Incomplete Equilibrium in Long-Range Interacting Systems

Baldovin, Fulvio; Orlandini, Enzo

doi:10.1103/physrevlett.97.100601

Cited by 57 publications

(59 citation statements)

References 28 publications

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“…As already noted, the dynamics of relaxation to equilibrium depends on the number of particles in the system and has been extensively studied in the recent literature [1][2][3][4][5]14,[17][18][19][20][21][22][23][24]. Its dependence on N can be obtained from collisional corrections to the Vlasov equation, i.e., by determining the relevant kinetic * marciano@fis.unb.br equation.…”

Section: Introductionmentioning

confidence: 99%

Scaling of the dynamics of a homogeneous one-dimensional anisotropic classical Heisenberg model with long-range interactions

Lourenço

Filho

2015

Phys. Rev. E

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The dynamics of quasistationary states of long-range interacting systems with N particles can be described by kinetic equations such as the Balescu-Lenard and Landau equations. In the case of one-dimensional homogeneous systems, two-body contributions vanish as two-body collisions in one dimension only exchange momentum and thus cannot change the one-particle distribution. Using a Kac factor in the interparticle potential implies a scaling of the dynamics proportional to N(δ) with δ=1 except for one-dimensional homogeneous systems. For the latter different values for δ were reported for a few models. Recently it was shown by Rocha Filho and collaborators [Phys. Rev. E 90, 032133 (2014)] for the Hamiltonian mean-field model that δ=2 provided that N is sufficiently large, while small N effects lead to δ≈1.7. More recently, Gupta and Mukamel [J. Stat. Mech. (2011) P03015] introduced a classical spin model with an anisotropic interaction with a scaling in the dynamics proportional to N(1.7) for a homogeneous state. We show here that this model reduces to a one-dimensional Hamiltonian system and that the scaling of the dynamics approaches N(2) with increasing N. We also explain from theoretical consideration why usual kinetic theory fails for small N values, which ultimately is the origin of noninteger exponents in the scaling.

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Section: Introductionmentioning

confidence: 99%

Scaling of the dynamics of a homogeneous one-dimensional anisotropic classical Heisenberg model with long-range interactions

Lourenço

Filho

2015

Phys. Rev. E

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“…Since the temperature is fixed, the fundamental statistical description of the BMF model is the canonical ensemble. It has been demonstrated in [39] that Hamiltonian reservoirs microscopically coupled with the system [40,41] and Langevin thermostats [17] provide equivalent descriptions even out-of-equilibrium. Therefore, the BMF model has many applications.…”

Section: Introductionmentioning

confidence: 99%

The Brownian mean field model

Chavanis

2014

Eur. Phys. J. B

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Abstract. We discuss the dynamics and thermodynamics of the Brownian Mean Field (BMF) model which is a system of N Brownian particles moving on a circle and interacting via a cosine potential. It can be viewed as the canonical version of the Hamiltonian Mean Field (HMF) model. The BMF model displays a second order phase transition between a homogeneous phase and an inhomogeneous phase below a critical temperature Tc = 1/2. We first complete the description of this model in the mean field approximation valid for N → +∞. In the strong friction limit, the evolution of the density towards the mean field Boltzmann distribution is governed by the mean field Smoluchowski equation. For T < Tc, this equation describes a process of self-organization from a non-magnetized (homogeneous) phase to a magnetized (inhomogeneous) phase. We obtain an analytical expression for the temporal evolution of the magnetization close to Tc. Then, we take fluctuations (finite N effects) into account. The evolution of the density is governed by the stochastic Smoluchowski equation. From this equation, we derive a stochastic equation for the magnetization and study its properties both in the homogenous and inhomogeneous phases. We show that the fluctuations diverge close to the critical point so that the mean field approximation ceases to be valid. Actually, the limits N → +∞ and T → Tc do not commute. The validity of the mean field approximation requires N (T − Tc) → +∞ so that N must be larger and larger as T approaches Tc. We show that the direction of the magnetization changes rapidly close to Tc but its amplitude takes a long time to relax. We also indicate that, for systems with long-range interactions, the lifetime of metastable states scales as e N except close to a critical point. The BMF model shares many analogies with other systems of Brownian particles with long range interactions such as self-gravitating Brownian particles, the Keller-Segel model describing the chemotaxis of bacterial populations, and the Kuramoto model describing the collective synchronization of coupled oscillators.

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“…The use of a mean first passage time approach proves to be successful in explaining the non-trivial scaling at stake here, and sheds some light on another case, where lifetimes diverging as e N above some critical energy have been reported. Explaining the emergence of quasistationary states (QSSs), predicting their characteristics or determining their lifetimes are still puzzling issues in the active research program [1][2][3][4][5][6][7] on long-range almost collisionless systems. Such systems are widely present in the Universe, since they range from assemblies of charged particles interacting via Coulomb interaction to self-gravitating massive objects such as globular clusters or stars in galaxies.…”

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confidence: 99%

Action diffusion and lifetimes of quasistationary states in the Hamiltonian mean-field model

Ettoumi

Firpo

2013

Phys. Rev. E

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Out-of-equilibrium quasistationary states (QSSs) are one of the signatures of a broken ergodicity in long-range interacting systems. For the widely studied Hamiltonian Mean-Field model, the lifetime of some QSSs has been shown to diverge with the number N of degrees of freedom with a puzzling N 1.7 scaling law, contradicting the otherwise widespread N scaling law. It is shown here that this peculiar scaling arises from the locality properties of the dynamics captured through the computation of the diffusion coefficient in terms of the action variable. The use of a mean first passage time approach proves to be successful in explaining the non-trivial scaling at stake here, and sheds some light on another case, where lifetimes diverging as e N above some critical energy have been reported. 05.70.Ln,05.20.Dd Explaining the emergence of quasistationary states (QSSs), predicting their characteristics or determining their lifetimes are still puzzling issues in the active research program [1][2][3][4][5][6][7] on long-range almost collisionless systems. Such systems are widely present in the Universe, since they range from assemblies of charged particles interacting via Coulomb interaction to self-gravitating massive objects such as globular clusters or stars in galaxies. Toy models have become a favorite tool to address those problems. For instance, the peculiar relaxation properties of long-range interacting systems began to be uncovered [8] through numerical simulations of the one-dimensional gravitational system, showing notably its reluctance to thermalize due to the existence of QSSs [9]. Introducing periodic boundary conditions produced even simpler models permitting convenient computations in a compact space. Because it only retains the lowest Fourier mode of the gravitational potential, the well-known Hamiltonian Mean Field (HMF) model may be viewed as the simplest relevant toy model to address the intricate relationships between dynamics and statistical mechanics of long-range interacting systems. It is defined by the following Hamiltonianwhere N is the number of particles, and q i and p i denote respectively the position and momentum of the i th particle. A useful collective quantity to introduce is the mean-field (also called magnetization) vector (M x , M y ) with M x = 1/N i cos q i and M y = 1/N i sin q i . The average energy per particle U = H/N reads thenwhere M ≡ M x 2 + M y 2 denotes the modulus of the magnetization vector. Equilibrium statistical mechanics [10] can be rather easily derived and shows that a second order phase transition takes place at U c = 3/4. As N tends to infinity, the ensemble average of the magnetization is accordingly positive for U < U c whether it is null for U > U c . However, contrary to short-range interacting systems, thermodynamic equilibrium may not be reached in the thermodynamic limit, which amounts, in the HMF model, to a Vlasov limit. Hence, QSSs are a signature of a broken ergodicity [11,12] in long-range interacting systems. Within the HMF model, their existen...

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Incomplete Equilibrium in Long-Range Interacting Systems

Cited by 57 publications

References 28 publications

Scaling of the dynamics of a homogeneous one-dimensional anisotropic classical Heisenberg model with long-range interactions

Scaling of the dynamics of a homogeneous one-dimensional anisotropic classical Heisenberg model with long-range interactions

The Brownian mean field model

Action diffusion and lifetimes of quasistationary states in the Hamiltonian mean-field model

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