Dissipative Version Research Articles

Dissipative fractional standard maps: Riemann-Liouville and Caputo.

In this study, given the inherent nature of dissipation in realistic dynamical systems, we explore the effects of dissipation within the context of fractional dynamics. Specifically, we consider the dissipative versions of two well known fractional maps: the Riemann-Liouville (RL) and the Caputo (C) fractional standard maps (fSMs). Both fSMs are two-dimensional nonlinear maps with memory given in action-angle variables (In,θn), with n being the discrete iteration time of the maps. In the dissipative versions, these fSMs are parameterized by the strength of nonlinearity K, the fractional order of the derivative α∈(1,2], and the dissipation strength γ∈(0,1]. In this work, we focus on the average action ⟨In⟩ and the average squared action ⟨In2⟩ when K≫1, i.e., along strongly chaotic orbits. We first demonstrate, for |I0|>K, that dissipation produces the exponential decay of the average action ⟨In⟩≈I0exp⁡(-γn) in both dissipative fSMs. Then, we show that while ⟨In2⟩RL-fSM barely depends on α (effects are visible only when α→1), any α<2 strongly influences the behavior of ⟨In2⟩C-fSM. We also derive an analytical expression able to describe ⟨In2⟩RL-fSM(K,α,γ).

Shearless and periodic attractors in the dissipative Labyrinthic map.

The Labyrinthic map is a two-dimensional area-preserving map that features a robust transport barrier known as the shearless curve. In this study, we explore a dissipative version of this map, examining how dissipation affects the shearless curve and leads to the emergence of quasi-periodic or chaotic attractors, referred to as shearless attractors. We present a route to chaos of the shearless attractor known as the Curry-Yorke route. To investigate the multi-stability of the system, we employ basin entropy and boundary basin entropy analyses to characterize diverse scenarios. Additionally, we numerically investigate the dynamic periodic structures known as "shrimps" and "Arnold tongues" by varying the parameters of the system.

Stochastic dissipative Euler’s equations for a free body

Abstract Intrinsic thermal fluctuations within a real solid challenge the rigid body assumption that is central to Euler’s equations for the motion of a free body. Recently, we have introduced a dissipative and stochastic version of Euler’s equations in a thermodynamically consistent way (European Journal of Mechanics – A/Solids 103, 105,184 (2024)). This framework describes the evolution of both orientation and shape of a free body, incorporating internal thermal fluctuations and their concomitant dissipative mechanisms. In the present work, we demonstrate that, in the absence of angular momentum, the theory predicts that the principal axes unit vectors of a body undergo an anisotropic Brownian motion on the unit sphere, with the anisotropy arising from the body’s varying moments of inertia. The resulting equilibrium time correlation function of the principal eigenvectors decays exponentially. This theoretical prediction is confirmed in molecular dynamics simulations of small bodies. The comparison of theory and equilibrium MD simulations allow us to measure the orientational diffusion tensor. We then use this information in the Stochastic Dissipative Euler’s Equations, to describe a non-equilibrium situation of a body spinning around the unstable intermediate axis. The agreement between theory and simulations is excellent, offering a validation of the theoretical framework.

Weakly interacting Bose gas with two-body losses

We study the many-body dynamics of weakly interacting Bose gases with two-particle losses. We show that both the two-body interactions and losses in atomic gases may be tuned by controlling the inelastic scattering process between atoms by an optical Feshbach resonance. Interestingly, the low-energy behavior of the scattering amplitude is governed by a single parameter, i.e. the complex ss-wave scattering length a_cac. The many-body dynamics are thus described by a Lindblad master equation with complex scattering length. We solve this equation by applying the Bogoliubov approximation in analogy to the closed systems. Various peculiar dynamical properties are discovered, some of them may be regarded as the dissipative counterparts of the celebrated results in closed Bose gases. For example, we show that the next-order correction to the mean-field particle decay rate is to the order of |n a_c^{3}|^{1/2}|nac3|1/2, which is an analogy of the Lee-Huang-Yang correction of Bose gases. It is also found that there exists a dynamical symmetry of symplectic group Sp(4,\mathbb{C})(4,ℂ) in the quadratic Bogoliubov master equation, which is an analogy of the SU(1,1) dynamical symmetry of the corresponding closed system. We further confirmed the validity of the Bogoliubov approximation by comparing its results with a full numerical calculation in a double-well toy model. Generalizations of other alternative approaches such as the dissipative version of the Gross-Pitaevskii equation and hydrodynamic theory are also discussed in the last.

Symmetry analysis and hidden variational structure of Westervelt’s equation in nonlinear acoustics

Westervelt’s equation is a nonlinear wave equation that is widely used to model the propagation of sound waves in a compressible medium, with one important application being ultra-sound in human tissue. Two fundamental aspects of the general dissipative version of Westervelt’s equation – symmetries and conservation laws – are studied in the present work by modern methods. Numerous results are obtained: new conserved integrals; potential systems yielding hidden symmetries and nonlocal conservation laws; mapping of Westervelt’s equation in the undamped case into a linear wave equation; hidden variational structures, including a Lagrangian and a Hamiltonian; a recursion operator and a Noether operator; contact symmetries; higher-order symmetries and higher-order conservation laws.

Topological Atomic Spin Wave Lattices by Dissipative Couplings.

Recent experimental advances in creating dissipative couplings provide a new route for engineering exotic lattice systems and exploring topological dissipation. Using the spatial lattice of atomic spin waves in a vacuum vapor cell, where purely dissipative couplings arise from diffusion of atoms, we experimentally realize a dissipative version of the Su-Schrieffer-Heeger (SSH) model. We construct the dissipation spectrum of the topological or trivial lattices via electromagnetically induced-transparency spectroscopy. The topological dissipation spectrum is found to exhibit edge modes within a dissipative gap. We validate chiral symmetry of the dissipative SSH couplings and also probe topological features of the generalized dissipative SSH model. This work paves the way for realizing non-Hermitian topological quantum optics via dissipative couplings.

Chaotic saddles and interior crises in a dissipative nontwist system.

We consider a dissipative version of the standard nontwist map. Nontwist systems present a robust transport barrier, called the shearless curve, that becomes the shearless attractor when dissipation is introduced. This attractor can be regular or chaotic depending on the control parameters. Chaotic attractors can undergo sudden and qualitative changes as a parameter is varied. These changes are called crises, and at an interior crisis the attractor suddenly expands. Chaotic saddles are nonattracting chaotic sets that play a fundamental role in the dynamics of nonlinear systems; they are responsible for chaotic transients, fractal basin boundaries, and chaotic scattering, and they mediate interior crises. In this work we discuss the creation of chaotic saddles in a dissipative nontwist system and the interior crises they generate. We show how the presence of two saddles increases the transient times and we analyze the phenomenon of crisis induced intermittency.

Bifurcations, relaxation time, and critical exponents in a dissipative or conservative Fermi model.

We investigated the time evolution for the stationary state at different bifurcations of a dissipative version of the Fermi-Ulam accelerator model. For local bifurcations, as period-doubling bifurcations, the convergence to the inactive state is made using a homogeneous and generalized function at the bifurcation parameter. It leads to a set of three critical exponents that are universal for such bifurcation. Near bifurcation, an exponential decay describes convergence whose relaxation time is characterized by a power law. For global bifurcation, as noticed for a boundary crisis, where a chaotic transient suddenly replaces a chaotic attractor after a tiny change of control parameters, the survival probability is described by an exponential decay whose transient time is given by a power law.

DISSIPATION AND THE INFORMATION CONTENT OF THE DEVIATION FROM HAMILTONIAN DYNAMICS

"We explain a dissipative version of hamiltonian mechanics, based on the information content of the deviation from hamiltonian dynam¬ics. From this formulation we deduce minimal dissipation principles, dynamical inclusions, or constrained evolution with hamiltonian drift reformulations. Among applications we recover a dynamics generaliza¬tion of Mielke et al quasistatic rate-independent processes. This article gives a clear and unitary presentation of the theory of hamiltonian inclusions with convex dissipation or symplectic Brezis- Ekeland-Nayroles principle, presented under various conventions first in [3] arXiv:0810.1419, then in [4] arXiv:1408.3102 and, for the ap¬pearance of bipotentials in relation to the symplectic duality, in [2] arXiv:1902.04598v1."

Invariant tori of the weakly dissipative version of the Ginzburg-Landau equation

Invariant tori of the weakly dissipative version of the Ginzburg-Landau equation

Cauchy Processes, Dissipative Benjamin–Ono Dynamics and Fat-Tail Decaying Solitons

In this paper, a dissipative version of the Benjamin–Ono dynamics is shown to faithfully model the collective evolution of swarms of scalar Cauchy stochastic agents obeying a follow-the-leader interaction rule. Due to the Hilbert transform, the swarm dynamic is described by nonlinear and non-local dynamics that can be solved exactly. From the mutual interactions emerges a fat-tail soliton that can be obtained in a closed analytic form. The soliton median evolves nonlinearly with time. This behaviour can be clearly understood from the interaction of mutual agents.

Destruction and resurgence of the quasiperiodic shearless attractor.

We consider a dissipative version of the standard nontwist map. It is known that nontwist systems may present a robust transport barrier, called shearless curve, that gives rise to an attractor that retains some of its properties when dissipation is introduced. This attractor is known as shearless attractor, and it may be quasiperiodic or chaotic depending on the control parameters. We describe a route for the destruction and resurgence of the quasiperiodic shearless attractor by analyzing the manifolds of the unstable periodic orbits (UPOs) which are fixed points of the map. We show that the shearless attractor is destroyed by a collision with the UPOs and it resurges after the reconnection of the unstable manifolds of different UPOs.

Exact Liouvillian Spectrum of a One-Dimensional Dissipative Hubbard Model.

A one-dimensional dissipative Hubbard model with two-body loss is shown to be exactly solvable. We obtain an exact eigenspectrum of a Liouvillian superoperator by employing a non-Hermitian extension of the Bethe-ansatz method. We find steady states, the Liouvillian gap, and an exceptional point that is accompanied by the divergence of the correlation length. A dissipative version of spin-charge separation induced by the quantum Zeno effect is also demonstrated. Our result presents a new class of exactly solvable Liouvillians of open quantum many-body systems, which can be tested with ultracold atoms subject to inelastic collisions.

Curry-Yorke route to shearless attractors and coexistence of attractors in dissipative nontwist systems.

The routes to chaos play an important role in predictions about the transitions from regular to irregular behavior in nonlinear dynamical systems, such as electrical oscillators, chemical reactions, biomedical rhythms, and nonlinear wave coupling. Of special interest are dissipative systems obtained by adding a dissipation term in a given Hamiltonian system. If the latter satisfies the so-called twist property, the corresponding dissipative version can be called a "dissipative twist system." Transitions to chaos in these systems are well established; for instance, the Curry-Yorke route describes the transition from a quasiperiodic attractor on torus to chaos passing by a chaotic banded attractor. In this paper, we study the transitions from an attractor on torus to chaotic motion in dissipative nontwist systems. We choose the dissipative standard nontwist map, which is a non-conservative version of the standard nontwist map. In our simulations, we observe the same transition to chaos that happens in twist systems, known as a soft one, where the quasiperiodic attractor becomes wrinkled and then chaotic through the Curry-Yorke route. By the Lyapunov exponent, we study the nature of the orbits for a different set of parameters, and we observe that quasiperiodic motion and periodic and chaotic behavior are possible in the system. We observe that they can coexist in the phase space, implying in multistability. The different coexistence scenarios were studied by the basin entropy and by the boundary basin entropy.

Testing dissipative collapse models with a levitated micromagnet

We present experimental tests of dissipative extensions of spontaneous wave function collapse models based on a levitated micromagnet with ultralow dissipation. The spherical micromagnet, with radius $R=27$ $\mu$m, is levitated by Meissner effect in a lead trap at $4.2$ K and its motion is detected by a SQUID. We perform accurate ringdown measurements on the vertical translational mode with frequency $57$ Hz, and infer the residual damping at vanishing pressure $\gamma/2\pi<9$ $\mu$Hz. From this upper limit we derive improved bounds on the dissipative versions of the CSL (continuous spontaneous localization) and the DP (Di\'{o}si-Penrose) models with proper choices of the reference mass. In particular, dissipative models give rise to an intrinsic damping of an isolated system with the effect parameterized by a temperature constant; the dissipative CSL model with temperatures below 1 nK is ruled out, while the dissipative DP model is excluded for temperatures below $10^{-13}$ K. Furthermore, we present the first bounds on dissipative effects in a more recent model, which relates the wave function collapse to fluctuations of a generalized complex-valued spacetime metric.

Bounds on the elastic threshold for problems of dissipative strain-gradient plasticity

This work is concerned with the purely dissipative version of a well-established model of rate-independent strain-gradient plasticity. In the conventional theory of plasticity the approach to determining plastic flow is local, and based on the stress distribution in the body. For the dissipative problem of strain-gradient plasticity such an approach is not valid as the yield function depends on microstresses that are not known in the elastic region. Instead, yield and plastic flow must be considered at the global level. This work addresses the problem of determining the elastic threshold by formulating primal and dual versions of the global problem and, motivated by techniques used in limit analysis for perfect plasticity, establishing conditions for lower and upper bounds to the threshold. The general approach is applied to two examples: of a plate under plane stress, and subjected to a prescribed displacement; and of a bar subjected to torsion.

Well-posedness for a perturbation of the KdV equation

This is a continuation of Carvajal and Panthee (Q Appl Math 74:571–594, 2016). In the present paper we study a dissipative versions of the Korteweg de Vries equation with rough initial data. By working in Bourgain’s type spaces we prove local well posedness results in Sobolev spaces of negative order.

Non-stationary coherent quantum many-body dynamics through dissipation

The assumption that quantum systems relax to a stationary state in the long-time limit underpins statistical physics and much of our intuitive understanding of scientific phenomena. For isolated systems this follows from the eigenstate thermalization hypothesis. When an environment is present the expectation is that all of phase space is explored, eventually leading to stationarity. Notable exceptions are decoherence-free subspaces that have important implications for quantum technologies and have so far only been studied for systems with a few degrees of freedom. Here we identify simple and generic conditions for dissipation to prevent a quantum many-body system from ever reaching a stationary state. We go beyond dissipative quantum state engineering approaches towards controllable long-time non-stationarity typically associated with macroscopic complex systems. This coherent and oscillatory evolution constitutes a dissipative version of a quantum time crystal. We discuss the possibility of engineering such complex dynamics with fermionic ultracold atoms in optical lattices.

Information scrambling in chaotic systems with dissipation

Chaotic dynamics in closed local quantum systems scrambles quantum information, which is manifested quantitatively in the decay of the out-of-time-ordered correlators (OTOC) of local operators. How is information scrambling affected when the system is coupled to the environment and suffers from dissipation? In this paper, we address this question by defining a dissipative version of OTOC and numerically study its behavior in a prototypical chaotic quantum chain in the presence of dissipation. We find that dissipation leads to not only the overall decay of the scrambled information due to leaking, but also structural changes so that the `information light cone' can only reach a finite distance even when the effect of overall decay is removed. Based on this observation we conjecture a modified version of the Lieb-Robinson bound in dissipative systems.

Persistent currents by reservoir engineering

We demonstrate that persistent currents can be induced in a quantum system in contact with a structured reservoir, without the need of any applied gauge field. The working principle of the mechanism leading to their presence is based on the extension to the many-body scenario of non-reciprocal Lindblad dynamics, recently put forward by Metelmann and Clerk in Phys. Rev. X 5, 021025 (2015): Non-reciprocity can be generated by suitably balancing coherent interactions with their corresponding dissipative version, induced by the coupling to a common structured environment, such to make total interactions directional. Specifically, we consider an interacting spin/boson model in a ring-shaped one-dimensional lattice coupled to an external bath. By employing a combination of cluster mean-field, exact diagonalization and matrix-product-operator techniques, we show that solely dissipative effects suffice to engineer steady states with a persistent current that survives in the limit of large systems. We also verify the robustness of such current in the presence of additional dissipative or Hamiltonian perturbation terms.