download report - Sapienza
download report - Sapienza
download report - Sapienza
Create successful ePaper yourself
Turn your PDF publications into a flip-book with our unique Google optimized e-Paper software.
Scientific Report 2007-2009<br />
Theoretical physics<br />
T18. Equilibrium statistical mechanics for one dimensional long range<br />
systems<br />
It is well known that one dimensional spin systems<br />
with long range interactions decaying as |x − y| −σ can<br />
give rise to a phase transition for 1 < σ ≤ 2. It is<br />
a conjecture suggested by Anderson that the range of<br />
the interactions (i.e. 1/σ) should play the role of the<br />
dimensionality,so that , varying a , these models could<br />
mimic the properties of more realistic higher dimensional<br />
systems (e.g. the spin glasses where a satisfactory theory<br />
is still lacking). This is the motivation for a rigorous<br />
analysis of the models belonging to this class. Starting<br />
from a geometrical description of the energy fluctuations<br />
it is possible, when the interactions are ferromagnetic<br />
and the temperature is sufficiently small, to prove :<br />
1) the existence of a phase transition and the convergence<br />
of a cluster expansion that allows to study the<br />
behaviour of the separation point between two coexisting<br />
phases<br />
2) the persistence of a phase transition for σ > 3/2<br />
when a stochastic magnetic field is present. (cfr. ref 1).<br />
The actual project is to study a one dimensional system<br />
of particles interacting via long range attractive potentials.<br />
In this case the motivation is different but we<br />
plan to exploit the techniques developed for spin systems<br />
on a lattice. A central problem in equilibrium statistical<br />
mechanics is the derivation of the phase diagram of fluids<br />
where gas ,liquid and solid regions are present and separated<br />
by coexistence curves . The van der Waals theory<br />
gives a qualitative reasonable description of the liquid<br />
-vapour coexistence curve but the mean field assumed in<br />
this approach is far away from any realistic interaction<br />
and to get a result consistent with thermodynamics it is<br />
necessary to introduce the so called Maxwell construction.<br />
The first rigorous version of the van der Waals<br />
theory in statistical mechanics is due to Kac . Its main<br />
assumption is a sharp separation of the scales between<br />
the attractive and repulsive forces. In d dimensions the<br />
basic model has an attractive pair interaction of strenght<br />
γ d and range 1/γ and an hard core of lenght 1. In the<br />
limit γ going to zero it is possible to obtain the van der<br />
Waals results with the Maxwell construction included.<br />
From a physical point of view this is not yet what desired<br />
as the phase diagram is only derived in the limit γ going<br />
to zero which does not correspond to any reasonable<br />
interaction among particles. The problem is to verify if<br />
the convergence of this limit is strong enough to ensure<br />
that before the limit ( when γ is finite and the interaction<br />
is reasonable) this structure of the phase diagram<br />
is preserved. In the last decades this analysis has been<br />
successfully performed for spin systems on a lattice for<br />
dimensions larger than 1. The general strategy is to perform<br />
a coarse graining on a scale smaller then 1/γ. The<br />
limit for γ going to zero of the effective hamiltonian is a<br />
non local functional where locally the interaction is mean<br />
field . The basic idea is to consider perturbations respect<br />
to the mean field equilibrium configurations rather then<br />
the original ground state. This allow to describe the<br />
configurations in term of contours and to prove a Peierls<br />
bound for all temperatures smaller than the mean field<br />
critical temperature and γ sufficiently small. This bound<br />
not only proves the existence of a phase transition but<br />
also the convergence of a cluster expansion that allows<br />
to fully describe the system for all temperatures smaller<br />
then the mean field critical temperature. The implementation<br />
of this strategy for a system of particles in<br />
the continuum for dimensions larger then one is so far<br />
not possible . The reason is technical and related to the<br />
actual control of the hard core component. In fact, in the<br />
region where we expect to have the transition, the liquid<br />
phase is ”close” to an hard core system with an effective<br />
fugacity exceeding the value for which the convergence<br />
of the cluster expansion has been proved.<br />
In one dimension this specific problem disappears because<br />
the hard core system is isomorphic to an ideal gas<br />
but it is necessary to add an attractive long range interaction<br />
to give rise to a phase transition.<br />
We study one dimensional hard rods interacting via a<br />
finite range Kac potential plus a long range decreasing<br />
tail:<br />
1<br />
J(r) = γ 1 rγ −1<br />
r σ (1)<br />
The one dimensional nature of our system allows to<br />
control the hard core contribution. Coupling the coarse<br />
graining techniques developed for Kac potentials and<br />
a definition of contours suitable to describe the energy<br />
fluctuations in one dimensional long range systems ,<br />
we expect to obtain ,via the Pirogov-Sinai approach<br />
, a Peierls bound and fully implement the strategy<br />
developed for spin systems on a lattice.<br />
References<br />
1. M.Cassandro et al, Comm. Math. Phys 2, 731 (2009)<br />
Authors<br />
M.Cassandro<br />
http://w3.uniroma1.it/neqphecq/<br />
<strong>Sapienza</strong> Università di Roma 41 Dipartimento di Fisica