Giambattista Giacomin

Phase segregation dynamics in particle systems with long range interactions. I. Macroscopic limits

We present and discuss the derivation of a nonlinear nonlocal integrodifferential equation for the macroscopic time evolution of the conserved order parameter ρ(r, t) of a binary alloy undergoing phase segregation. Our model is ad-dimensional lattice gas evolving via Kawasaki exchange with respect to the Gibbs measure for a Hamiltonian which includes both short-range (local) and long-range (nonlocal) interactions. The nonlocal part is given by a pair potential γdJ(γ|x−y|), γ>0 x and y in ℤd, in the limit γ→0. The macroscopic evolution is observed on the spatial scale γ−1 and time scale γ−2, i.e., the density ρ(r, t) is the empirical average of the occupation numbers over a small macroscopic volume element centered atr=γx. A rigorous derivation is presented in the case in which there is no local interaction. In a subsequent paper (Part II) we discuss the phase segregation phenomena in the model. In particular we argue that the phase boundary evolutions, arising as sharp interface limits of the family of equations derived in this paper, are the same as the ones obtained from the corresponding limits for the Cahn-Hilliard equation.

Macroscopic evolution of particle systems with short- and long-range interactions

We consider a lattice gas with general short-range interactions and a Kac potential Jγ(r) of range γ-1, γ>0, evolving via particles hopping to nearest-neighbour empty sites with rates which satisfy detailed balance with respect to the equilibrium measure. Scaling spacelike γ-1 and timelike γ-2, we prove that in the limit γ→0 the macroscopic density profile ρ(r,t) satisfies the equation Here σs(ρ) is the mobility of the reference system, that with J≡0, and (ρ) = ∫[fs(ρ(r))-½ρ(r)∫J(r-r)ρ(r)xa0drxa0dr], where fs(ρ) is the (strictly convex) free energy density of the reference system. Beside a regularity condition on J, the only requirement for this result is that the reference system satisfy the hypotheses of the Varadhan-Yau theorem leading to (*) for J≡0. Therefore, (*) also holds if achieves its minimum on non-constant density profiles and this includes the cases in which phase segregation occurs. Using the same techniques we also derive hydrodynamic equations for the densities of a two-component A-B mixture with long-range repulsive interactions between A and B particles. The equations for the densities ρA and ρB are of the form (*). They describe, at low temperatures, the demixing transition in which segregation takes place via vacancies, i.e.xa0jumps to empty sites. In the limit of very few vacancies the problem becomes similar to phase segregation in a continuum system in the so-called incompressible limit.

Entropic repulsion for massless fields

We consider the anharmonic crystal, or lattice massless field, with 0-boundary conditions outside and N a large natural number, that is the finite volume Gibbs measure on for every x[negated set membership]DN} with Hamiltonian [summation operator]x~yV([phi]x-[phi]y), V a strictly convex even function. We establish various bounds on , where [Omega]+(DN)={[phi]:[phi]x[greater-or-equal, slanted]0 for all x[set membership, variant]DN}. Then we extract from these bounds the asymptotics (N-->[infinity]) of : roughly speaking we show that the field is repelled by a hard-wall to a height of in d[greater-or-equal, slanted]3 and of O(log N) in d=2. If we interpret [phi]x as the height at x of an interface in a (d+1)-dimensional space, our results on the conditioned measure clarify some aspects of the effect of a hard-wall on an interface. Besides classical techniques, like the FKG inequalities and the Brascamp-Lieb inequalities for log-concave measures, we exploit a representation of the random field in term of a random walk in dynamical random environment (Helffer-Sjostrand representation).

On Entropic Reduction of Fluctuations

We point out that there is no general relation between ground state degeneracy and finite-temperature fluctuations for tilted interfaces.

The Sherrington-Kirkpatrick model with short range ferromagnetic interactions

Abstract We study a model of Ising spins with short range ferromagnetic and long range SK interactions. We generalize the results obtained for the standard SK model, computing in particular the high temperature pressure.