#### Keywords

Long
range interactions, Self gravitating systems, Plasma
physics, Geophysical Flows, Wave particle interactions,
Large deviations, Ensemble Inequivalence, Phase
transitions, Kinetic theory, Vlasov equation, Stochastic
processes.

Context

Research
Summary

Main
Results

Classification
of phase transitions

Large
deviations for systems with long range interactions

Kinetic
theory and stochastic processes

Polemic
against non extensive statistical mechanics

Generic
ergodicity breaking

Bicluster
dynamics

Long
range interactions in cold atom experiments

### Context

In
a large number of physical systems, any single particle
experiences a force which is dominated by interactions
with far away particles. For instance, in a system with
algebraic decay of the inter-particle potential V(r) ∼ 1/r^{α} for
large r, when α is less than the spatial
dimension of the system, the interaction is long range
(such interactions are sometimes called
``non-integrable’’). Such long range interacting systems
are not-additive, as the interaction of any macroscopic
part of the system with the whole is not negligible with
respect to the internal energy of the given part.

The main physical examples of non-additive, long range interacting systems are: astrophysical self-gravitating systems, two-dimensional or geophysical fluid dynamics (vortices have a logarithmic interaction), some of the plasma physics models, models for wave-particle interactions, interaction due to multiple light scattering in cold atoms, or due to the dipolar force in optics, etc ... Spin systems and toy models with long range interactions have also been widely studied.

As a consequence of the lack of additivity, peculiar thermodynamic behaviors are likely to be observed in such Hamiltonian systems. For instance, the usual proof of the validity of the canonical ensemble, for a system in contact with a thermostat, or for a part of a bigger isolated system, uses explicitly this additivity property. Hertel and Thirring provided a toy model, mimicking self-gravitating dynamics, which displays inequivalence between canonical and microcanonical solutions, with negative specific heat regions in the microcanonical ensemble. Negative specific heat and ensemble inequivalence, previously known to astrophysicists, were then found in various fields: plasma physics and geophysical fluid dynamics. These examples show that new types of phase transitions are found in long range interacting systems.

Recently, a new light was shed on the equilibrium statistical mechanics of these systems. The first reason is that a mathematical characterization (Ellis and col. 2001) of ensemble inequivalence and the study of several simple models have illustrated different types of inequivalence that might exist between the microcanonical and the canonical ensembles. The second is the appearance of a very useful technique, namely the large deviation theory, to compute the microcanonical number of microstates and thus the associated microcanonical entropy (Barré, Bouchet, Dauxois and Ruffo 2005). The third is a full classification of phase transitions and of ensemble inequivalence in such systems (Bouchet and Barré 2005). The last, but not the least reason, is the understanding that the broad spectrum of applications should be considered simultaneously since significant advances were performed independently in the different domains.

The dynamics of all of these systems also share deep analogies between each other. For instance, the analogies between the Vlasov equation and the 2D Euler equation have repeatedly been illustrated in the last forty years. The studies of the equilibrium states linear and nonlinear stabilities, is very similar for each of these models (Arnold’s type theorems) and are deeply related to the thermodynamics stability. Moreover the equilibrium fluctuations and the relaxation towards equilibrium is described by similar kinetic theories.

Recently,
we
have
emphasized the role of Quasi-Stationnary-States in such
systems, and explained their long time stability
(Yamaguchi and col. 2003) in the framework of usual
kinetic theory. We have theoretically explained and
predicted, previously observed anomalous diffusion and non
exponential relaxation, in these systems (Bouchet and
Dauxois 2005). Both of these works, led in the framework
of usual statistical mechanics, led to a strong
controversy with the defenders of Tsallis non extensive
statistics (see for instance Bouchet, Dauxois and Ruffo
2006). We have also proved that the relaxation towards
equilibrium of 1D systems with long range interaction is
not described by the usual quasi-linear kinetic theory
(Lenard-Balescu equation) (Bouchet and Dauxois 2005).

Classification of phase transitions and ensemble
inequivalence

Systems
with
long
range interactions in general are not additive, which can
lead to an inequivalence of the microcanonical and
canonical ensembles. The microcanonical ensemble may show
richer behavior than the canonical one, including negative
specific heats and other non-common behaviors.

By analogy with the classical Landau classification of
phase transitions, valid for systems with short range
interactions ; in this work we proposed a classification
of microcanonical phase transitions, of their link to
canonical ones, and of the possible situations of ensemble
inequivalence, in systems with long range interactions.

This classification is thus an exhaustive list of all
possible phase transitions and situations of ensemble
inequivalence. We emphasize on impossible situations, due
to thermodynamical constraints.

We discuss previously observed phase transitions and
inequivalence in self-gravitating, two-dimensional fluid
dynamics and non-neutral plasmas. We note a number of
generic situations that have not yet been observed in such
systems. This opens the quest for observing such phase
transitions in real physical systems or in models.

(see Bouchet-Barre-JStatPhys-2005).

Large deviations and systems with long range interactions

We
have used large deviation estimations for the probability
of macrostates in the microcanonical ensemble, in order to
solve models with long-range interactions in the
microcanonical and canonical ensemble. We show how this
can be adapted to obtain the solution of a large class of
simple models, which can show ensemble inequivalence. The
model Hamiltonian can have both discrete (Ising,Potts) and
continuous (HMF, Free Electron Laser) state variables. We
treat both infinite range and slowly decreasing
interactions and, in particular, we present the solution
of the α−Ising model in onedimension with 0 ≤α < 1.
This extremely simple model already displays ensemble
inequivalence.

Large deviation estimations is a very powerful tool that
has been then used in a number of other domains.

(see Barre-Bouchet-Dauxois-Ruffo-JStatPhys-2005).

Kinetic theory and nonlinear dynamics in systems with
long range interactions

We
have proposed a kinetic description of the Hamiltonian
Mean Field model, which is paradigmatic for dynamical
systems with long-range interactions, following the
classical quasi-linear theory.

(see Yamaguchi-Barre-Bouchet-Dauxois-Ruffo-PhysicaD-2003, Bouchet-PRE-2004, Bouchet-Dauxois-PRE-2004, Yamaguchi-Bouchet-Dauxois-JStatMech-2007 and Jain-Bouchet-Mukamel-JStatMech-2007.).

In
this framework, we have shown that generically, the Fokker
Planck equation which governs fluctuations, has a non
exponential relaxation and leads to anomalous behaviors.
This mechanism is not limited to the HMF model. (see Bouchet-Dauxois-PRE-2004)

Studying this Fokker-Planck equation in details, we have
predicted algebraic tails for the momentum
auto-correlations and anomalous diffusion for the angles.
We derive analytically the corresponding laws in the limit
of a large number of particles. We argue that the
mechanism for such an anomalous transport does not depend
on some complex structure of the phase space: indeed, the
transport is anomalous for out-of-equilibrium
distributions but also for the equilibrium microcanonical
distribution. (see Bouchet-Dauxois-PRE-2004
and Yamaguchi-Bouchet-Dauxois-JStatMech-2007)

We have also proved that the relaxation towards
equilibrium of 1D systems with long range interaction is
not described by the usual quasi-linear kinetic theory
(Lenard-Balescu equation). (see Bouchet-Dauxois-PRE-2004)

With K. Jain and D. Mukamel, we have addressed the
validity time for the Vlasov description of a N-body
system with long range interactions. We have proved that
the known lower bound t ∝ log(N) (Braun and Hepp) is
actually optimal. We have exemplified this both in systems
with continuous state variables (Hamiltonian dynamics) and
discrete systems with Monte Carlo type dynamics. (see Jain-Bouchet-Mukamel-JStatMech-2007).

Polemic with C. Tsallis group against the “Non extensive
statistical mechanics”

The
classical
equilibrium
statistical mechanics used in all the previous works, is
based on the classical and natural hypothesis of averaging
with respect to a uniform distribution, on an energy
shell, of the N particle phase space (microcanonical
distribution). For out of equilibrium problems, our
approach deeply rely on the actual dynamics of the system
(kinetic theory).

By contrast, an other school uses maximization of a
baseless functional, called “non extensive entropy”. Then
by logically meaningless considerations, but using
analogies, they develop a whole “theory” similar to the
classical statistical mechanics.

This point of view concerning the "Non extensive
statistical mechanics" may seem extreme, but it is in
accordance with the reality, at least as far as I
understand it.

Because our works led to contradictions with
previous statements of C. Tsallis’ group, the publication
of the latter has been very difficult. This also led to a
comment to an article published in Europhysics news.

(see Yamaguchi-Barre-Bouchet-Dauxois-Ruffo-PhysicaD-2003, Bouchet-PRE-2004, Bouchet-Dauxois-PRE-2004 and Bouchet-Dauxois-Ruffo-EurophysicsNews-2006)

Generic ergodicity breaking in systems with long range
interactions

(see Bouchet-Dauxois-Mukamel-Ruffo-PRE-2008)

(see Barre-Bouchet-Dauxois-Ruffo-PRL-2002
and Barre-Bouchet-Dauxois-Ruffo-EPJB-2002)