I. Merola | University of L'Aquila (original) (raw)
Papers by I. Merola
We prove a liquid vapor phase transition for a quantum system of particles in the continuum. The ... more We prove a liquid vapor phase transition for a quantum system of particles in the continuum. The system is the quantum version with Boltzmann statistics of the point particles model introduced in (8). AMS classification: 82B26, 82B10
We consider an Ising system in d ≥ 2 dimensions with ferromagnetic spinspin interactions −J γ (x,... more We consider an Ising system in d ≥ 2 dimensions with ferromagnetic spinspin interactions −J γ (x, y)σ(x)σ(y), x, y ∈ Z d , where J γ (x, y) scales like a Kac potential. We prove that when the temperature is below the mean field critical value, for any γ 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 [7], the proof follows that in [12] 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 [2] which reduce the proof of relative uniqueness to the absence of percolation of "bad events".
Journal of Statistical Physics, 2007
We consider the Q-state Potts model on Z d , Q ≥ 3, d ≥ 2, with Kac ferromagnetic interactions an... more We consider the Q-state Potts model on Z d , Q ≥ 3, d ≥ 2, with Kac ferromagnetic interactions and scaling parameter γ. We prove the existence of a first order phase transition for large but finite potential ranges. More precisely we prove that for γ small enough there is a value of the temperature at which coexist Q + 1 Gibbs states. The proof is obtained by a perturbation around mean-field using Pirogov-Sinai theory. The result is valid in particular for d = 2, 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.
Journal of Mathematical Physics, 2005
Following Fröhlich and Spencer, [8], we study one dimensional Ising spin systems with ferromagnet... more Following Fröhlich and Spencer, [8], 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.
Journal of Mathematical Physics, 2005
Inverse Problems, 1994
In the framework of the reduction technique for Poisson-Nijenhuis structures, we derive a new hie... more In the framework of the reduction technique for Poisson-Nijenhuis structures, we derive a new hierarchy of integrable lattice, whose continuum limit is the AKNS hierarchy.
Journal of Statistical Physics, 2000
We consider extended Pirogov-Sinai models including lattice and continuum particle systems with K... more We consider extended Pirogov-Sinai models including lattice and continuum particle systems with Kac potentials. Calling λ an intensive variable conjugate to an extensive quantity α appearing in the Hamiltonian via the additive term −λα, we prove that if a Pirogov-Sinai phase transition with order parameter α occurs at λ = 0, then this is the only point in an interval of values of λ centered at 0, where phase transitions occur.
We prove a liquid vapor phase transition for a quantum system of particles in the continuum. The ... more We prove a liquid vapor phase transition for a quantum system of particles in the continuum. The system is the quantum version with Boltzmann statistics of the point particles model introduced in (8). AMS classification: 82B26, 82B10
We consider an Ising system in d ≥ 2 dimensions with ferromagnetic spinspin interactions −J γ (x,... more We consider an Ising system in d ≥ 2 dimensions with ferromagnetic spinspin interactions −J γ (x, y)σ(x)σ(y), x, y ∈ Z d , where J γ (x, y) scales like a Kac potential. We prove that when the temperature is below the mean field critical value, for any γ 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 [7], the proof follows that in [12] 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 [2] which reduce the proof of relative uniqueness to the absence of percolation of "bad events".
Journal of Statistical Physics, 2007
We consider the Q-state Potts model on Z d , Q ≥ 3, d ≥ 2, with Kac ferromagnetic interactions an... more We consider the Q-state Potts model on Z d , Q ≥ 3, d ≥ 2, with Kac ferromagnetic interactions and scaling parameter γ. We prove the existence of a first order phase transition for large but finite potential ranges. More precisely we prove that for γ small enough there is a value of the temperature at which coexist Q + 1 Gibbs states. The proof is obtained by a perturbation around mean-field using Pirogov-Sinai theory. The result is valid in particular for d = 2, 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.
Journal of Mathematical Physics, 2005
Following Fröhlich and Spencer, [8], we study one dimensional Ising spin systems with ferromagnet... more Following Fröhlich and Spencer, [8], 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.
Journal of Mathematical Physics, 2005
Inverse Problems, 1994
In the framework of the reduction technique for Poisson-Nijenhuis structures, we derive a new hie... more In the framework of the reduction technique for Poisson-Nijenhuis structures, we derive a new hierarchy of integrable lattice, whose continuum limit is the AKNS hierarchy.
Journal of Statistical Physics, 2000
We consider extended Pirogov-Sinai models including lattice and continuum particle systems with K... more We consider extended Pirogov-Sinai models including lattice and continuum particle systems with Kac potentials. Calling λ an intensive variable conjugate to an extensive quantity α appearing in the Hamiltonian via the additive term −λα, we prove that if a Pirogov-Sinai phase transition with order parameter α occurs at λ = 0, then this is the only point in an interval of values of λ centered at 0, where phase transitions occur.