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We model an overdamped Brownian particle that is subject to resetting facilitated by a ratchet potential on a spatially periodic domain. This asymmetric potential switches on with a constant rate, but switches off again only upon the particle's first passage to a resetting point at the minimum of the potential. Repeating this cycle sustains a non-equilibrium steady-state, as well as a directed steady-state current which can be harnessed to perform useful work. We derive exact analytic expressions for the probability densities of the free-diffusion and resetting phases, the associated currents for each phase, and an efficiency parameter that quantifies the return in current for given power input. These expressions allow us to fully characterise the system and obtain experimentally relevant results such as the optimal current and efficiency. Our results are corroborated by simulations, and have implications for experimentally viable finite-time resetting protocols. [ABSTRACT FROM AUTHOR]

Partial resetting, whereby a state variable x (t) is reset at random times to a value a x (t) , 0 ⩽ a ⩽ 1 , generalizes conventional resetting by introducing the resetting strength a as a parameter. Partial resetting generates a broad family of non-equilibrium steady states (NESS) that interpolates between the conventional NESS at strong resetting (a = 0) and a Gaussian distribution at weak resetting (a → 1). Here such resetting processes are studied from a thermodynamic perspective, and the mean cost associated with maintaining such NESS are derived. The resetting phase of the dynamics is implemented by a resetting potential Φ (x) that mediates the resets in finite time. By working in an ensemble of trajectories with a fixed number of resets, we study both the steady-state properties of the propagator and its moments. The thermodynamic work needed to sustain the resulting NESS is then investigated. We find that different resetting traps can give rise to rates of work with widely different dependencies on the resetting strength a. Surprisingly, in the case of resets mediated by a harmonic trap with otherwise free diffusive motion, the asymptotic rate of work is insensitive to the value of a. For general anharmonic traps, the asymptotic rate of work can be either increasing or decreasing as a function of the strength a, depending on the degree of anharmonicity. Counter to intuition, the rate of work can therefore in some cases increase as the resetting becomes weaker (a → 1) although the work vanishes at a = 1. Work in the presence of a background potential is also considered. Numerical simulations confirm our findings. [ABSTRACT FROM AUTHOR]

We construct exact ring soliton-like solutions of the cylindrically symmetric (i.e. radial) Gross-Pitaevskii equation with a potential, using the similarity transformation method. Depending on the choice of the allowed free functions, the solutions can take the form of stationary dark or bright rings whose time dependence is in the phase dynamics only, or oscillating and bouncing solutions, related to the second Painlevé transcendent. In each case the potential can be chosen to be time independent. [ABSTRACT FROM AUTHOR]

Mixed quantum–classical spin systems have been proposed in spin chain theory and, more recently, in magnon spintronics. However, current models of quantum–classical dynamics beyond mean-field approximations typically suffer from long-standing consistency issues, and, in some cases, invalidate Heisenberg's uncertainty principle. Here, we present a fully Hamiltonian theory of quantum–classical spin dynamics that appears to be the first to ensure an entire series of consistency properties, including positivity of both the classical and the quantum density, so that Heisenberg's principle is satisfied at all times. We show how this theory may connect to recent energy-balance considerations in measurement theory and we present its Poisson bracket structure explicitly. After focusing on the simpler case of a classical Bloch vector interacting with a quantum spin observable, we illustrate the extension of the model to systems with several spins, and restore the presence of orbital degrees of freedom. [ABSTRACT FROM AUTHOR]

We present a theoretical study of resonant frequencies and spatial correlations of Josephson phases in frustrated arrays of Josephson junctions. Two types of one-dimensional arrays, namely, the diamond and sawtooth chains, are discussed in detail. For these arrays in the linear regime the Josephson phase dynamics is characterized by multiband dispersion relation , and the lowest band becomes completely flat at a critical value of frustration, f   =  f c. In a strongly nonlinear regime such critical value of frustration determines the crossover from non-frustrated (0  <  f   <  f c) to frustrated (f c  <  f   <  1) regimes. The crossover is characterized by the thermodynamic spatial correlation functions of phases on vertices, , i.e. displaying the transition from long- to short-range spatial correlations. We find that higher-order correlations functions, e.g. p   =  2 and p   =  3, restore the long-range behavior deeply in the frustrated regime, . Monte-Carlo simulations of the thermodynamics of frustrated arrays of Josephson junctions are in good agreement with analytical results. We also outline the extension of our results to the case of kagome lattice, prototypical 2D frustrated lattice, and other higher dimensional lattices. [ABSTRACT FROM AUTHOR]

Motivated by Gutzwiller’s semiclassical quantization, in which unstable periodic orbits of low-dimensional deterministic dynamics serve as a WKB ‘skeleton’ for chaotic quantum mechanics, we construct the corresponding deterministic skeleton for infinite-dimensional lattice-discretized scalar field theories. In the field-theoretical formulation, there is no evolution in time, and there is no ‘Lyapunov horizon’; there is only an enumeration of lattice states that contribute to the theory’s partition sum, each a global spatiotemporal solution of system’s deterministic Eulerâ€"Lagrange equations. The reformulation aligns ‘chaos theory’ with the standard solid state, field theory, and statistical mechanics. In a spatiotemporal, crystallographer formulation, the time-periodic orbits of dynamical systems theory are replaced by periodic d -dimensional Bravais cell tilings of spacetime, each weighted by the inverse of its instability, its Hill determinant. Hyperbolic shadowing of large cells by smaller ones ensures that the predictions of the theory are dominated by the smallest Bravais cells. The form of the partition function of a given field theory is determined by the group of its spatiotemporal symmetries, that is, by the space group of its lattice discretization, best studied on its reciprocal lattice. Already one-dimensional lattice discretization is of sufficient interest to be the focus of this paper. In particular, from a spatiotemporal field theory perspective, ‘time’-reversal is a purely crystallographic notion, a reflection point group, leading to a novel, symmetry quotienting perspective of time-reversible theories and associated topological zeta functions. [ABSTRACT FROM AUTHOR]

In the past few years, stochastic resetting has become a subject of immense interest. Most of the theoretical studies so far focused on instantaneous resetting which is, however, a major impediment to practical realisation or experimental verification in the field. This is because in the real world, taking a particle from one place to another requires finite time and thus a generalization of the existing theory to incorporate non-instantaneous resetting is very much in need. In this paper, we propose a method of resetting which involves non-instantaneous returns facilitated by an external confining trap potential U(x) centered at the resetting location. We consider a Brownian particle that starts its random motion from the origin. Upon resetting, the trap is switched on and the particle starts experiencing a force towards the center of the trap which drives it to return to the origin. The return phase ends when the particle makes a first passage to this center. We develop a general framework to study such a set up. Importantly, we observe that the system reaches a non-equilibrium steady state which we analyze in full details for two choices of U(x), namely, (i) linear and (ii) harmonic. Finally, we perform numerical simulations and find an excellent agreement with the theory. The general formalism developed here can be applied to more realistic return protocols opening up a panorama of possibilities for further theoretical and experimental applications. [ABSTRACT FROM AUTHOR]

We investigate non-autonomous solitons in a general coherently coupled nonlinear Schrödinger (CCNLS) system with temporally modulated nonlinearities and with an external harmonic oscillator potential. This general CCNLS system encompasses three distinct types of CCNLS equations that describe the dynamics of beam propagation in an inhomogeneous Kerr-like nonlinear optical medium for different choices of nonlinear polarizations owing to the anisotropy of the medium. We identify a generalized similarity transformation to relate the considered model into the standard integrable homogeneous coupled nonlinear evolution equations with constant nonlinearities, accompanied by a constraint relation expressed in the form of the Riccati equation. With the help of a non-standard Hirota's bilinearization method and exact soliton solutions, we explore the impact of varying nonlinearities and refractive index in the propagation and collisions analytically by reverse engineering. Interestingly, we show the emergence of several modulated solitonic phenomena such as periodic oscillation, amplification, compression, tunneling/cross-over, excitons, as well as their combined effect in the single-soliton propagation and two-soliton collisions with appropriate forms of nonlinearity. Notably, we identify a tool to transform the nature of soliton collisions with certain type of inhomogeneous nonlinearities. The results could be of significant interest to the studies on management of nonlinear waves in the contexts like nonlinear optics and can also be extended to Bose–Einstein condensates and super-fluids. [ABSTRACT FROM AUTHOR]

This paper describes the effect of quenched disorder on the synchronization properties of a class of circuits known as Josephson junction arrays. The methodology extends the ‘dynamical systems approach’ used to analyze identical-element arrays to a ‘statistical mechanics approach’ to incorporate disorder, a topic of great importance from the perspective of device applications. The essential step is to map the circuit equations onto a version of the fabled Kuramoto model, whose tractability allows one to quantify the system-wide quality of synchronization. [ABSTRACT FROM AUTHOR]

We develop a linear response theory by computing the asymptotic value of the order parameter from the linearized equation of continuity around the nonsynchronized reference state using the Laplace transform in time. The proposed theory is applicable to a wide class of coupled phase oscillator systems and allows for any coupling functions, any natural frequency distributions, any phase-lag parameters, and any values for the time-delay parameter. This generality is in contrast to the limitation of the previous methods of the Ott–Antonsen ansatz and the self-consistent equation for an order parameter, which are restricted to a model family whose coupling function consists of only a single sinusoidal function. The theory is verified by numerical simulations. [ABSTRACT FROM AUTHOR]

We study an extension of the Winfree model of coupled phase oscillators in which both natural frequencies and phase-response curves (PRCs) are heterogeneous. In the first part of the paper we resort to averaging and derive an approximate model, in which the oscillators are coupled through their phase differences. Remarkably, this simplified model is the ‘Kuramoto model with distributed shear’ (Montbrió and Pazó 2011 Phys. Rev. Lett. 106 254101). Using this approximation, we find that above a critical level of PRC heterogeneity the incoherent state is always stable. In the second part of the paper we perform the analysis of the full model for Lorentzian heterogeneities, resorting to the Ott–Antonsen ansatz. As expected, the results obtained using the full model are fully consistent with those obtained using the averaging approximation. However, we find that the critical level of PRC heterogeneity obtained within the averaging approximation has a different manifestation in the full model depending on the sign of the center of the distribution of PRCs. [ABSTRACT FROM AUTHOR]

We develop a geometric approach to quantum mechanics based on the concept of the Tulczyjew triple. Our approach is genuinely infinite-dimensional, i.e. we do not restrict considerations to finite-dimensional Hilbert spaces, contrary to many other works on the geometry of quantum mechanics, and include a Lagrangian formalism in which self-adjoint (Schrödinger) operators are obtained as Lagrangian submanifolds associated with the Lagrangian. As a byproduct we also obtain results concerning coadjoint orbits of the unitary group in infinite dimensions, embedding of pure states in the unitary group, and self-adjoint extensions of symmetric relations. [ABSTRACT FROM AUTHOR]

We survey some applications of SU(2) covariant maps to the phase space quantum mechanics of systems with fixed or variable spin. A generalization to SU(3) symmetry is also briefly discussed in framework of the axiomatic Stratonovich–Weyl formulation. [ABSTRACT FROM AUTHOR]

Real-world systems can be strongly influenced by time delays occurring in self-coupling interactions, due to unavoidable finite signal propagation velocities. When the delays become significantly long, complicated high-dimensional phenomena appear and a simple extension of the methods employed in low-dimensional dynamical systems is not feasible. We review the general theory developed in this case, describing the main destabilization mechanisms, the use of visualization tools, and commenting on the most important and effective dynamical indicators as well as their properties in different regimes. We show how a suitable approach, based on a comparison with spatio-temporal systems, represents a powerful instrument for disclosing the very basic mechanism of long-delay systems. Various examples from different models and a series of recent experiments are reported. [ABSTRACT FROM AUTHOR]

We take a soliton solution of a classical non-linear integrable equation and quench (suddenly change) its non-linearity parameter. For that we multiply the amplitude or the width of a soliton by a numerical factor η and take the obtained profile as a new initial condition. We find the values of η for which the post-quench solution consists of only a finite number of solitons. The parameters of these solitons are found explicitly. Our approach is based on solving the direct scattering problem analytically. We demonstrate how it works for Korteweg–de Vries, sine-Gordon and non-linear Schrödinger integrable equations. [ABSTRACT FROM AUTHOR]

We study the nonlinear dynamics of globally coupled nonidentical oscillators in the framework of two order parameter (mean field and amplitude-frequency correlator) reduction. The main result of the paper is the exact solution of a corresponding nonlinear system on a two-dimensional invariant manifold. We present a complete classification of phase portraits and bifurcations, obtain explicit expressions for invariant manifolds (a limit cycle among them) and derive analytical solutions for arbitrary initial data and different regimes. [ABSTRACT FROM AUTHOR]

A comprehensive theory describing spin dynamics in a time-dependent magnetic field is constructed on the basis of a Maurer–Cartan (MC) form selected following the exponential Wei–Norman ordering technique. We demonstrate that the purely diffusive SU(2) problem described by a MC form yields the Fokker–Planck equation for the probability density function. The Fokker–Planck-like operator is observed to correspond to the Casimir operator. The case of a multichannel Coulomb model is considered for illustration. [ABSTRACT FROM AUTHOR]

We study the asymptotic behavior of an ensemble of identical Lohe oscillators on the unit sphere in the presence of small time delay interaction effects. When there is no time delay, the ensemble of identical Lohe oscillators collapses asymptotically to a one-cluster ensemble on the sphere; its asymptotic dynamics are governed by linear motion on the unit sphere with a constant natural velocity. We show that the presence of a small time delay can induce rich dynamical features such as asymptotic changes in the velocity and asymptotic low-dimensional dynamics in high-dimensional cases. For d = 1, the Lohe dynamics is equivalent to the Kuramoto dynamics via polar coordinates. In this case, the modified asymptotic frequency is uniquely determined by an implicit relation based on the natural frequency, coupling strength, and time delay. For d = 3, we show that the dynamics of identical Lohe oscillators converges to the Kuramoto dynamics for properly chosen initial configurations. We also provide several numerical simulations to confirm our analytical results. [ABSTRACT FROM AUTHOR]

In this paper we develop a geometric approach to higher order mechanics on graded bundles in both, the Lagrangian and Hamiltonian formalism, via the recently discovered weighted algebroids. We present the corresponding Tulczyjew triple for this higher order situation and derive in this framework the phase equations from an arbitrary (also singular) Lagrangian or Hamiltonian, as well as the Euler–Lagrange equations. As important examples, we geometrically derive the classical higher order Euler–Lagrange equations and analogous reduced equations for invariant higher order Lagrangians on Lie groupoids. [ABSTRACT FROM AUTHOR]

Real space condensation is known to occur in stochastic models of mass transport in the regime in which the globally conserved mass density is greater than a critical value. It has been shown within models with factorized stationary states that the condensation can be understood in terms of sums of independent and identically distributed random variables: these exhibit condensation when they are conditioned to a large deviation of their sum. It is well understood that the condensation, whereby one of the random variables contributes a finite fraction to the sum, occurs only if the underlying probability distribution (modulo exponential) is heavy-tailed, i.e. decaying slower than exponential. Here we study a similar phenomenon in which condensation is exhibited for non-heavy-tailed distributions, provided random variables are additionally conditioned on a large deviation of certain linear statistics. We provide a detailed theoretical analysis explaining the phenomenon, which is supported by Monte Carlo simulations (for the case where the additional constraint is the sample variance) and demonstrated in several physical systems. Our results suggest that the condensation is a generic phenomenon that pertains to both typical and rare events. [ABSTRACT FROM AUTHOR]