Core-halo distribution in the Hamiltonian mean-field model

We study a paradigmatic system with long-range interactions: the Hamiltonian mean-field (HMF) model. It is shown that in the thermodynamic limit this model does not relax to the usual equilibrium Maxwell-Boltzmann distribution. Instead, the final stationary state has a peculiar core-halo structure. In the thermodynamic limit, HMF is neither ergodic nor mixing. Nevertheless, we find that using dynamical properties of Hamiltonian systems it is possible to quantitatively predict both the spin distribution and the velocity distribution functions in the final stationary state, without any adjustable parameters. We also show that HMF undergoes a nonequilibrium first-order phase transition between paramagnetic and ferromagnetic states.     

Phase transitions of quasistationary states in the Hamiltonian Mean Field model

The out-of equilibrium dynamics of the Hamiltonian Mean Field (HMF) model is studied in presence of an externally imposed magnetic field h. Lynden-Bell’s theory of violent relaxation is revisited and shown to adequately capture the system dynamics, as revealed by direct Vlasov based numerical simulations in the limit of vanishing field. This includes the existence of an out-of-equilibrium phase transition separating magnetized and non magnetized phases. We also monitor the fluctuations in time of the magnetization, which allows us to elaborate on the choice of the correct order parameter when challenging the performance of Lynden-Bell’s theory. The presence of the field h removes the phase transition, as it happens at equilibrium. Moreover, regions with negative susceptibility are numerically found to occur, in agreement with the predictions of the theory.     

Dynamics and thermodynamics of a simple model similar to self-gravitating systems: the HMF model

We discuss the dynamics and thermodynamics of the Hamiltonian Mean Field model (HMF) which is a prototypical system with long-range interactions. The HMF model can be seen as the one Fourier component of a one-dimensional self-gravitating system. Interestingly, it exhibits many features of real self-gravitating systems (violent relaxation, persistence of metaequilibrium states, slow collisional dynamics, phase transitions,...) while avoiding complicated problems posed by the singularity of the gravitational potential at short distances and by the absence of a large-scale confinement. We stress the deep analogy between the HMF model and self-gravitating systems by developing a complete parallel between these two systems. This allows us to apply many technics introduced in plasma physics and astrophysics to a new problem and to see how the results depend on the dimension of space and on the form of the potential of interaction. This comparative study brings new light in the statistical mechanics of self-gravitating systems. We also mention simple astrophysical applications of the HMF model in relation with the formation of bars in spiral galaxies.     

Metastability in the Hamiltonian mean field model and Kuramoto model

We briefly discuss the state of the art on the anomalous dynamics of the Hamiltonian mean field (HMF) model. We stress the important role of the initial conditions for understanding the microscopic nature of the intriguing metastable quasi-stationary states (QSS) observed in the model and the connections to Tsallis statistics and glassy dynamics. We also present new results on the existence of metastable states in the Kuramoto model and discuss the similarities with those found in the HMF model. The existence of metastability seems to be quite a common phenomenon in fully coupled systems, whose origin could be also interpreted as a dynamical mechanism preventing or hindering synchronization.     

Nonlinear Instability of Inhomogeneous Steady States Solutions to the HMF Model

Journal of Statistical Physics - In this work we prove the nonlinear instability of inhomogeneous steady states solutions to the Hamiltonian mean field (HMF) model. We first study the linear...     

Dynamical stability of systems with long-range interactions: application of the Nyquist method to the HMF model

We apply the Nyquist method to the Hamiltonian mean field (HMF) model in order to settle the linear dynamical stability of a spatially homogeneous distribution function with respect to the Vlasov equation. We consider the case of Maxwell (isothermal) and Tsallis (polytropic) distributions and show that the system is stable above a critical kinetic temperature T_c and unstable below it. Then, we consider a symmetric double-humped distribution, made of the superposition of two decentered Maxwellians, and show the existence of a re-entrant phase in the stability diagram. When we consider an asymmetric double-humped distribution, the re-entrant phase disappears above a critical value of the asymmetry factor Δ > 1.09. We also consider the HMF model with a repulsive interaction. In that case, single-humped distributions are always stable. For asymmetric double-humped distributions, there is a re-entrant phase for 1 ≤ Δ < 25.6, a double re-entrant phase for 25.6 < Δ < 43.9 and no re-entrant phase for Δ > 43.9. Finally, we extend our results to arbitrary potentials of interaction and mention the connexion between the HMF model, Coulombian plasmas and gravitational systems. We discuss the relation between linear dynamical stability and formal nonlinear dynamical stability and show their equivalence for spatially homogeneous distributions. We also provide a criterion of dynamical stability for spatially inhomogeneous systems.     

Dynamics and thermodynamics of rotators interacting with both long- and short-range couplings

The effect of nearest-neighbor coupling on the thermodynamic and dynamical properties of the ferromagnetic Hamiltonian mean field model (HMF) is studied. For a range of antiferromagnetic nearest-neighbor coupling, a canonical first-order transition is observed, and the canonical and microcanonical ensembles are non-equivalent. In studying the relaxation time of non-equilibrium states it is found that as in the HMF model, a class of non-magnetic states is quasi-stationary, with an algebraic divergence of their lifetime with the number of degrees of freedom N. The lifetime of metastable states is found to increase exponentially with N as expected.     

Nonlinear coupling and relaxation of fluctuations and the central peak

A simple model Hamiltonian for a structural phase transition is discussed. It is shown that using the modecular field approximation one obtains a nonlinear coupling of the order parameter and entropy fluctuations. This results in a central peak in the order parameter fluctuation spectrum. The width of the central peak is proportional to the entropy relaxation rate.     

Local atomic arrangement in mechanosynthesized Co x Fe1−x−y Ni y alloys studied by Mössbauer spectroscopy

Mechanosynthesized Co _x Fe_1?x?y Ni _y alloys were examined using X-ray diffraction (XRD) and Mössbauer spectroscopy. In order to explain the shape of hyperfine magnetic field (HMF) distributions for the alloys, a local environment model based on a multinomial distribution was proposed. The model was in agreement with the XRD data and confirmed that the studied alloys were disordered solid solutions. It was successfully applied to describe the samples with bcc and fcc crystalline lattice type within the relatively broad range of components concentration. The results showed that the change of the crystalline lattice type does not cause an abrupt change of the HMF value. Moreover, a mean number of unpaired spins for the first coordination sphere may be used as a parameter to describe the HMF value experienced by ⁵⁷Fe nucleus. Finally, a set of the most probable atomic configurations and their corresponding contributions to the HMF distribution were obtained.     

Space Dependent Mean Field Approximation Modelling

It is shown that the self-consistency condition which is the basic equation for calculating the mean-field order parameter of any mean-field model Hamiltonian can be replaced by the standard Metropolis Monte Carlo scheme. The advantage of this method is its ease of implementation for both the homogeneous mean-field order parameter and the heterogeneous one. To be specific, the mean-field version of the Ising model spin system is discussed in detail and the resulting magnetization is the same as in the case of solving the respective mean-field self-consistency equation. In addition, it is shown that if a high temperature phase of such system is quenched below critical temperature then the mean field experienced by spins develops into a network of domains in analogous way as it happens with the spins in the case of the exact many-body Hamiltonian system and the coarsening processes start to take place. To show that the introduced Metropolis Monte Carlo method works also in case of the continuous variables the order parameter for the Maier-Saupe model for nematic liquid crystals has been calculated.     

On a pathology in indefinite metric inner product space

A pathology related to an indefinite metric, which has been pointed out by Ito in connection with construction of a two dimensional quantum field model at a finite cutoff, is mathematically analyzed in a simple model. It is found for a model Hamiltonian with a parameter in an indefinite metric inner product space that eigenvalues with a complete set of eigenvectors changes suddenly from positive integers to negative integers as a parameter crosses a critical value (the Hamiltonian being skew selfadjoint with absolutely continuous spectrum on a pure imaginary axis at the critical value of the parameter), if a fixed (positive definite Hilbert space) topology is used in the completion of the underlying indefinite metric inner product space. However it is also found that if the topology is varied with the parameter of the Hamiltonian in the manner similar to analytic continuation, then the Hamiltonian keeps positive integer eigenvalues with a complete set of eigenvectors.     

Inhomogeneous Tsallis distributions in the HMF model

We study the maximization of the Tsallis functional at fixed mass and energy in the Hamiltonian Mean Field (HMF) model. We give a thermodynamical and a dynamical interpretation of this variational principle. This leads to q-distributions known as stellar polytropes in astrophysics. We study phase transitions between spatially homogeneous and spatially inhomogeneous equilibrium states. We show that there exists a particular index q _c = 3 playing the role of a canonical tricritical point separating first and second order phase transitions in the canonical ensemble and marking the occurence of a negative specific heat region in the microcanonical ensemble. We apply our results to the situation considered by Antoni and Ruffo [Phys. Rev. E 52, 2361 (1995)] and show that the anomaly displayed on their caloric curve can be explained naturally by assuming that, in this region, the QSSs are polytropes with critical index q _c = 3. We qualitatively justify the occurrence of polytropic (Tsallis) distributions with compact support in terms of incomplete relaxation and inefficient mixing (non-ergodicity). Our paper provides an exhaustive study of polytropic distributions in the HMF model and the first plausible explanation of the surprising result observed numerically by Antoni and Ruffo (1995). In the course of our analysis, we also report an interesting situation where the caloric curve presents both microcanonical first and second order phase transitions.     

General linear response formula for non integrable systems obeying the Vlasov equation

Long-range interacting N-particle systems get trapped into long-living out-of-equilibrium stationary states called quasi-stationary states (QSS). We study here the response to a small external perturbation when such systems are settled into a QSS. In the N → ∞ limit the system is described by the Vlasov equation and QSS are mapped into stable stationary solutions of such equation. We consider this problem in the context of a model that has recently attracted considerable attention, the Hamiltonian mean field (HMF) model. For such a model, stationary inhomogeneous and homogeneous states determine an integrable dynamics in the mean-field effective potential and an action-angle transformation allows one to derive an exact linear response formula. However, such a result would be of limited interest if restricted to the integrable case. In this paper, we show how to derive a general linear response formula which does not use integrability as a requirement. The presence of conservation laws (mass, energy, momentum, etc.) and of further Casimir invariants can be imposed a posteriori. We perform an analysis of the infinite time asymptotics of the response formula for a specific observable, the magnetization in the HMF model, as a result of the application of an external magnetic field, for two stationary stable distributions: the Boltzmann-Gibbs equilibrium distribution and the Fermi-Dirac one. When compared with numerical simulations the predictions of the theory are very good away from the transition energy from inhomogeneous to homogeneous states.     

Curious behaviour of the diffusion coefficient and friction force for the strongly inhomogeneous HMF model

We present first elements of kinetic theory appropriate to the inhomogeneous phase of the Hamiltonian Mean Field (HMF) model. In particular, we investigate the case of strongly inhomogeneous distributions for T→0 and exhibit curious behaviour of the force auto-correlation function and friction coefficient. The temporal correlation function of the force has an oscillatory behaviour which averages to zero over a period. By contrast, the effects of friction accumulate with time and the friction coefficient does not satisfy the Einstein relation. On the contrary, it presents the peculiarity to increase linearly with time. Motivated by this result, we provide analytical solutions of a simplified kinetic equation with a time dependent friction. Analogies with self-gravitating systems and other systems with long-range interactions are also mentioned.     

Josephson effect between nanoclusters in resonance conditions

A general expression is derived for the Josephson current between nanoclusters. It is shown that, in the resonance conditions between electron levels of clusters, the expression for the current obtained in the tunnel Hamiltonian model becomes invalid. In the case of degeneracy or close to degeneracy of energy levels in isolated clusters, the critical Josephson current may exceed the value obtained in the model of tunnel Hamiltonian in the large parameter, viz., the ratio of the order parameter |Δ| to the distance between the resonance level and the levels closest to it.     

Similarity renormalization of the electron-phonon coupling

We study the problem of the phonon-induced electron-electron interaction in a solid. Starting with a Hamiltonian that contains an electron-phonon interaction, we perform a similarity renormalization transformation to calculate an effective Hamiltonian. Using this transformation singularities due to degeneracies are avoided explicitly. The effective interactions are calculated to second order in the electronphonon coupling. It is shown that the effective interaction between two electrons forming a Cooper pair is attractive in the whole parameter space. For a simple Einstein model we calculate the renormalization of the electronic energies and the critical temperature of superconductivity.     

Lack of phase transition in the Dicke model with external fields

We show that the addition of external fields to the Dicke Hamiltonian removes the critical behaviour of the model. We analyse this result in terms of order parameter and conjugated field and compare with recent papers.     

Monte-Carlo study of the Maier-Saupe model on square and triangle lattices

The Monte-Carlo method is used to investigate the statistical physics of the Maier-Saupe model on square and triangle lattices. These systems are found to exhibit higher than first order phase transitions except for the negative coupling square lattice case. The transition is signaled by a pronounced maximum in the specific heat as a function of temperature. The mechanism of the transition is shown to be associated with the onset of partial ordering of the vectors in individual members of the ensemble. A temperature dependent internal order parameter is introduced to characterize the degree of internal order in the system since the ordering process does not break the rotational symmetry of the Hamiltonian and no macroscopic order parameter exists.     

Generalization of the Hellmann-Feynman theorem

The well-known Hellmann-Feynman theorem of quantum mechanics connected with the derivative of the eigenvalues with respect to a parameter upon which the Hamiltonian depends, is generalized to include cases in which the domain of definition of the Hamiltonian of the system also depends on that parameter.     

A note on the Holst action, the time gauge, and the Barbero–Immirzi parameter

In this note, we review the canonical analysis of the Holst action in the time gauge, with a special emphasis on the Hamiltonian equations of motion and the fixation of the Lagrange multipliers. This enables us to identify at the Hamiltonian level the various components of the covariant torsion tensor, which have to be vanishing in order for the classical theory not to depend upon the Barbero–Immirzi parameter. We also introduce a formulation of three-dimensional gravity with an explicit phase space dependency on the Barbero–Immirzi parameter as a potential way to investigate its fate and relevance in the quantum theory.