In the context of the Kuramoto model of coupled oscillators with distributed natural frequencies ... more In the context of the Kuramoto model of coupled oscillators with distributed natural frequencies interacting through a time-delayed mean-field, we derive as a function of the delay exact results for the stability boundary between the incoherent and the synchronized state and the nature in which the latter bifurcates from the former at the critical point. Our results are based on an unstable manifold expansion in the vicinity of the bifurcation, which we apply to both the kinetic equation for the single-oscillator distribution function in the case of a generic frequency distribution and the corresponding Ott-Antonsen(OA)-reduced dynamics in the special case of a Lorentzian distribution. Besides elucidating the effects of delay on the nature of bifurcation, we show that the approach due to Ott and Antonsen, although an ansatz, gives an amplitude dynamics of the unstable modes close to the bifurcation that remarkably coincides with the one derived from the kinetic equation. Further more, quite interestingly and remarkably, we show that close to the bifurcation, the unstable manifold derived from the kinetic equation has the same form as the OA manifold, implying thereby that the OA-ansatz form follows also as a result of the unstable manifold expansion. We illustrate our results by showing how delay can affect dramatically the bifurcation of a bimodal distribution.
Journal of Statistical Mechanics: Theory and Experiment, 2010
Long-range interacting systems, while relaxing towards equilibrium, may get trapped in nonequilib... more Long-range interacting systems, while relaxing towards equilibrium, may get trapped in nonequilibrium quasistationary states (QSS) for a time which diverges algebraically with the system size. These intriguing non-Boltzmann states have been observed under deterministic Hamiltonian evolution of a paradigmatic system, the Hamiltonian Mean-Field (HMF) model. We study here the robustness of QSS with respect to stochastic processes beyond deterministic dynamics within a microcanonical ensemble. To this end, we generalize the HMF model by allowing for stochastic three-particle collision dynamics in addition to the deterministic ones. By analyzing the resulting Boltzmann equation for the phase space density, we demonstrate that in the presence of stochasticity, QSS occur only as a crossover phenomenon over a finite time determined by the strength of the stochastic process. In particular, we argue that the relaxation time to equilibrium does not scale algebraically with the system size. We propose a scaling form for the relaxation time which is in very good agreement with results of extensive numerical simulations. The broader validity of these results is tested on a different stochastic HMF model involving microcanonical Monte Carlo dynamical moves.
Systems with long-range interactions, while relaxing towards equilibrium, sometimes get trapped i... more Systems with long-range interactions, while relaxing towards equilibrium, sometimes get trapped in long-lived non-Boltzmann quasistationary states (QSS) which have lifetimes that grow algebraically with the system size. Such states have been observed in models of globally coupled particles that move under Hamiltonian dynamics either on a unit circle or on a unit spherical surface. Here, we address the ubiquity of QSS in longrange systems by considering a different dynamical setting. Thus, we consider an anisotropic Heisenberg model consisting of classical Heisenberg spins with mean-field interactions and evolving under classical spin dynamics. Our analysis of the corresponding Vlasov equation for time evolution of the phase space distribution shows that in a certain energy interval, relaxation of a class of initial states occurs over a timescale which grows algebraically with the system size. We support these findings by extensive numerical simulations. This work further supports the generality of occurrence of QSS in long-range systems evolving under Hamiltonian dynamics.
A paradigmatic framework to study the phenomenon of spontaneous collective synchronization is pro... more A paradigmatic framework to study the phenomenon of spontaneous collective synchronization is provided by the Kuramoto model comprising a large collection of limit-cycle oscillators of distributed frequencies that are globally coupled through the sine of their phase differences. We study here a variation of the model by including nearest-neighbor interactions on a one-dimensional lattice. While the mean-field interaction resulting from the global coupling favors global synchrony, the nearest-neighbor interaction may have cooperative or competitive effects depending on the sign and the magnitude of the nearest-neighbor coupling. For unimodal and symmetric frequency distributions, we demonstrate that as a result, the model in the stationary state exhibits in contrast to the usual Kuramoto model both continuous and first-order transitions between synchronized and incoherent phases, with the transition lines meeting at a tricritical point. Our results are based on numerical integration of the dynamics as well as an approximate theory involving appropriate averaging of fluctuations in the stationary state.
Journal of Physics A: Mathematical and Theoretical, 2018
We obtain exact results on autocorrelation of the order parameter in the nonequilibrium stationar... more We obtain exact results on autocorrelation of the order parameter in the nonequilibrium stationary state of a paradigmatic model of spontaneous collective synchronization, the Kuramoto model of coupled oscillators, evolving in presence of Gaussian, white noise. The method relies on an exact mapping of the stationarystate dynamics of the model in the thermodynamic limit to the noisy dynamics of a single, non-uniform oscillator, and allows to obtain besides the Kuramoto model the autocorrelation in the equilibrium stationary state of a related model of longrange interactions, the Brownian mean-field model. Both the models show a phase transition between a synchronized and an incoherent phase at a critical value of the noise strength. Our results indicate that in the two phases as well as at the critical point, the autocorrelation for both the model decays as an exponential with a rate that increases continuously with the noise strength.
For mean-field classical spin systems exhibiting a second-order phase transition in the stationar... more For mean-field classical spin systems exhibiting a second-order phase transition in the stationary state, we obtain within the corresponding phase space evolution according to the Vlasov equation the values of the critical exponents describing power-law behavior of response to a small external field. The exponent values so obtained significantly differ from the ones obtained on the basis of an analysis of the static phase-space distribution, with no reference to dynamics. This work serves as an illustration that cautions against relying on a static approach, with no reference to the dynamical evolution, to extract critical exponent values for mean-field systems.
Journal of Statistical Mechanics: Theory and Experiment, 2019
For a model long-range interacting system of classical Heisenberg spins, we study how fluctuation... more For a model long-range interacting system of classical Heisenberg spins, we study how fluctuations, such as those arising from having a finite system size or through interaction with the environment, affect the dynamical process of relaxation to Boltzmann-Gibbs equilibrium. Under deterministic spin precessional dynamics, we unveil the full range of quasistationary behavior observed during relaxation to equilibrium, whereby the system is trapped in nonequilibrium states for times that diverge with the system size. The corresponding stochastic dynamics, modeling interaction with the environment and constructed in the spirit of the stochastic Landau-Lifshitz-Gilbert equation, however shows a fast relaxation to equilibrium on a sizeindependent timescale and no signature of quasistationarity, provided the noise is strong enough. Similar fast relaxation is also seen in Glauber Monte Carlo dynamics of the model, thus establishing the ubiquity of what has been reported earlier in particle dynamics (hence distinct from the spin dynamics considered here) of longrange interacting systems, that quasistationarity observed in deterministic dynamics is washed away by fluctuations induced through contact with the environment.
Due to an unfortunate error occurred during production, two references are wrongly quoted within ... more Due to an unfortunate error occurred during production, two references are wrongly quoted within the text, i.e., ref. [9] should be ref. [7] on page 3, left column, line 3 and on page 4 on the line above eq. ( ). We deeply apologize to the authors for the unwanted mistake.
Monthly Notices of the Royal Astronomical Society, 2017
Observations strongly suggest that filaments in galactic molecular clouds are in a non-thermal st... more Observations strongly suggest that filaments in galactic molecular clouds are in a non-thermal state. As a simple model of a filament, we study a two-dimensional system of self-gravitating point particles by means of numerical simulations of the dynamics, with various methods: direct N-body integration of the equations of motion, particle-in-cell simulations, and a recently developed numerical scheme that includes multiparticle collisions in a particle-in-cell approach. Studying the collapse of Gaussian overdensities, we find that after the damping of virial oscillations the system settles in a non-thermal steady state whose radial density profile is similar to the observed ones, thus suggesting a dynamical origin of the non-thermal states observed in real filaments. Moreover, for sufficiently cold collapses, the density profiles are anticorrelated with the kinetic temperature, i.e. exhibit temperature inversion, again a feature that has been found in some observations of filaments. The same happens in the state reached after a strong perturbation of an initially isothermal cylinder. Finally, we discuss our results in the light of recent findings in other contexts (including non-astrophysical ones) and argue that the same kind of non-thermal states may be observed in any physical system with long-range interactions.
For a many-particle system with long-range interactions and evolving under stochastic dynamics, w... more For a many-particle system with long-range interactions and evolving under stochastic dynamics, we study for the first time the out-of-equilibrium fluctuations of the work done on the system by a time-dependent external force. For equilibrium initial conditions, the work distributions for a given protocol of variation of the force in time and the corresponding time-reversed protocol exhibit a remarkable scaling and a symmetry when expressed in terms of the average and the standard deviation of the work. The distributions of the work per particle predict, by virtue of the Crooks fluctuation theorem, the equilibrium free-energy density of the system. For a large number N of particles, the latter is in excellent agreement with the value computed by considering the Langevin dynamics of a single particle in a self-consistent mean field generated by its interaction with other particles. The agreement highlights the effective mean-field nature of the original many-particle dynamics for large N. For initial conditions in non-equilibrium stationary states (NESSs), we study the distribution of a quantity similar to dissipated work that satisfies the non-equilibrium generalization of the Clausius inequality, namely, the Hatano-Sasa equality, for transitions between NESSs. Besides illustrating the validity of the equality, we show that the distribution has exponential tails that decay differently on the left and on the right.
We study the dynamics of overdamped Brownian particles diffusing in conservative force fields and... more We study the dynamics of overdamped Brownian particles diffusing in conservative force fields and undergoing stochastic resetting to a given location with a generic space-dependent rate of resetting. We present a systematic approach involving path integrals and elements of renewal theory that allows to derive analytical expressions for a variety of statistics of the dynamics such as (i) the propagator prior to first reset; (ii) the distribution of the first-reset time, and (iii) the spatial distribution of the particle at long times. We apply our approach to several representative and hitherto unexplored examples of resetting dynamics. A particularly interesting example for which we find analytical expressions for the statistics of resetting is that of a Brownian particle trapped in a harmonic potential with a rate of resetting that depends on the instantaneous energy of the particle. We find that using energy-dependent resetting processes is more effective in achieving spatial confinement of Brownian particles on a faster timescale than by performing quenches of parameters of the harmonic potential.
In recent years, studies of long-range interacting (LRI) systems have taken center stage in the a... more In recent years, studies of long-range interacting (LRI) systems have taken center stage in the arena of statistical mechanics and dynamical system studies, due to new theoretical developments involving tools from as diverse a field as kinetic theory, non-equilibrium statistical mechanics, and large deviation theory, but also due to new and exciting experimental realizations of LRI systems. In the first, introductory, Section 1, we discuss the general features of long-range interactions, emphasizing in particular the main physical phenomenon of non-additivity, which leads to a plethora of distinct effects, both thermodynamic and dynamic, that are not observed with short-range interactions: Ensemble inequivalence, slow relaxation, broken ergodicity. In Section 2, we discuss several physical systems with long-range interactions: mean-field spin systems, self-gravitating systems, Euler equations in two dimensions, Coulomb systems, one-component electron plasma, dipolar systems, free-el...
Journal of Physics A: Mathematical and Theoretical, 2017
In the context of the celebrated Kuramoto model of globally-coupled phase oscillators of distribu... more In the context of the celebrated Kuramoto model of globally-coupled phase oscillators of distributed natural frequencies, which serves as a paradigm to investigate spontaneous collective synchronization in many-body interacting systems, we report on a very rich phase diagram in presence of thermal noise and an additional non-local interaction on a one-dimensional periodic lattice. Remarkably, the phase diagram involves both equilibrium and non-equilibrium phase transitions. In two contrasting limits of the dynamics, we obtain exact analytical results for the phase transitions. These two limits correspond to (i) the absence of thermal noise, when the dynamics reduces to that of a non-linear dynamical system, and (ii) the oscillators having the same natural frequency, when the dynamics becomes that of a statistical system in contact with a heat bath and relaxing to a statistical equilibrium state. In the former case, our exact analysis is based on the use of the so-called Ott-Antonsen ansatz to derive a reduced set of nonlinear partial differential equations for the macroscopic evolution of the system. Our results for the case of statistical equilibrium are on the other hand obtained by extending the well-known transfer matrix approach for nearestneighbor Ising model to consider non-local interactions. The work offers a case study of exact analysis in many-body interacting systems. The results obtained underline the crucial role of additional non-local interactions in either destroying or enhancing the possibility of observing synchrony in mean-field systems exhibiting spontaneous synchronization.
What happens when one of the parameters governing the dynamics of a long-range interacting system... more What happens when one of the parameters governing the dynamics of a long-range interacting system of particles in thermal equilibrium is abruptly changed (quenched) to a different value? While a short-range system, under the same conditions, will relax in time to a new thermal equilibrium with a uniform temperature across the system, a long-range system shows a fast relaxation to a nonequilibrium quasistationary state (QSS). The lifetime of such an off-equilibrium state diverges with the system size, and the temperature is non-uniform across the system. Quite surprisingly, the density profile in the QSS obtained after the quench is anticorrelated with the temperature profile in space, thus exhibiting the phenomenon of temperature inversion: denser regions are colder than sparser ones. We illustrate with extensive molecular dynamics simulations the ubiquity of this scenario in a prototypical longrange interacting system subject to a variety of quenching protocols, and in a model that mimics an experimental setup of atoms interacting with light in an optical cavity. We further demonstrate how a procedure of iterative quenching combined with filtering out the high-energy particles in the system may be employed to cool the system. Temperature inversion is observed in nature in some astrophysical settings; our results imply that such a phenomenon should be observable, and could even be exploitable to advantage, also in controlled laboratory experiments.
Quantum measurements are crucial to observe the properties of a quantum system, which however una... more Quantum measurements are crucial to observe the properties of a quantum system, which however unavoidably perturb its state and dynamics in an irreversible way. Here we study the dynamics of a quantum system while being subject to a sequence of projective measurements applied at random times. In the case of independent and identically distributed intervals of time between consecutive measurements, we analytically demonstrate that the survival probability of the system to remain in the projected state assumes a large-deviation (exponentially decaying) form in the limit of an infinite number of measurements. This allows us to estimate the typical value of the survival probability, which can therefore be tuned by controlling the probability distribution of the random time intervals. Our analytical results are numerically tested for Zeno-protected entangled states, which also demonstrates that the presence of disorder in the measurement sequence further enhances the survival probability when the Zeno limit is not reached (as it happens in experiments). Our studies provide a new tool for protecting and controlling the amount of quantum coherence in open complex quantum systems by means of tunable stochastic measurements.
We study a generic model of globally coupled rotors that includes the effects of noise, phase shi... more We study a generic model of globally coupled rotors that includes the effects of noise, phase shift in the coupling, and distributions of moments of inertia and natural frequencies of oscillation. As particular cases, the setup includes previously studied Sakaguchi-Kuramoto, Hamiltonian and Brownian mean-field, and Tanaka-Lichtenberg-Oishi and Acebrón-Bonilla-Spigler models. We derive an exact solution of the self-consistent equations for the order parameter in the stationary state, valid for arbitrary parameters in the dynamics, and demonstrate nontrivial phase transitions to synchrony that include reentrant synchronous regimes.
Journal of Statistical Mechanics: Theory and Experiment, 2015
We present a novel method to compute the phase space distribution in the nonequilibrium stationar... more We present a novel method to compute the phase space distribution in the nonequilibrium stationary state of a wide class of mean-field systems involving rotators subject to quenched disordered external drive and dissipation. The method involves a series expansion of the stationary distribution in inverse of the damping coefficient; the expansion coefficients satisfy recursion relations whose solution requires computing a matrix where about three quarters of the elements vanish, making numerical evaluation simple and efficient. We illustrate our method for the paradigmatic Kuramoto model of spontaneous collective synchronization and for its two mode generalization, in presence of noise and inertia, and demonstrate an excellent agreement between simulations and theory for the phase space distribution.
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Papers by Shamik Gupta