Outofequilibrium phase transitions in the Hamiltonian meanfield model: a closer look.
ABSTRACT We provide a detailed discussion of outofequilibrium phase transitions in the Hamiltonian meanfield (HMF) model in the framework of LyndenBell's statistical theory of the Vlasov equation. For twolevel initial conditions, the caloric curve β(E) only depends on the initial value f(0) of the distribution function. We evidence different regions in the parameter space where the nature of the phase transitions between magnetized and nonmagnetized states changes: (i) For f(0)>0.10965, the system displays a secondorder phase transition; (ii) for 0.109497<f(0)<0.10965, the system displays a secondorder phase transition and a firstorder phase transition; (iii) for 0.10947<f(0)<0.109497, the system displays two secondorder phase transitions; and (iv) for f(0)<0.1047, there is no phase transition. The passage from a firstorder to a secondorder phase transition corresponds to a tricritical point. The sudden appearance of two secondorder phase transitions from nothing corresponds to a secondorder azeotropy. This is associated with a phenomenon of phase reentrance. When metastable states are taken into account, the problem becomes even richer. In particular, we find another situation of phase reentrance. We consider both microcanonical and canonical ensembles and report the existence of a tiny region of ensemble inequivalence. We also explain why the use of the initial magnetization M(0) as an external parameter, instead of the phase level f(0), may lead to inconsistencies in the thermodynamical analysis. Finally, we mention different causes of incomplete relaxation that could be a limitation to the application of LyndenBell's theory.

Article: Gravitational instabilities of isothermal spheres in the presence of a cosmological constant
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ABSTRACT: Gravitational instabilities of isothermal spheres are studied in the presence of a positive or negative cosmological constant, in the Newtonian limit. In gravity, the statistical ensembles are not equivalent. We perform the analysis both in the microcanonical and the canonical ensembles, for which the corresponding instabilities are known as `gravothermal catastrophe' and `isothermal collapse', respectively. In the microcanonical ensemble, no equilibria can be found for radii larger than a critical value, which is increasing with increasing cosmological constant. In contrast, in the canonical ensemble, no equilibria can be found for radii smaller than a critical value, which is decreasing with increasing cosmological constant. For a positive cosmological constant, characteristic reentrant behaviour is observed.Nuclear Physics B 02/2013; 871(1). · 4.33 Impact Factor 
Article: The quantum HMF model: I. Fermions
[Show abstract] [Hide abstract]
ABSTRACT: We study the thermodynamics of quantum particles with longrange interactions at T = 0. Specifically, we generalize the Hamiltonian meanfield (HMF) model to the case of fermions. We consider the Thomas–Fermi approximation that becomes exact in a proper thermodynamic limit with a coupling constant k ~ N. The equilibrium configurations, described by the meanfield Fermi (or waterbag) distribution, are equivalent to polytropes of index n = 1/2. We show that the homogeneous phase, which is unstable in the classical regime, becomes stable in the quantum regime. The homogeneous phase is stabilized by the Pauli exclusion principle. This takes place through a firstorder phase transition where the control parameter is the normalized Planck constant. The homogeneous phase is unstable for , metastable for and stable for . The inhomogeneous phase is stable for , metastable for and disappears for (for , there exists an unstable inhomogeneous phase with magnetization ). We point out analogies between the fermionic HMF model and the concept of fermion stars in astrophysics. Finally, as a byproduct of our analysis, we obtain new results concerning the Vlasov dynamical stability of the waterbag distribution which is the ground state of the LyndenBell distribution in the theory of violent relaxation of the classical HMF model. We show that spatially homogeneous waterbag distributions are Vlasovstable iff ≥ c = 1/3 and spatially inhomogeneous waterbag distributions are Vlasovstable iff ≤ * = 0.379 and b ≥ b* = 0.37, where and b are the normalized energy and magnetization. The magnetization curve displays a firstorder phase transition at t = 0.352 and the domain of metastability ranges from c to *.Journal of Statistical Mechanics Theory and Experiment 08/2011; 2011(08):P08002. · 1.87 Impact Factor  [Show abstract] [Hide abstract]
ABSTRACT: We present here how the gravothermal or Antonov's instability, which was originally formulated in the microcanonical ensemble, is modified in the presence of a cosmological constant and in the canonical ensemble. In contrast to the microcanonical ensemble, there is a minimum, and not maximum, radius for which metastable states exist. In addition this critical radius is decreasing, and not increasing, with increasing cosmological constant. The minimum temperature for which metastable states exist is decreasing with increasing cosmological constant, while above some positive value of the cosmological constant, there appears a second critical temperature. For lower temperatures than the second critical temperature value, metastable states reappear, indicating a typical reentrant phase transition. The two critical temperatures merge when the cosmological density equals one half the mean density of the system.Journal of Physics Conference Series 02/2013; 410(1):2130.
Page 1
arXiv:1012.1566v1 [condmat.statmech] 7 Dec 2010
Outofequilibrium phase transitions in the HMF model: a closer look
F. Staniscia,1,2P.H. Chavanis,3and G. De Ninno2,4
1Dipartimento di Fisica, Universit` a di Trieste, 34127 Trieste, Italy
2Sincrotrone Trieste, S.S. 14 km 163.5, Basovizza, 34149 Trieste, Italy
3Laboratoire de Physique Th´ eorique (IRSAMC), Universit´ e de Toulouse (UPS) and CNRS, F31062 Toulouse, France
4Physics Department, Nova Gorica University 5001 Nova Gorica, Slovenia
We provide a detailed discussion of outofequilibrium phase transitions in the Hamiltonian Mean
Field (HMF) model in the framework of LyndenBell’s statistical theory of the Vlasov equation.
For twolevels initial conditions, the caloric curve β(E) only depends on the initial value f0 of the
distribution function. We evidence different regions in the parameter space where the nature of phase
transitions between magnetized and nonmagnetized states changes: (i) for f0 > 0.10965, the system
displays a second order phase transition; (ii) for 0.109497 < f0 < 0.10965, the system displays a
second order phase transition and a first order phase transition; (iii) for 0.10947 < f0 < 0.109497, the
system displays two second order phase transitions; (iv) for f0 < 0.10947, there is no phase transition.
The passage from a first order to a second order phase transition corresponds to a tricritical point.
The sudden appearance of two second order phase transitions from nothing corresponds to a second
order azeotropy. This is associated with a phenomenon of phase reentrance. When metastable states
are taken into account, the problem becomes even richer. In particular, we find a new situation
of phase reentrance. We consider both microcanonical and canonical ensembles and report the
existence of a tiny region of ensembles inequivalence. We also explain why the use of the initial
magnetization M0as an external parameter, instead of the phase level f0, may lead to inconsistencies
in the thermodynamical analysis.
PACS numbers:
I. INTRODUCTION
Systems with longrange interactions have recently
been the object of an intense activity [1–4].
systems are numerous in nature and concern different
disciplines such as astrophysics (galaxies) [5–7], two
dimensional turbulence (vortices) [8–10], biology (chemo
taxis) [11], plasma physics [12–14] and modern technolo
gies such as Free Electron Lasers (FEL) [15–17]. In ad
dition, their study is interesting at a conceptual level
because it obliges to go back to the foundations of sta
tistical mechanics and kinetic theory [4, 18–20]. Indeed,
systems with longrange interactions exhibit a number
of unusual features that are not present in systems with
shortrange interactions. For example, their equilibrium
statistical mechanics is marked by the existence of spa
tially inhomogeneous equilibrium states [1], unusual ther
modynamic limits [21–23], inequivalence of statistical en
sembles [5, 7, 24], negative specific heats [25, 26], vari
ous kinds of phase transitions [7, 27] etc. Their dynam
ics is also very interesting because these systems can be
found in longlived quasi stationary states (QSS) that
are different from Boltzmann equilibrium states. These
QSSs can be interpreted as stable steady states of the
Vlasov equation which governs the evolution of the sys
tem for sufficiently “short” times before correlations have
developed [12, 28]. In fact, for systems with longrange
interactions, the collisional relaxation time towards the
Boltzmann distribution increases rapidly with the num
ber of particles N and diverges at the thermodynamic
limit N → +∞ [4, 19, 20, 28]. Therefore, the domain
of validity of the Vlasov equation is huge and the QSSs
These
have very long lifetimes. In many cases, they are the
only observable structures in a longrange system, so that
they are often more physically relevant than the Boltz
mann equilibrium state itself. A question that naturally
emerges is whether one can predict the QSS actually
reached by the system. This is not an easy task since
the Vlasov equation admits an infinite number of stable
steady states in which the system can be trapped [28]. In
a seminal paper, LyndenBell [29] proposed to determine
the QSS eventually reached by the system by develop
ing a statistical mechanics of the Vlasov equation. To
that purpose, he introduced the notions of phase mixing,
violent relaxation and coarsegrained distributions. He
obtained the most probable distribution by maximizing
a Boltzmanntype entropy while conserving all the con
straints of the Vlasov equation (in particular the infinite
class of Casimirs). By definition, this “most mixed state”
is the statistical equilibrium state of the Vlasov equation
(at a coarsegrained scale). Whether or not the system
truly reaches this equilibrium state relies on an assump
tion of ergodicity and efficient mixing. This ergodicity
assumption is not always fulfilled in the process of vio
lent relaxation and the LyndenBell prediction may fail.
In that case, the QSS can be another stable steady state
of the Vlasov equation that is incompletely mixed. This
is referred to as incomplete relaxation (see, e.g. [30], for
discussion and further references). In case of incomplete
relaxation, the prediction of the QSS is very difficult,
and presumably impossible. Nevertheless, in many cases,
the LyndenBell approach gives a fine first order predic
tion of the achieved QSS and allows one to predict out
ofequilibrium phase transitions between different types
of structures that can be compared with direct simula
Page 2
2
tions or experiments. Before addressing this problem in
a specific situation, namely the Hamiltonian Mean Field
(HMF) model [31, 32], let us first briefly review the suc
cesses and the weaknesses of the LyndenBell approach.
LyndenBell’s statistical theory of violent relaxation
was elaborated in the context of 3D stellar systems. Un
fortunately, this is the worse situation for its practical
application. Indeed, the predicted distribution function
has infinite mass (the spatial density decreases at large
distances like r−2). In other words, this means that there
is no entropy maximum for a stellar system in an infinite
domain [28]. This is a clear evidence of the fact that
galaxies have necessarily reached a state of incomplete
violent relaxation. In fact, the LyndenBell theory is able
to explain the isothermal core of elliptical galaxies with
out recourse to collisions that operate on a much longer
timescale (of the order of the Chandrasekhar relaxation
time [33]). This is usually recognized as a major suc
cess of the theory. Unfortunately, it fails at predicting
the structure of the halo whose velocity distribution is
anisotropic and whose spatial density decreases like r−4
[28]. Models of incomplete violent relaxation have been
elaborated by Bertin & Stiavelli [34], Stiavelli & Bertin
[35] and Hjorth & Madsen [36]. These models are able
to reproduce the de Vaucouleurs law of elliptical galax
ies and provide a very good agreement with numerical
simulations up to nine orders of magnitude [37]. An
other possibility to describe incomplete relaxation is to
develop a kinetic theory of violent relaxation in order
to understand what limits mixing [38–40]. The idea is
that, in case of incomplete relaxation (nonergodicity),
the prediction of the QSS is impossible without consid
ering the dynamics [30]. Finally, in more academic stud
ies [41], one can confine the system within an artificial
spherical box and assume a complete relaxation inside
the box. Since the LyndenBell distribution is similar to
the FermiDirac statistics (in the twolevels approxima
tion), the problem is mathematically equivalent to the
study of a gas of selfgravitating fermions in a box. This
theoretical problem has been studied in detail by Chava
nis [42]. The caloric curve β(E) displays a rich variety of
microcanonical and canonical phase transitions (zeroth
and first order) between gaseous (non degenerate) and
condensed (degenerate) states, depending on the value
of a degeneracy parameter related to the initial distribu
tion function f0in the LyndenBell theory. In particular,
there exists two critical points in the phase diagram, one
in each ensemble, at which the phase transitions are sup
pressed. For details about these phase transitions, and
for an extended bibliography, we refer to the review [7].
The LyndenBell prediction has also been tested in 1D
and 2D gravity [43, 44] where the infinite mass problem
does not arise [45]. However, it is found again that relax
ation is incomplete and that the LyndenBell prediction
fails [89]. Finally, Arad & LyndenBell [46] have shown
that the theory itself presents some inconsistencies aris
ing from its nontransitive nature. These negative results
have led many astrophysicists to the conclusion that the
LyndenBell theory does not work in practice [28].
A similar statistical theory has been developed by
Miller [47], and independently by Robert & Sommeria
[48], in 2D turbulence in order to explain the robust
ness of longlived vortices in astrophysical and geophys
ical flows (a notorious example being Jupiter’s great red
spot). Largescale vortices are interpreted as quasi sta
tionary states of the 2D Euler equation in the same way
that galaxies are quasi stationary states of the Vlasov
equation (see [10, 49] for a discussion of the numerous
analogies between the statistical mechanics and the ki
netic theory of 2D vortices and stellar systems). Miller
RobertSommeria (MRS) developed a statistical theory
of the 2D Euler equation in order to predict the most
probable state achieved by the system. Although situa
tions of incomplete relaxation have also been evidenced
in 2D turbulence [50–52], the MRS theory has met a lot
of success. For example, it is able to account for geom
etry induced phase transitions between monopoles and
dipoles as we change the aspect ratio of the domain [53–
57]. Phase transitions and bifurcations between different
types of flows have also been studied in [58–60]. On the
other hand, when applied to geophysical and astrophys
ical flows, the MRS theory is able to account for the
structure and the organization of largescale flows such
as jovian jets and vortices [61–64] and Fofonoff flows in
oceanic basins [55, 65]. This theory has also been applied
to more complicated situations such as 2D magnetohy
drodynamics (MHD) [66, 67] and axisymmetric flows (the
celebrated von K´ arm´ an flow) [68].
A toy model of systems with longrange interactions,
called the Hamiltonian Mean Field (HMF) model, has
been introduced in statistical physics [31, 32] and exten
sively studied [4]. It can be viewed as a XY spin system
with infinite range interactions or as a one dimensional
model of particles moving on a ring and interacting via a
longrange potential truncated to one Fourier mode (co
sine potential). In that second interpretation, it shares
many analogies with selfgravitating systems [31, 32, 69]
but is much simpler to study since it avoids difficulties
linked with the singular nature of the gravitational po
tential at the origin and the absence of a natural confine
ment [5–7]. The observation of quasi stationary states in
the HMF model [32, 70] was a surprise in the community
of statistical mechanics working on systems with long
range interactions. It was recognized early that these
QSSs are outofequilibrium structures and that they are
nonBoltzmannian. They were first interpreted [70] in
terms of Tsallis generalized thermodynamics [71] with
the argument that the system is nonextensive so that
Boltzmann statistical mechanics is not applicable. Later,
inspired by analogies with stellar systems and 2D vor
tices reported in [10], different groups started to interpret
these QSSs in terms of stable steady states of the Vlasov
equation and statistical equilibrium states in the sense
of LyndenBell [69, 72, 73]. Chavanis [74] studied out
ofequilibrium phase transitions in the HMF model by
analogy with similar studies in astrophysics and hydro
Page 3
3
dynamics [41, 53] and obtained a phase diagram in the
(f0,E) plane [90] between magnetized (M ?= 0) and non
magnetized (M = 0) states. These regions are separated
by a critical line Ec(f0) that marks the domain of stabil
ity of the homogeneous phase. This critical line displays
a turning point at ((f0)∗,E∗) ≃ (0.10947,0.608) leading
to a phenomenon of phase reentrance (as we reduce the
energy, the homogeneous phase is successively stable, un
stable and stable again). Antoniazzi et al. [75] studied
the validity of the LyndenBell prediction by performing
careful comparisons with direct Nbody simulations at
E = 0.69 and found a good agreement for initial magne
tizations M0< (M0)crit(E) ≃ 0.897 leading to spatially
homogeneous LyndenBell distributions [91]. Antoniazzi
et al. [78] obtained a phase diagram in the (M0,E) plane
and showed that the system exhibits first and second or
der phase transitions separated by a tricritical point. Fi
nally, Antoniazzi et al. [79] performed numerical simu
lations of the Vlasov equation and found a good agree
ment with direct Nbody simulations and LyndenBell’s
prediction for the explored range of parameters. A syn
thesis of these results was published in [80]. In this pa
per, a more detailed discussion of phase transitions in
the (f0,E) plane was given, showing the lines of first
and second order phase transitions and the domains of
metastability. On the other hand, a comparison between
the phase diagrams in the (f0,E) and (M0,E) planes was
made. It was stated, without rigorous justification, that
the tricritical point in the (M0,E) plane corresponds to
the turning point of the critical line Ec(f0) in the (f0,E)
plane, i.e. the point where the phase reentrance starts.
These results were confronted to numerical simulations
by Staniscia et al. [81]. These simulations confirmed the
existence of a reentrant phase in the very narrow region
predicted by the theory [74] but also showed discrepan
cies with the LyndenBell prediction (such as an addi
tional reentrant phase and a persistence of magnetized
states in the a priori nonmagnetized region) that were
interpreted as a result of incomplete relaxation. Stanis
cia et al. [81] also determined the physical caloric curve
βkin(E), where βkin= 1/Tkinis the inverse kinetic tem
perature, in the region of the phase diagram displaying
first and second order phase transitions, and reported
the existence of a region of negative kinetic specific heat
Ckin= dE/dTkin< 0. In a recent paper [82], the thermo
dynamical caloric curve β(E) was determined in the same
region of parameters and it was shown that the thermo
dynamical specific heat C = dE/dT is always positive,
even in the region where the kinetic specific heat is neg
ative. In particular, it is argued that the ensembles are
equivalent although the experimentally measured specific
heat is negative [92].
These various results show that the description of out
ofequilibrium phase transitions in the HMF model is
extremely rich and subtle. In this paper, we describe
in more detail the phase transitions between magnetized
and nonmagnetized states in the (f0,E) plane. In par
ticular, we plot the series of equilibria β(E) for differ
ent values of f0 and determine the caloric curve cor
responding to fully stable states. This completes and
illustrates our previous study [81] where only the final
phase diagram was reported. We evidence different re
gions in the parameter space where the nature of phase
transitions changes: (i) for f0 > (f0)t ≃ 0.10965, the
system displays a second order phase transition; (ii) for
(f0)1 ≃ 0.109497 < f0 < (f0)t ≃ 0.10965, the system
displays a second order phase transition and a first or
der phase transition; (iii) for (f0)∗ ≃ 0.10947 < f0 <
(f0)1≃ 0.109497, the system displays two second order
phase transitions; (iv) for f0< (f0)∗≃ 0.10947, there is
no phase transition. The passage from a first order phase
transition to a second order phase transition corresponds
to a tricritical point. The sudden appearance of two sec
ond order phase transitions from nothing corresponds to
a second order azeotropy. This is associated with a phe
nomenon of phase reentrance. When we take into account
metastable states, the description is even richer and seven
regions must be considered (see Sec. III). In particular,
we find a new situation of phase reentrance. We also
stress two unexpected results that were not reported (or
incorrectly reported) in previous works: (i) Contrary to
what is stated in [82], there exists a region of ensembles
inequivalence but it concerns an extremely narrow range
of parameters so that the conclusions of [82] are not al
tered; (ii) the tricritical point separating second and first
order phase transitions does not exactly coincide with
the turning point of the stability line Ec(f0), contrary to
what is stated in [80], but is slightly different. Again,
the difference is small so that the main results of previ
ous works are not affected. However, this slight difference
leads to an even richer variety of phase transitions. We
may be fascinated by the fact that so many things hap
pen in such a very narrow range of parameters (typically
(f0)m ≃ 0.1075 < f0 < (f0)c ≃ 0.11253954) although
f0can take a priori any value between 0 and +∞! Fi
nally, we make clear in this paper (see Sec. II) that the
relevant control parameters associated with the Lynden
Bell theory are (f0,E) [74] while the use of the variables
(M0,E) [78, 79] may lead to physical inconsistencies in
the thermodynamical analysis.
II.
CHOICE OF THE CONTROL PARAMETERS
THE LYNDENBELL THEORY AND THE
The HMF model [31, 32], which shares many similar
ities with gravitational and charged sheet models, de
scribes the onedimensional motion of N particles of unit
mass moving on a unit circle and coupled through a mean
field cosine interaction. The system Hamiltonian reads
H =1
2
N
?
i=1
v2
i+
1
2N
N
?
i,j=1
[1 − cos(θi− θj)],(1)
where θirepresents the angle that particle i makes with
an axis of reference and vi stands for its velocity. The
Page 4
4
1/N factor in front of the potential energy corresponds
to the Kac prescription to make the system extensive
and justify the validity of the mean field approximation
in the limit N → +∞. The relevant order parameter
is the magnetization defined as M = (?
mi = (cosθi,sinθi). In the N → +∞ limit, the time
evolution of the one body distribution function f(θ,v,t)
is governed by the Vlasov equation
imi)/N where
∂f
∂t+ v∂f
∂θ− (Mx[f]sinθ − My[f]cosθ)∂f
∂v= 0, (2)
where Mx[f] =
?f(θ,v,t)sinθdθdv are the two components of the mag
netization.
The statistical theory of the Vlasov equation, intro
duced by LyndenBell [29], has been reviewed in several
papers [4, 18, 74, 75, 80, 81] so that we shall here only
recall the main lines that are important to understand
the sequel. We assume that the initial distribution func
tion takes only to values f(θ,v,t = 0) ∈ {0,f0}. For
example, it can be made of one or several patches of
uniform distribution f(θ,v,0) = f0surrounded by “vac
uum” f(θ,v,0) = 0. We note that the number and the
shape of these patches can be completely arbitrary. For
such initial conditions, the quantities conserved by the
Vlasov equation are: (i) the value f0 of the initial dis
tribution; (ii) the normalization M =?f dθdv = 1; (iii)
the energy E =
2
?fv2dθdv +1
grained distribution function f(θ,v,t) is stirred in phase
space but conserves its two values f0and 0 at any time,
i.e. f(θ,v,t) ∈ {0,f0} ∀t. However, as time goes on,
the two levels values f0and 0 become more and more in
termingled as a result of a mixing process (filamentation)
in phase space. The coarsegrained distribution f(θ,v,t),
which can be viewed as a local average of the finegrained
distribution function, takes values intermediate between
0 and f0, i.e. 0 ≤ f(θ,v,t) ≤ f0. It is expected to achieve
a steady state f(θ,v) as a result of violent relaxation on a
relatively short timescale (a few dynamical times). This
corresponds to the QSS observed in the simulations. The
most probable, or most mixed state, is obtained by max
imizing the LyndenBell entropy
?f(θ,v,t)cosθdθdv and My[f] =
1
2(1 − M2). The fine
S = −
? ?f
f0lnf
f0
+
?
1 −f
f0
?
ln
?
1 −f
f0
??
dθdv,
(3)
while conserving E and M (for a given value of f0).
This determines the statistical equilibrium state of the
Vlasov equation. Note that the whole theory relies on
an assumption of ergodicity, i.e. efficient mixing. Our
aim here is not to determine the range of validity of the
LyndenBell theory, so that we shall assume that this as
sumption is fulfilled (see, e.g. [77], for a discussion of
incomplete relaxation in the HMF model). We are led
therefore to considering the maximization problem
max
f
?S[f]E[f] = E,M[f] = 1?, (4)
for a given value of f0. The critical points of (4), cancel
ing the first order variations of the constrained entropy,
are given by the variational principle
δS − βδE − αδM = 0, (5)
where β and α are Lagrange multipliers. This yields the
LyndenBell distribution
fLB(θ,v) =
f0
1 + eβf0ǫ(θ,v)+f0α,(6)
where ǫ(θ,v) = v2/2 − Mx[fLB]cosθ − My[fLB]sinθ
is the individual energy. In the twolevels approxima
tion, the LyndenBell distribution is formally identical
to the FermiDirac statistics [29]. Note that T = β−1=
(∂S/∂E)−1is the thermodynamical temperature. Since
the distribution function (6) is nonBoltzmannian, the
thermodynamical temperature differs from the classical
kinetic temperature Tkin =
been studied specifically in [82].
The maximization problem (4) corresponds to the mi
crocanonical ensemble (MCE). Since the LyndenBell
theory is based on the Vlasov equation that describes
an isolated system, the microcanonical ensemble is the
relevant ensemble to consider (the energy is fixed). We
can, however, formally define a canonical ensemble. We
introduce the free energy functional J[f] = S[f]−βE[f]
[93] and consider the maximization problem
?fv2dθdv. This point has
max
f
?J[f]M[f] = 1?,(7)
for a given value of f0. The maximization problems (4)
and (7) have the same critical points since the variational
principle
δJ − αδM = 0(8)
returns Eq. (5) (recall that β is fixed in the canonical
ensemble). In addition, it can be shown at a general
level [24] that a solution of the canonical problem (7) is
always a solution of the more constrained dual micro
canonical problem (4), but that the reciprocal is wrong
in case of ensembles inequivalence [94]. Therefore, even
if the canonical ensemble is not physically justified in
the context of LyndenBell’s theory of violent relaxation,
it provides nevertheless a sufficient condition of micro
canonical thermodynamical stability. It is therefore use
ful in that respect. In addition, it is interesting on a
conceptual point of view to study possible inequivalence
between microcanonical and canonical ensembles. There
fore, we shall study in this paper the two maximization
problems (4) and (7), while emphasis and illustrations
will be given for the more physical microcanonical case.
Before that, let us recall general notions that will be
useful in the sequel (for an extended account, see e.g.
[7]). For a given value of f0, the series of equilibria is
the curve β(E) containing all the critical points of (4)
Page 5
5
or (7) (as we have seen, they are the same). The sta
ble part of this curve, in each ensemble, gives the corre
sponding caloric curve. In MCE, the control parameter
is the energy and the stable states are maxima of en
tropy S at fixed energy and normalization. This defines
the microcanonical caloric curve β(E). In CE, the con
trol parameter is the inverse temperature and the stable
states are maxima of free energy J at fixed normaliza
tion. This defines the canonical caloric curve E(β). The
strict caloric curve contains only fully stable states (S)
that are global entropy maxima at fixed energy and nor
malization in MCE or global free energy maxima at fixed
normalization in CE. The physical caloric curve contains
fully stable and metastable states (M), that are local en
tropy maxima at fixed energy and normalization in MCE
or local free energy maxima at fixed normalization in
CE. The unstable states (U), that are minima or sad
dle points of the thermodynamical potential, must be
rejected. Note that for systems with longrange inter
actions, metastable states can have very long lifetimes
so that they are very important in practice. By study
ing the caloric curve β(E) for a given value of f0, we
can describe phase transitions. Microcanonical first or
der phase transition are marked by the discontinuity of
the inverse temperature β(E) at some energy Et. This
corresponds to a discontinuity of the first derivative of
entropy S′(E) = β(E) at Et in the energy vs entropy
curve. There can exist metastable branches around Et
that possibly end at microcanonical spinodal points. Mi
crocanonical second order phase transitions are marked
by the discontinuity of β′(E) at some energy Ec. This
corresponds to a discontinuity of the second derivative of
entropy S′′(E) = β′(E) at Ec. Similarly, canonical first
order phase transitions are marked by the discontinuity
of energy E(β) at some inverse temperature βt. This
corresponds to a discontinuity of the first derivative of
free energy J′(E) = −E(β) at βtin the inverse temper
ature vs free energy curve. There can exist metastable
branches around βtthat possibly end at canonical spin
odal points. Canonical second order phase transitions
are marked by the discontinuity of E′(β) at some inverse
temperature βc. This corresponds to a discontinuity of
the second derivatives of free energy J′′(β) = −E′(β) at
βc. Finally, by varying the external parameter f0, we can
describe changes from different kinds of phase transitions
at some critical values of f0and plot the corresponding
phase diagrams (f0,E) and (f0,β) in microcanonical and
canonical ensembles. This is the programm that we shall
follow in this paper.
We emphasize that these general results are valid for
the caloric curve β(E) where β is the inverse thermo
dynamical temperature, not the inverse kinetic temper
ature. In particular, the thermodynamical specific heat
C = dE/dT is always positive in the canonical ensemble
while the kinetic specific heat Ckin = dE/dTkin can be
positive or negative in the canonical ensemble. This has
been illustrated in [82].
A last comment is in order. If we consider a waterbag
initial condition in which f(θ,v,t = 0) = f0in the rect
angle [θmin,θmax] × [−vmin,vmax] and f(θ,v,t = 0) = 0
outside, it seems convenient to take as control param
eters the initial magnetization M0and the energy E as
done in [78, 79]. Indeed, the specification of these param
eters determines f0= φ(E,M0) and E and thus allows
to compute the corresponding LyndenBell state. There
fore, it seems that the choice of the control parameters
(E,M0) or (E,f0) is just a question of commodity. In
fact, this is not the case, and we would like to point out
some difficulties in taking (E,M0) as control parameters
in the thermodynamical analysis:
(i) The control parameters (E,M0) are less general
than (E,f0) because they assume that the initial con
dition is a waterbag distribution, whereas the control
parameters (E,f0) are valid for any initial distribution
with two levels, whatever the number of patches and their
shape. They allow therefore to describe a wider class of
situations.
(ii) The variables (E,M0) may lead to redundancies
because there may exist two (or more) couples (E,M(1)
and (E,M(2)
0) that correspond to the same (E,f0) and,
consequently, to the same LyndenBell state (recall that
the LyndenBell prediction only depends on E and f0)
[95]. This has been illustrated in [80, 81].
(iii) More importantly, the use of M0as an external pa
rameter (instead of f0) leads to physical inconsistencies
in the thermodynamical analysis. Indeed, if we work in
terms of the variables (E,M0), the initial value of the dis
tribution f0becomes a function f0= φ(E,M0) of these
variables. As a result, the LyndenBell entropy functional
0)
S = −
? ?
?
f
φ(E,M0)ln
f
φ(E,M0)
??
+
?
1 −
f
φ(E,M0)
?
ln1 −
f
φ(E,M0)
dθdv, (9)
depends not only of the external parameter M0but also
on the energy E. This is clearly a very unconventional
situation. Indeed, if we want to apply the standard re
sults recalled above, the entropic functional can depend
on an external parameter but it cannot explicitly depend
on the energy. Therefore, these general results [24] are
not valid for functionals of the form (9). In particular,
the “improper” caloric curve β(E) at fixed M0can dis
play a region of negative specific heat while the proper
caloric curve β(E) at fixed f0does not. This is exempli
fied in Fig. 1(b) of [79] where the entropy versus energy
is plotted at fixed M0. This curve has a convex dip (re
vealing a negative specific heat region), while the curve
S(E) at fixed f0 has no convex dip and the ensembles
are equivalent [82].
Finally, in the other contexts where the LyndenBell
theory has been applied [41, 52, 58–60, 63], the control
parameters that have been taken are E and f0. It is
therefore important to describe the phase transitions in
terms of these parameters as initiated in [74].