic lattices in the past, but using a virtualpower statement it was recently shown that QC approaches can also be used for spring and beam lattice models that include dissipation. Recent results have shown that QC approaches for planar beam lattices experiencing in‐plane and out‐of‐plane deformation require higherorder interpolation. Higher‐order QC frameworks are scarce nevertheless. In this contribution, the possibilities of a second‐order and third‐order QC framework are investigated for an elastoplastic spring lattice. The higher‐order QC frameworks are compared to the results of the direct lattice computations and to those of a linear QC scheme. Examples are chosen so that both a macroscale and a microscale quantity influences the results. The two multiscale examples focused on are (i) macroscopically prescribed uniaxial deformation and (ii) macroscopically prescribed pure bending. Furthermore, the examples include an individual inclusion in a large lattice and hence, are concurrent in nature. (© 2015 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Electric-Field-Driven Resistive Switching in the Dissipative Hubbard Model.

We study how strongly correlated electrons on a dissipative lattice evolve out of equilibrium under a constant electric field, focusing on the extent of the linear regime and hysteretic nonlinear effects at higher fields. We access the nonequilibrium steady states, nonperturbatively in both the field and the electronic interactions, by means of a nonequilibrium dynamical mean-field theory in the Coulomb gauge. The linear response regime, limited by Joule heating, breaks down at fields much smaller than the quasiparticle energy scale. For large electronic interactions, strong but experimentally accessible electric fields can induce a resistive switching by driving the strongly correlated metal into a Mott insulator. We predict a nonmonotonic upper switching field due to an interplay of particle renormalization and the field-driven temperature. Hysteretic I-V curves suggest that the nonequilibrium current is carried through a spatially inhomogeneous metal-insulator mixed state.

Random attractors for partly dissipative stochastic lattice dynamical systems with multiplicative white noises

The present paper is devoted to the existence of the random attractor for partly dissipative stochastic lattice dynamical systems with multiplicative white noises.

Harnessing intrinsic localized modes to identify impurities in nonlinear periodic systems

Intrinsic localized modes (ILMs) are concentrations of vibrational energy in periodic systems/lattices due to the combined influences of nonlinearity and discreteness. Moreover, ILMs can move within the system and may strongly interact with an impurity, such as a stiffness change, mass variation, etc. Numerous scientific fields have uncovered examples and evidence of ILMs, motivating a multidisciplinary pursuit to rigorously understand the underlying principles. In spite of the diverse technical studies, a characterization of ILM interaction behaviors with multiple impurities in dissipative lattices remains outstanding. The insights on such behaviors may be broadly useful when dynamic measurements are the only accessible features of the periodic system. For instance, one may guide an ILM within the lattice using a deliberately applied and steered impurity and harness the observed interaction behaviors with a second, static (immovable) impurity/defect to identify how the underlying lattice is different at the second, defected site, whether or not one knew the position of the defect a priori. In this spirit, this research studies, analyzes, and characterizes the interaction types amongst an ILM and multiple impurities, and devises a method to identify a static defect impurity using quantitatively and qualitatively distinct interaction phenomena. The method is found to be robust to moderate levels of lattice stiffness heterogeneity and is applicable to monitor various property changes that represent impurities. Finally, experimental studies verify that ILMs interact with multiple impurities in unique ways such that defect features may be effectively identified.

Infinite density for cold atoms in shallow optical lattices

Infinite densities are non-normalizable quasi-probability distributions that can describe the long-time properties of systems when ergodicity is broken and the equilibrium Boltzmann-Gibbs distribution fails. Their experimental observation has remained elusive so far. We here perform semiclassical Monte Carlo simulations of cold atoms in dissipative optical lattices with realistic parameters. We show that the momentum infinite density, as well as its scale invariance, should be observable in shallow potentials. We further evaluate the momentum autocorrelation function both in the stationary and nonstationary, aging regime.

Antiferromagnetic long-range order in dissipative Rydberg lattices

We study the dynamics of dissipative spin lattices with power-law interactions, realized via few-level atoms driven by coherent laser-coupling and decoherence processes. Using Monte Carlo simulations, we determine the phase diagram in the steady state and analyze its generation dynamics. As opposed to mean-field predictions and nearest-neighbor models, there is no transition to long-range-ordered phases for realistic interactions and resonant driving. However, for finite laser detunings, we demonstrate the emergence of crystalline order. Although the static and dynamical critical exponents of the revealed dissipative phase transition fall into the two-dimensional Ising universality class, the found steady states differ considerably from those of an equilibrium Ising magnet. Two complementary schemes for an experimental implementation with cold Rydberg atoms are discussed.

Uniform exponential attractor for nonautonomous partly dissipative lattice dynamical system

In this paper, we consider the existence of a uniform exponential attractor for the nonautonomous partly dissipative lattice dynamical system with quasiperiodic external forces.

Nonlinear periodic waves solutions of the nonlinear self-dual network equations

The new classes of periodic solutions of nonlinear self-dual network equations describing the breather and soliton lattices, expressed in terms of the Jacobi elliptic functions have been obtained. The dependences of the frequencies on energy have been found. Numerical simulations of soliton lattice demonstrate their stability in the ideal lattice and the breather lattice instability in the dissipative lattice. However, the lifetime of such structures in the dissipative lattice can be extended through the application of ac driving terms.

Two-dimensional solitons and clusters in dissipative lattices

We study the dynamics of two-dimensional spatial solitons in the structured optical medium modeled by the complex Ginzburg–Landau equation with cubic–quintic nonlinearity and a spatially periodic modulation of the local gain–loss coefficient [a dissipative lattice (DL)]. The analysis, following the variation of the DL’s amplitude and period, reveals several dynamical scenarios: stable or unstable propagation of a single dissipative soliton (the unstable propagation entails generation of an irregular multisoliton cluster), transformation of the input soliton into stable or unstable regular clusters patterned as the underlying DL, and decay of the input. Most results are obtained by means of systematic simulations, but the boundary of the single-soliton stability domain is explained analytically.

Nondecaying linear and nonlinear modes in a periodic array of spatially localized dissipations

We demonstrate the existence of extremely weakly decaying linear and nonlinear modes (i.e. modes immune to dissipation) in the one-dimensional periodic array of identical spatially localized dissipations, where the dissipation width is much smaller than the period of the array. We consider wave propagation governed by the one-dimensional Schrödinger equation in the array of identical Gaussian-shaped dissipations with three parameters, the integral dissipation strength Γ0, the width σ and the array period d. In the linear case, setting σ → 0, while keeping Γ0 fixed, we get an array of zero-width dissipations given by the Dirac delta-functions, i.e. the complex Kronig–Penney model, where an infinite number of nondecaying modes appear with the Bloch index being either at the center, k = 0, or at the boundary, k = π/d, of an analogue of the Brillouin zone. By using numerical simulations we confirm that the weakly decaying modes persist for σ such that σ/d ≪ 1 and have the same Bloch index. The nondecaying modes persist also if a real-valued periodic potential is added to the spatially periodic array of dissipations, with the period of the dissipative array being a multiple of that of the periodic potential. We also consider evolution of the soliton-shaped pulses in the nonlinear Schrödinger equation with the spatially periodic dissipative lattice and find that when the pulse width is much larger than the lattice period and its wave number k is either at the center, k = 2π/d, or at the boundary, k = π/d, a significant fraction of the pulse escapes the dissipation forming a stationary nonlinear mode with the soliton-shaped envelope and the Fourier spectrum consisting of two peaks centered at k and −k.

Computer simulation techniques for food science and engineering: Simulating atomic scale and coarse-grained models

Abstract We describe computer simulation techniques that have been, or can be, used in Food Science and Engineering. We describe models which cannot be utilized without employing computer simulation but do not explicitly address models such as self consistent field approaches or DLVO theory. We describe techniques which either continue to play fundamental roles in computer simulation of soft matter and fluids, or newer developments which have shown increased use in the last decade. Here we outline the background to statistical mechanics followed by descriptions with selected examples of Molecular Dynamics, Coarse-Grained modeling, Monte Carlo techniques, Dissipative Particle Dynamics and Lattice–Boltzmann theory.

Stability bands for moving solitons in uniform and inhomogeneous damped ac-driven circular Toda lattices

Abstract By means of systematic simulations, we study the motion of discrete solitons in weakly dissipative Toda lattices (TLs) with periodic boundary conditions, resonantly driven by a spatially staggered time-periodic (ac) force. A complex set of alternating stability bands and instability gaps, including scattered isolated stability points, is revealed in the parametric plane of the soliton’s velocity and forcing amplitude for a given size of the circular lattice. The analysis is also reported for the circular TL including a single light- or heavymass defect. The stability chart as a whole shrinks and eventually disappears with the increase of the lattice’s size and strength of the mass defect. Qualitative explanations to these findings are proposed. We also report the dependence of the stability area on the initial position of the soliton, finding that the area is largest for some intersite position. For a pair of solitons traveling in opposite directions, there exist regimes where both solitons survive periodic collisions in small-size lattices.

A multiscale quasicontinuum method for dissipative lattice models and discrete networks

Lattice models and discrete networks naturally describe mechanical phenomena at the mesoscale of fibrous materials. A disadvantage of lattice models is their computational cost. The quasicontinuum (QC) method is a suitable multiscale approach that reduces the computational cost of lattice models and allows the incorporation of local lattice defects in large-scale problems. So far, all QC methods are formulated for conservative (mostly atomistic) lattice models. Lattice models of fibrous materials however, often require non-conservative interactions. In this paper, a QC formulation is derived based on the virtual-power of a non-conservative lattice model. By using the virtual-power statement instead of force-equilibrium, errors in the governing equations of the force-based QC formulations are avoided. Nevertheless, the non-conservative interaction forces can still be directly inserted in the virtual-power QC framework. The summation rules for energy-based QC methods can still be used in the proposed framework as shown by two multiscale examples.

Beyond Boltzmann–Gibbs statistical mechanics in optical lattices

Cold atoms in dissipative optical lattices exhibit an unusual transport behaviour that cannot be described within Boltzmann–Gibbs statistical mechanics. New theoretical tools and concepts need thus be developed to account for their observable macroscopic properties. Here we review recent progress achieved in the study of these processes. We emphasize the generality of the findings for a broad class of physical, chemical and biological systems, and discuss open questions and perspectives for future work. Cold atoms trapped in dissipative optical lattices can behave in ways that cannot be described within the framework of Boltzmann–Gibbs statistical mechanics. Recent theoretical and experimental developments may lead to a better understanding of these processes.

Discrete breathers in integrate-and-fire oscillator networks

A number of novel dynamical patterns are found in an integrate-and-fire network, including periodic breathers with single, double, and triple wavefronts, and chaotic breathers. It is found that the periodic breathers always propagate whereas the chaotic ones always remain stationary. A novel route to chaotic breathers through intermittency is identified, as opposed to the paths through quasi-periodicity and period-doubling mostly discussed in dissipative lattice systems. The system is found to display prominent bistability, resulting in “hysteresis” loops in single-oscillator inter-spike intervals as the parameter varies.

Nondecaying Bloch modes of a dissipative lattice

We consider a Bose-Einstein condensate in a periodic optical lattice having a periodic sublattice of dissipative sites (which we call the dissipative lattice). The optical analog of this system consisting of lossy optical waveguides arranged in a checkerboard order is also presented. Using numerical simulations within the mean-field theory we demonstrate the existence of nondecaying (or propagating, in optical applications) Bloch-like coherent modes, which are eigenstates of the non-Hermitian operator with the dissipative lattice serving as a spatially periodic potential. The nondecaying modes have the Bloch index confined to the boundary of the Brillouin zone of the dissipative lattice. Besides the numerical simulations for a finite lattice, we use the Fourier space perturbation theory for a weak dissipative lattice and the single-band tight-binding approximation for a strong lattice to show that the effect is a universal property of the dissipative lattice.