Effective spin-glass Hamiltonian for the anomalous dynamics of the HMF model

We discuss an effective spin-glass Hamiltonian which can be used to study the glassy-like dynamics observed in the metastable states of the Hamiltonian mean field (HMF) model. By means of the Replica formalism, we were able to find a self-consistent equation for the glassy order parameter which reproduces, in a restricted energy region below the phase transition, the microcanonical simulations for the polarization order parameter recently introduced in the HMF model.     

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 and the Ising model

We discuss a recent theorem which establishes a precise connection between (i) the approximate degeneracy of the zero eigenvalue for the generator of the Glauber dynamics of the Ising model in a small nonzero field and below the critical temperature, (ii) the existence of a partition of the configuration space into a normal region and a metastable region. This enables us to demonstrate that the recent approach to metastability of Davies and Martin may be viewed as a simple (although in some ways fairly crude) approximation to the conventional approach. We also obtain what appear to be the first results concerning the stability of metastable states under small perturbations.     

Barriers in the p-spin interacting spin-glass model: the dynamical approach

We investigate the barriers separating metastable states in the spherical p-spin glass model using the instanton method. We show that the problem of finding the barrier heights can be reduced to the causal two-real-replica dynamics. We find the probability for the system to escape one of the highest energy metastable states and the energy barrier corresponding to this process.     

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.     

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.     

Thermodynamics of the HMF model with a magnetic field

We study the thermodynamics of the Hamiltonian mean field (HMF) model with an external potential playing the role of a “magnetic field”. If we consider only fully stable states, the caloric curve does not present any phase transition. However, if we take into account metastable states (for a restricted class of perturbations), we find a very rich phenomenology. In particular, the caloric curve displays a region of negative specific heat in the microcanonical ensemble in which the temperature decreases as the energy increases. This leads to ensembles inequivalence and to zeroth order phase transitions similar to the “gravothermal catastrophe” and to the “isothermal collapse” of self-gravitating systems. In the present case, they correspond to the reorganization of the system from an “anti-aligned” phase (magnetization pointing in the direction opposite to the magnetic field) to an “aligned” phase (magnetization pointing in the same direction as the magnetic field). We also find that the magnetic susceptibility can be negative in the microcanonical ensemble so that the magnetization decreases as the magnetic field increases. The magnetic curves can take various shapes depending on the values of energy or temperature. We describe first order phase transitions and hysteretic cycles involving positive or negative susceptibilities. We also show that this model exhibits gaps in the magnetization at fixed energy, resulting in ergodicity breaking.     

The Vlasov equation and the Hamiltonian mean-field model

We show that the quasi-stationary states of homogeneous (zero magnetization) states observed in the N-particle dynamics of the Hamiltonian mean-field (HMF) model are nothing but Vlasov stable homogeneous states. There is an infinity of Vlasov stable homogeneous states corresponding to different initial momentum distributions. Tsallis q-exponentials in momentum, homogeneous in angle, distribution functions are possible, however, they are not special in any respect, among an infinity of others. All Vlasov stable homogeneous states lose their stability because of finite N effects and, after a relaxation time diverging with a power-law of the number of particles, the system converges to the Boltzmann–Gibbs equilibrium.     

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.     

Interfaces and ripple states in ferroelastic crystals—A simple model

A simple atomistic model with two harmonic potentials between layers of atoms describes surface relaxations equivalent to those found as kink solitary waves in ferroelastic crystals. The parameters of the model are expressed in terms of the entropy term in a Landau potential and the Ginzburg gradient energy. We discuss the possibility that metastable ripple states exist in which a modulation is superimposed to an underlying uniform order parameter. Such ripple states can occur at temperatures close to the transition point between a ferroelastic phase and an incommensurate phase. In the ripple state, domain walls consist of kinks with modulations on either side of the kink.     

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.     

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.     

Organization and energy properties of metastable states for the random-field Ising model

Random-field Ising model (RFIM) systems are characterized by a large number of metastable states corresponding to local minima of the system energy with respect to single spin flip. We classified the minima in a hierarchical way based on the possibility of a given state to escape from a basin of mutually reachable states. We investigate the energy properties of the metastable states in relation to the basin they belong to: states of particularly high energy, obtained by fast-quenching randomly initial spin configurations, tend to have access to a complex structure of correlated basins, opposite to what is found for low-energy states. The purpose of this paper is to investigate the connection between the properties of the basin oriented graph and the energy of the corresponding states.     

Relaxation Height in Energy Landscapes: An Application to Multiple Metastable States

The study of systems with multiple (not necessarily degenerate) metastable states presents subtle difficulties from the mathematical point of view related to the variational problem that has to be solved in these cases. We prove sufficient conditions to identify multiple metastable states. Since this analysis typically involves non-trivial technical issues, we give different conditions that can be chosen appropriately depending on the specific model under study. We show how these results can be used to attack the problem of multiple metastable states via the use of the modern approaches to metastability. We finally apply these general results to the Blume–Capel model for a particular choice of the parameters for which the model happens to have two multiple not degenerate in energy metastable states. We estimate in probability the time for the transition from the metastable states to the stable state. Moreover we identify the set of critical configurations that represent the minimal gate for the transition.     

Competition between folding and aggregation in a model for protein solutions

We study the thermodynamic and kinetic consequences of the competition between single-protein folding and protein-protein aggregation using a phenomenological model, in which the proteins can be in the unfolded (U), misfolded (M) or folded (F) states. The phase diagram shows the coexistence between a phase with aggregates of misfolded proteins and a phase of isolated proteins (U or F) in solution. The spinodal at low protein concentrations shows non-monotonic behavior with temperature, with implications for the stability of solutions of folded proteins at low temperatures. We follow the dynamics upon “quenching” from the U-phase (cooling) or the F-phase (heating) to the metastable or unstable part of the phase diagram that results in aggregation. We describe how interesting consequences to the distribution of aggregate size, and growth kinetics arise from the competition between folding and aggregation.     

Heterogenous mean-field analysis of a generalized voter-like model on networks

We propose a generalized framework for the study of voter models in complex networks at the heterogeneous mean-field (HMF) level that (i) yields a unified picture for existing copy/invasion processes and (ii) allows for the introduction of further heterogeneity through degree-selectivity rules. In the context of the HMF approximation, our model is capable of providing straightforward estimates for central quantities such as the exit probability and the consensus/fixation time, based on the statistical properties of the complex network alone. The HMF approach has the advantage of being readily applicable also in those cases in which exact solutions are difficult to work out. Finally, the unified formalism allows one to understand previously proposed voter-like processes as simple limits of the generalized model.     

Quantum mechanical look at the radioactive-like decay of metastable dark energy

We derive the Shafieloo, Hazra, Sahni and Starobinsky (SHSS) phenomenological formula for the radioactive-like decay of metastable dark energy directly from the principles of quantum mechanics. To this aim we use the Fock–Krylov theory of quantum unstable states. We obtain deeper insight on the decay process as having three basic phases: the phase of radioactive decay, the next phase of damping oscillations, and finally the phase of power-law decay. We consider the cosmological model with matter and dark energy in the form of decaying metastable dark energy and study its dynamics in the framework of non-conservative cosmology with an interacting term determined by the running cosmological parameter. We study the cosmological implications of metastable dark energy and estimate the characteristic time of ending of the radioactive-like decay epoch to be \(2.2\times 10^4\) of the present age of the Universe. We also confront the model with astronomical data which show that the model is in good agreement with the observations. Our general conclusion is that we are living in the epoch of the radioactive-like decay of metastable dark energy which is a relict of the quantum age of the Universe.     

Excitonic effects in core-excitation spectra of semiconductors

Computer simulations of a model glass-forming system are presented, which study the correlation between the dynamics in real space and the topography of the potential energy landscape. This analysis clearly reveals that in the supercooled regime the dynamics is strongly influenced by the presence of deep valleys in the energy landscape, corresponding to long-lived metastable amorphous states. We explicitly relate nonexponential relaxation effects and dynamic heterogeneities to these metastable states and thus to the specific topography of the energy landscape.