# Exploring the thermodynamic limit of Hamiltonian models: convergence to the Vlasov equation.

**ABSTRACT** We here discuss the emergence of quasistationary states (QSS), a universal feature of systems with long-range interactions. With reference to the Hamiltonian mean-field model, numerical simulations are performed based on both the original N-body setting and the continuum Vlasov model which is supposed to hold in the thermodynamic limit. A detailed comparison unambiguously demonstrates that the Vlasov-wave system provides the correct framework to address the study of QSS. Further, analytical calculations based on Lynden-Bell's theory of violent relaxation are shown to result in accurate predictions. Finally, in specific regions of parameters space, Vlasov numerical solutions are shown to be affected by small scale fluctuations, a finding that points to the need for novel schemes able to account for particle correlations.

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**ABSTRACT:**Systems with long-range interactions (LRI) display unusual thermodynamical and dynamical properties that stem from the non-additive character of the interaction potential. We focus in this work on the lack of relaxation to thermal equilibrium when a LRI system is started out-of-equilibrium. Several attempts have been made at predicting the so-called quasi-stationary state (QSS) reached by the dynamics and at characterizing the resulting transition between magnetized and non-magnetized states. We review in this work recent theories and interpretations about the QSS. Several theories exist but none of them has provided yet a full account of the dynamics found in numerical simulations.10/2012; -
##### Article: Stepwise transition to higher degrees of coherence in a random network of phase oscillators

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**ABSTRACT:**We consider a model system of phase oscillators which are connected in a random network. The network favors the connection of oscillators with close values of phases. We extend the order parameter used in the study of synchronization of phase oscillators and define generalized order parameters for the model system. We investigate the equilibrium properties of the model and reveal a phenomenon of stepwise transitions to higher degrees of coherence as the system goes through a series of second-order phase transitions for the order parameters. We also discuss a possible realization of the model in real physical systems.EPL (Europhysics Letters) 01/2012; 99(1). · 2.26 Impact Factor - SourceAvailable from: Tarcisio Nunes Teles06/2008, Degree: Master in Physics, Supervisor: Yan Levin and Renato Pakter

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arXiv:cond-mat/0612219v2 [cond-mat.stat-mech] 13 Jan 2007

Exploring the thermodynamic limit of Hamiltonian models: convergence to the

Vlasov equation.

Andrea Antoniazzi1, Francesco Califano2, Duccio Fanelli1,3, Stefano Ruffo1

1. Dipartimento di Energetica and CSDC, Universit` a di Firenze,

and INFN, via S. Marta, 3, 50139 Firenze, Italy

2.Dipartimento di Fisica ”E.Fermi” and CNISM,

Universit` a di Pisa, Largo Pontecorvo, 3 56127 Pisa, Italy

3. Theoretical Physics, School of Physics and Astronomy,

University of Manchester, Manchester M13 9PL, United Kingdom

(Dated: February 6, 2008)

We here discuss the emergence of Quasi Stationary States (QSS), a universal feature of systems

with long-range interactions. With reference to the Hamiltonian Mean Field (HMF) model, numeri-

cal simulations are performed based on both the original N-body setting and the continuum Vlasov

model which is supposed to hold in the thermodynamic limit. A detailed comparison unambiguously

demonstrates that the Vlasov-wave system provides the correct framework to address the study of

QSS. Further, analytical calculations based on Lynden-Bell’s theory of violent relaxation are shown

to result in accurate predictions. Finally, in specific regions of parameters space, Vlasov numerical

solutions are shown to be affected by small scale fluctuations, a finding that points to the need for

novel schemes able to account for particles correlations.

PACS numbers:

dynamical systems; 05.20.-y Classical statistical mechanics;

52.65.Ff Fokker Planck and Vlasov equations; 05.45.-a Nonlinear dynamics and nonlinear

The Vlasov equation constitutes a universal theoreti-

cal framework and plays a role of paramount importance

in many branches of applied and fundamental physics.

Structure formation in the universe is for instance a

rich and fascinating problem of classical physics: The

fossile radiation that permeates the cosmos is a relic

of microfluctuation in the matter created by the Big

Bang, and such a small perturbation is believed to have

evolved via gravitational instability to the pronounced

agglomerations that we see nowdays on the galaxy clus-

ter scale. Within this scenario, gravity is hence the en-

gine of growth and the Vlasov equation governs the dy-

namics of the non baryonic “dark matter” [1]. Further-

more, the continuous Vlasov description is the reference

model for several space and laboratory plasma applica-

tions, including many interesting regimes, among which

the interpretation of coherent electrostatic structures ob-

served in plasmas far from thermodynamic equilibrium.

The Vlasov equation is obtained as the mean–field limit

of the N–body Liouville equation, assuming that each

particle interacts with an average field generated by all

plasma particles (i.e. the mean electromagnetic field de-

termined by the Poisson or Maxwell equations where the

charge and current densities are calculated from the par-

ticle distribution function) while inter–particle correla-

tions are completely neglected.

Numerical simulations are presently one of the most

powerful resource to address the study of the Vlasov

equation.In the plasma context, the Lagrangian

Particle-In-Cell approach is by far the most popular,

while Eulerian Vlasov codes are particularly suited for

analyzing specific model problems, due to the associated

low noise level which is secured even in the non–linear

regime [2]. However, any numerical scheme designed to

integrate the continuous Vlasov system involves a dis-

cretization over a finite mesh. This is indeed an unavoid-

able step which in turn affects numerical accuracy. A nu-

merical (diffusive and dispersive) characteristic length is

in fact introduced being at best comparable with the grid

mesh size: as soon as the latter matches the typical length

scale relative to the (dynamically generated) fluctuations

a violation of the continuous Hamiltonian characterof the

equations occurs (see Refs. [3]). It is important to em-

phasize that even if such non Vlasov effects are strongly

localized (in phase space), the induced large scale topo-

logical changes will eventually affect the system globally.

Therefore, aiming at clarifying the problem of the valid-

ity of Vlasov numerical models, it is crucial to compare a

continuous Vlasov, but numerically discretized, approach

to a homologous N-body model.

Vlasov equation has been also invoked as a reference

model in many interesting one dimensional problems, and

recurrently applied to the study of wave-particles inter-

acting systems. The Hamiltonian Mean Field (HMF)

model [4], describing the coupled motion of N rotators,

is in particular assimilated to a Vlasov dynamics in the

thermodynamic limit on the basis of rigorous results [5].

The HMF model has been historically introduced as rep-

resenting gravitational and charged sheet models and is

quite extensively analyzed as a paradigmatic representa-

tive of the broader class of systems with long-range inter-

actions [6]. A peculiar feature of the HMF model, shared

also by other long-range interacting systems, is the pres-

ence of Quasi Stationary States (QSS). During time evo-

lution, the system gets trapped in such states, which are

characterized by non Gaussian velocity distributions, be-

Page 2

2

fore relaxing to the final Boltzmann-Gibbs equilibrium

[7]. An attempt has been made [8] to interpret the emer-

gence of QSSs by invoking Tsallis statistics [9].

approach has been later on criticized in [10], where QSSs

were shown to correspond to stationary stable solutions

of the Vlasov equation, for a particular choice of the ini-

tial condition. More recently, an approximate analytical

theory, based on the Vlasov equation, which derives the

QSSs of the HMF model using a maximum entropy prin-

ciple, was developed in [11]. This theory is inspired by

the pioneering work of Lynden-Bell [12] and relies on

previous work on 2D turbulence by Chavanis [13]. How-

ever, the underlying Vlasov ansatz has not been directly

examined and it is recently being debated [14].

In this Letter, we shall discuss numerical simulations

of the continuous Vlasov model, the kinetic counterpart

of the discrete HMF model. By comparing these results

to both direct N-body simulations and analytical predic-

tions, we shall reach the following conclusions: (i) the

Vlasov formulation is indeed ruling the dynamics of the

QSS; (ii) the proposed analytical treatment of the Vlasov

equation is surprisingly accurate, despite the approxima-

tions involved in the derivation; (iii) Vlasov simulations

are to be handled with extreme caution when exploring

specific regions of the parameters space.

The HMF model is characterized by the following

Hamiltonian

This

H =1

2

N

?

j=1

p2

j+

1

2N

N

?

i,j=1

[1 − cos(θj− θi)](1)

where θjrepresents the orientation of the j-th rotor and

pjis its conjugate momentum. To monitor the evolution

of the system, it is customary to introduce the magneti-

zation, a macroscopic order parameter defined as M =

|M| = |?mi|/N, where mi = (cosθi,sinθi) stands

for the microscopic magnetization vector. As previously

reported [4], after an initial transient, the system gets

trapped into Quasi-Stationary States (QSSs), i.e. non-

equilibrium dynamical regimes whose lifetime diverges

when increasing the number of particles N. Importantly,

when performing the mean-field limit (N → ∞) before

the infinite time limit, the system cannot relax towards

Boltzmann–Gibbs equilibrium and remains permanently

confined in the intermediate QSSs. As mentioned above,

this phenomenology is widely observed for systems with

long-range interactions, including galaxy dynamics [15],

free electron lasers [16], 2D electron plasmas [17].

In the N → ∞ limit the discrete HMF dynamics re-

duces to the Vlasov equation

∂f/∂t + p∂f/∂θ − (dV/dθ)∂f/∂p = 0,(2)

where f(θ,p,t) is the microscopic one-particle distribu-

tion function and

V (θ)[f] = 1 − Mx[f]cos(θ) − My[f]sin(θ) ,

?π

−π

−∞

?π

−π

∞

(3)

Mx[f] =

?∞

f(θ,p,t) cosθdθdp, (4)

My[f] =

?∞

f(θ,p,t) sinθdθdp. (5)

The specific energy h[f] =

(M2

y− 1)/2

? ?pf(θ,p,t)dθdp functionals are conserved quanti-

ties. Homogeneous states are characterized by M = 0,

while non-homogeneous states correspond to M ?= 0.

Rigorous mathematical results [5] demonstrate that,

indeed, the Vlasov framework applies in the continuum

description of mean-field type models. This observation

corroborates the claim that any theoretical attempt to

characterize the QSSs should resort to the above Vlasov

based interpretative picture. Despite this, the QSS non-

Gaussian velocity distributions have been fitted [8] us-

ing Tsallis’ q–exponentials, and the Vlasov formalism as-

sumed valid only for the limiting case of homogeneous

initial conditions [14]. In a recent paper [11], the afore-

mentioned velocity distribution functions were instead

reproduced with an analytical expression derived from

the Vlasov scenario, with no adjustable parameters and

for a large class of initial conditions, including inhomo-

geneous ones. The key idea dates back to the seminal

work by Lynden-Bell [12] (see also [18], [19]) and con-

sists in coarse-graining the microscopic one-particle dis-

tribution function f(θ,p,t) by introducing a local aver-

age in phase space. It is then possible to associate an

entropy to the coarse-grained distribution¯f: The corre-

sponding statistical equilibrium is hence determined by

maximizing such an entropy, while imposing the conser-

vation of the Vlasov dynamical invariants, namely en-

ergy, momentum and norm of the distribution. We shall

here limit our discussion to the case of an initial single

particle distribution which takes only two distinct val-

ues: f0= 1/(4∆θ∆p), if the angles (velocities) lie within

an interval centered around zero and of half-width ∆θ

(∆p), and zero otherwise. This choice corresponds to the

so-called “water-bag” distribution which is fully specified

by energy h[f] = e, momentum P[f] = σ and the initial

magnetization M0 = (Mx0,My0).

tropy calculation is then performed analytically [11] and

results in the following form of the QSS distribution

? ?(p2/2)f(θ,p,t)dθdp −

momentum

x+ M2

andP[f]=

The maximum en-

¯f(θ,p) = f0

e−β(p2/2−My[¯ f]sinθ−Mx[¯ f]cosθ)−λp−µ

1 + e−β(p2/2−My[¯ f]sinθ−Mx[¯ f]cosθ)−λp−µ

(6)

where β/f0, λ/f0 and µ/f0 are rescaled Lagrange mul-

tipliers, respectively associated to the energy, momen-

tum and normalization. Inserting expression (6) into the

above constraints and recalling the definition of Mx[¯f],

My[¯f], one obtains an implicit system which can be

Page 3

3

solved numerically to determine the Lagrange multipliers

and the expected magnetization in the QSS. Note that

the distribution (6) differs from the usual Boltzmann-

Gibbs expression because of the “fermionic” denomina-

tor. Numerically computed velocity distributions have

been compared in [11] to the above theoretical predictions

(where no free parameter is used), obtaining an overall

good agreement. However, the central part of the distri-

butions is modulated by the presence of two symmetric

bumps, which are the signature of a collective dynamical

phenomenon [11]. The presence of these bumps is not

explained by our theory. Such discrepancies has been re-

cently claimed to be an indirect proof of the fact that

the Vlasov model holds only approximately true. We

shall here demonstrate that this claim is not correct and

that the deviations between theory and numerical obser-

vation are uniquely due to the approximations built in

the Lynden-Bell approach.

A detailed analysis of the Lynden-Bell equilibrium

(6) in the parameter plane (M0,e) enabled us to un-

ravel a rich phenomenology, including out of equilibrium

phase transitions between homogeneous (MQSS = 0)

and non-homogeneous (MQSS ?= 0) QSS states.

ond and first order transition lines are found that sep-

arate homogeneous and non homogeneous states and

merge into a tricritical point approximately located in

(M0,e) = (0.2,0.61). When the transition is second or-

der two extrema of the Lynden-Bell entropy are identified

in the inhomogeneous phase: the solution M = 0 corre-

sponds to a saddle point, being therefore unstable; the

global maximum is instead associated to M ?= 0, which

represents the equilibrium predicted by the theory. This

argument is important for what will be discussed in the

following.

Let us now turn to direct simulations, with the aim of

testing the above scenario, and focus first on the kinetic

model (2)–(5). The algorithm solves the Vlasov equation

in phase space and uses the so-called “splitting scheme”,

a widely adopted strategy in numerical fluid dynamics.

Such a scheme, pioneered by Cheng and Knorr [20], was

first applied to the study of the Vlasov-Poisson equations

in the electrostatic limit and then employed for a wide

spectrum of problems [3]. For different values of the pair

(M0,e), which sets the widths of the initial water-bag

profile, we performed a direct integration of the Vlasov

system (2)–(5). After a transient, magnetization is shown

to eventually attain a constant value, which corresponds

to the QSS value observed in the HMF, discrete, frame-

work. The asymptotic magnetizations are hence recorded

when varying the initial condition. Results (stars) are re-

ported in figure 1(a) where MQSSis plotted as function

of e. A comparison is drawn with the predictions of our

theory (solid line) and with the outcome of N-body sim-

ulation (squares) based on the Hamiltonian (1), finding

an excellent agreement. This observation enables us to

conclude that (i) the Vlasov equation governs the HMF

Sec-

dynamics for N → ∞ both in the homogeneous and non

homogeneous case; (ii) Lynden-Bell’s violent relaxation

theory allows for reliable predictions, including the tran-

sition from magnetized to non-magnetized states.

Deviations from the theory are detected near the tran-

sition. This fact has a natural explaination and raises

a number of fundamental questions related to the use

of Vlasov simulations. As confirmed by the inspection of

figure 1(b), close to the transition point, the entropy S of

the Lynden-Bell coarse-grained distribution takes almost

the same value when evaluated on the global maximum

(solid line) or on the saddle point (dashed line). The en-

tropy is hence substantially flat in this region, which in

turn implies that there exists an extended basin of states

accessible to the system. This interpretation is further

validated by the inset of figure 1(a), where we show the

probability distribution of MQSS computed via N-body

simulation. The bell-shaped profile presents a clear peak,

approximately close to the value predicted by our theory.

Quite remarkably, the system can converge to final mag-

netizations which are sensibly different from the expected

value. Simulations based on the Vlasov code running at

different resolutions (grid points) confirmed this scenario,

highlighting a similar degree of variability. These findings

point to the fact that in specific regions of the parame-

ter space, Vlasov numerics needs to be carefully analyzed

(see also Ref. [21]). Importantly, it is becoming nowadays

crucial to step towards an “extended Vlasov theoretical

model which enables to account for discreteness effects,

by incorporating at least two particles correlations inter-

action term.

0,50,550,6 0,65

0

0,2

0,4

0,6

MQSS

0,20,3 0,4

0

10

20

0,50,550,60,65

e

0

1

2

3

4

5

6

7

S

(a)

(b)

MQSS > 0

MQSS = 0

FIG. 1: Panel (a): The magnetization in the QSS is plotted

as function of energy, e, at M0 = 0.24. The solid line refers to

the Lynden-Bell inspired theory. Stars (resp. squares) stand

for Vlasov (resp. N-body) simulations. Inset: Probability

distribution of MQSS computed via N-body simulation (the

solid line is a Gaussian fit). Panel (b): Entropy S at the sta-

tionary points, as function of energy, e: magnetized solution

(solid line) and non–magnetized one (dashed line).

Qualitatively, one can track the evolution of the sys-

tem in phase space, both for the homogeneous and non

Page 4

4

FIG. 2: Phase space snapshots for (M0,e) = (0.5,0.69).

homogeneous cases. Results of the Vlasov integration

are displayed in figure 2 for (M0,e) = (0.5,0.69), where

the system is shown to evolve towards a non magnetized

QSS. The initial water-bag distribution splits into two

large resonances, which persist asymptotically: the lat-

ter acquire constant opposite velocities which are main-

tained during the subsequent time evolution, in agree-

ment with the findings of [11]. The two bumps are there-

fore an emergent property of the model, which is correctly

reproduced by the Vlasov dynamics. For larger values

of the initial magnetization (M0> 0.89), while keeping

e = 0.69, the system evolves towards an asymptotic mag-

netized state, in agreement with the theory. In this case

several resonances are rapidly developed which eventu-

ally coaelesce giving rise to complex patterns in phase

space. More quantitatively, one can compare the veloc-

ity distributions resulting from, respectively, Vlasov and

N-body simulations. The curves are diplayed in figure

3 (a),(b),(c) for various choices of the initial conditions

in the magnetized region. The agreement is excellent,

thus reinforcing our former conclusion about the validity

of the Vlasov model. Finally, let us stress that, when

e = 0.69, the two solutions (resp. magnetized and non

magnetized) [11] are associated to a practically indistin-

guishible entropy level (see figure 3 (d)). As previously

discussed, the system explores an almost flat entropy

landscape and can be therefore be stuck in local traps,

because of finite size effects. A pronounced variability of

the measured MQSSis therefore to be expected.

In this Letter, we have analyzed the emergence of QSS,

a universal feature that occurs in systems with long-range

interactions, for the specific case of the HMF model. By

comparing numerical simulations and analytical predic-

tions, we have been able to unambiguously demonstrate

that the Vlasov model provides an accurate framework

to address the study of the QSS. Working within the

Vlasov context one can develop a fully predictive theo-

retical approach, which is completely justified from first

principles. Finally, and most important, results of con-

ventional Vlasov codes are to be critically scrutinized,

-1,5

-1

-0,5

0

0,5

1

1,5

0,3

0,6

a)

-1,5

-1

-0,5

0

0,5

1

1,5

0,3

0,6

b)

-1,5

-1

-0,5

0

0,5

1

1,5

0,3

0,6

c)

0,85

0,9

0,95

1

5

10

d)

FIG. 3: Symbols: velocity distributions computed via N-

body simulations. Solid line: velocity distributions obtained

through a direct integration of the Vlasov equation. Here

e = 0.69 and M0 = 0.3 (a), M0 = 0.5 (b), M0 = 0.7 (c).

Panel (d): Entropy at the stationary points as a function of

the initial magnetization: the solid line refers to the global

maximum, while the dotted line to the saddle point.

especially in specific regions of parameters space close

to transitions from homogeneous to non homogeneous

states.

We acknowledge financial support from the PRIN05-

MIUR project Dynamics and thermodynamics of systems

with long-range interactions.

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