1,720,972 research outputs found
Lebowitz-Penrose limit for continuum particle systems
No full textWe consider a particle system in dimensions with
an attractive Kac potential whose scaling parameter is
and a repulsive Kac potential
with parameter
. They have the same form as
the corresponding potentials in \cite{LMP}, but in \cite{LMP}
the scaling parameter was \ga for both of them.
We compute
the \llp\ for the pressure and prove that the
minimizers
of the limit variational problem are spatially constant.
The limit phase diagram has a
liquid-vapour, of van der Waals type phase transition
as, in \cite{LMP
Asymptotic expansion of the pressure in the inverse interaction range
No full textWe consider an Ising system in d greater than or equal to 2 dimensions with a ferromagnetic Kac potential whose scaling parameter is denoted by gamma. We derive an asymptotic series for the thermodynamic pressure P-beta,P-gamma, in powers of gamma. The 0th-order term of the expansion is the mean-field pressure of the Lebowitz and Penrose theory
On the absence of non-translational invariant Gibbs states in two dimensions
No full textWe consider an Ising system in dimensions with a
ferromagnetic Kac potential whose scaling parameter
is denoted by . We will show that for any \ga
sufficiently small every Gibbs state is translational
invariant.
Taking account of the result on the phase diagram \cite{BMP},
our result implies that below the critical temperature,
for any \ga sufficiently small there exist only two
pure phase
First-order phase transition in Potts models with finite-range interactions
No full textWe consider the -state Potts model on , , , with Kac ferromagnetic interactions and scaling parameter \ga. We prove the existence of a first order phase transition for large but finite potential ranges. More precisely we prove that for \ga small enough there is a value of the temperature at which coexist Gibbs states. The proof is obtained by a perturbation around mean-field using Pirogov-Sinai theory. The result is valid in particular for , Q=3, in contrast with the case of nearest-neighbor interactions for which available results indicate a second order phase transition. Putting both results together provides an example of a system which undergoes a transition from second to first order phase transition by changing only the finite range of the interaction
Study of a Long Range Perturbation of a One-Dimensional Kac Model
No full textWe consider a one dimensional ferromagnetic Ising spin system with interactions
that correspond to a
long range perturbation of the usual Kac model. We apply a coarse
graining procedure widely used for higher-dimensional finite range Kac potentials to describe the basic properties of the system and the relation with the mean field theory
On the validity of the van der Waals theory in Ising systems with long range interactions
No full textWe consider an Ising system
in dimensions with ferromagnetic
spin-spin
interactions -J_\g(x,y)\s(x)\s(y), , , where
J_\g(x,y) scales like a Kac potential. We prove that when the
temperature is below the mean field critical value,
for any \g small enough (i.e. when the range of the interaction
is long but finite), there are only two pure homogeneous phases,
as stated by the van der Waals theory. After introducing
block spin variables and
relying on the Peierls estimates proved
in [\rcite{CP}], the proof follows that in
[\rcite{GM}] on the translationally invariant states at low
temperatures for nearest neighbor interactions, supplemented by
a ``relative
uniqueness criterion for Gibbs fields" which yields
uniqueness in a restricted ensemble of measures, in a
context where there is a phase transition. This
criterion is derived by introducing special couplings as in
[\rcite {BM}] which reduce the proof of relative
uniqueness to the absence of percolation of ``bad even
Renewal properties of the d = 1 Ising model
Full text linkWe consider the d = 1 Ising model with Kac potentials at inverse temperature β > 1 where the mean field predicts a phase transition with two possible equilibrium magnetizations ± mβ, mβ > 0. We show that when the Kac scaling parameter γ is sufficiently small, typical spin configurations are described (via a coarse graining) by an infinite sequence of successive plus and minus intervals where the empirical magnetization is "close" to mβ, and respectively, - mβ. We prove that the corresponding marginal of the unique DLR measure is a renewal process
Coexistence of ordered and disordered phases in Potts models in the continuum
No full textThis is the second of two papers on a continuum version of the Potts model, where
particles are points in Rd , d ≥ 2, with a spin which may take S ≥ 3 possible values. Particles
with different spins repel each other via a Kac pair potential of range γ
−1, γ >0. In this
paper we prove phase transition, namely we prove that if the scaling parameter of the Kac
potential is suitably small, given any temperature there is a value of the chemical potential
such that at the given temperature and chemical potential there exist S +1 mutually distinct
DLR measures
One-Dimensional Ising Models with Long Range Interactions: Cluster Expansion, Phase-Separating Point
No full textWe consider the phase separation problem for the one-dimensional ferromagnetic
Ising model with long-range two-body interaction, J (n) = n−2+α, where n ∈ N
denotes the distance of the two spins and α ∈ [0, α+[ with α+ = (log 3)/(log 2)−1.We
prove that when α = 0 the localization of the phase separation fluctuates macroscopically
with a non-uniform explicit limiting law, while when 0 < α < α+ the macroscopic
fluctuations disappear and mesoscopic ones appear with a gaussian behavior when conveniently
scaled. The mean magnetization profile is also given
Geometry of contours and Peierls estimates in d=1 Ising models with long range interactions
No full textFollowing Fröhlich and Spencer, we study one dimensional Ising spin systems with ferromagnetic, long range interactions which decay as ∣x−y∣−2+α, 0 ⩽ α ⩽ 1/2. We introduce a geometric description of the spin configurations in terms of triangles which play the role of contours and for which we establish Peierls bounds. This in particular yields a direct proof of the well-known result by Dyson about phase transitions at low temperatures
- …
