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