arXiv:cond-mat/0612219v2 [cond-mat.stat-mech] 13 Jan 2007
Exploring the thermodynamic limit of Hamiltonian models: convergence to the
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.
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” . 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 . 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. ). 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 , 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 .
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 . 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-
fore relaxing to the final Boltzmann-Gibbs equilibrium
. An attempt has been made  to interpret the emer-
gence of QSSs by invoking Tsallis statistics .
approach has been later on criticized in , 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 . This theory is inspired by
the pioneering work of Lynden-Bell  and relies on
previous work on 2D turbulence by Chavanis . How-
ever, the underlying Vlasov ansatz has not been directly
examined and it is recently being debated .
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
[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 , 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 ,
free electron lasers , 2D electron plasmas .
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) ,
f(θ,p,t) cosθdθdp, (4)
f(θ,p,t) sinθdθdp. (5)
The specific energy h[f] =
? ?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  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  us-
ing Tsallis’ q–exponentials, and the Vlasov formalism as-
sumed valid only for the limiting case of homogeneous
initial conditions . In a recent paper , 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  (see also , ) 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  and
results in the following form of the QSS distribution
? ?(p2/2)f(θ,p,t)dθdp −
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
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
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  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 . 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
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 , was
first applied to the study of the Vlasov-Poisson equations
in the electrostatic limit and then employed for a wide
spectrum of problems . 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
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. ). 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-
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
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 . 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)  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,
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
We acknowledge financial support from the PRIN05-
MIUR project Dynamics and thermodynamics of systems
with long-range interactions.
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