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arXiv:1012.1566v1 [cond-mat.stat-mech] 7 Dec 2010

Out-of-equilibrium 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, F-31062 Toulouse, France

4Physics Department, Nova Gorica University 5001 Nova Gorica, Slovenia

We provide a detailed discussion of out-of-equilibrium phase transitions in the Hamiltonian Mean

Field (HMF) model in the framework of Lynden-Bell’s statistical theory of the Vlasov equation.

For two-levels 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 non-magnetized 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 long-range 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 long-range interactions exhibit a number

of unusual features that are not present in systems with

short-range 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 long-lived 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 long-range

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 long-range 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, Lynden-Bell [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 coarse-grained distributions. He

obtained the most probable distribution by maximizing

a Boltzmann-type 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 coarse-grained 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 Lynden-Bell 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 Lynden-Bell approach gives a fine first order predic-

tion of the achieved QSS and allows one to predict out-

of-equilibrium phase transitions between different types

of structures that can be compared with direct simula-

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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 Lynden-Bell approach.

Lynden-Bell’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 Lynden-Bell 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 (non-ergodicity),

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 Lynden-Bell distribution is similar to

the Fermi-Dirac statistics (in the two-levels approxima-

tion), the problem is mathematically equivalent to the

study of a gas of self-gravitating 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 Lynden-Bell 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 Lynden-Bell 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 Lynden-Bell prediction

fails [89]. Finally, Arad & Lynden-Bell [46] have shown

that the theory itself presents some inconsistencies aris-

ing from its non-transitive nature. These negative results

have led many astrophysicists to the conclusion that the

Lynden-Bell 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 long-lived vortices in astrophysical and geophys-

ical flows (a notorious example being Jupiter’s great red

spot). Large-scale 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-

Robert-Sommeria (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 large-scale 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 long-range 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

long-range potential truncated to one Fourier mode (co-

sine potential). In that second interpretation, it shares

many analogies with self-gravitating 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 out-of-equilibrium structures and that they are

non-Boltzmannian. 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 Lynden-Bell [69, 72, 73]. Chavanis [74] studied out-

of-equilibrium phase transitions in the HMF model by

analogy with similar studies in astrophysics and hydro-

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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 Lynden-Bell prediction by performing

careful comparisons with direct N-body simulations at

E = 0.69 and found a good agreement for initial magne-

tizations M0< (M0)crit(E) ≃ 0.897 leading to spatially

homogeneous Lynden-Bell 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 N-body simulations and Lynden-Bell’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 Lynden-Bell prediction (such as an addi-

tional reentrant phase and a persistence of magnetized

states in the a priori non-magnetized 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-

of-equilibrium 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 non-magnetized 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 LYNDEN-BELL THEORY AND THE

The HMF model [31, 32], which shares many similar-

ities with gravitational and charged sheet models, de-

scribes the one-dimensional 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

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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 Lynden-Bell [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 coarse-grained distribution f(θ,v,t),

which can be viewed as a local average of the fine-grained

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 Lynden-Bell 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

Lynden-Bell 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

Lynden-Bell 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 two-levels approxima-

tion, the Lynden-Bell distribution is formally identical

to the Fermi-Dirac statistics [29]. Note that T = β−1=

(∂S/∂E)−1is the thermodynamical temperature. Since

the distribution function (6) is non-Boltzmannian, 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 Lynden-Bell

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 Lynden-Bell’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)

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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 long-range 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 Lynden-Bell 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 Lynden-Bell state (recall that

the Lynden-Bell 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 Lynden-Bell 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 Lynden-Bell

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].