Fourier Law, Phase Transitions and the Stationary Stefan Problem

  title={Fourier Law, Phase Transitions and the Stationary Stefan Problem},
  author={Anna De Masi and Errico Presutti and Dimitrios K. Tsagkarogiannis},
  journal={Archive for Rational Mechanics and Analysis},
We study the one-dimensional stationary solutions of the integro-differential equation which, as proved in Giacomin and Lebowitz (J Stat Phys 87:37–61, 1997; SIAM J Appl Math 58:1707–1729, 1998), describes the limit behavior of the Kawasaki dynamics in Ising systems with Kac potentials. We construct stationary solutions with non-zero current and prove the validity of the Fourier law in the thermodynamic limit showing that below the critical temperature the limit equilibrium profile has a… 
Fick's law in non-local evolution equations
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Current with “wrong” sign and phase transitions
We prove that under certain conditions, phase separation is enough to sustain a regime in which current flows along the concentration gradient, a phenomenon which is known in the literature as
Free Boundary Problems in PDEs and Particle Systems
In this volume a theory for models of transport in the presence of a free boundary is developed. Macroscopic laws of transport are described by PDEs. When the system is open, there are several
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AbstractWe study a system of particles in the interval [0,ϵ-1]∩Z,ϵ-1 a positive integer. The particles move as symmetric independent random walks (with reflections at the endpoints); simultaneously
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A Free Boundary Problem with Non Local Interaction
  • Jimyeong Lee
  • Mathematics
    Mathematical Physics, Analysis and Geometry
  • 2018
We prove local existence for classical solutions of a free boundary problem which arises in one of the biological selection models proposed by Brunet and Derrida, (Phys. Rev. E 56, 2597D2604, 1997)


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We study a one-dimensional lattice gas where particles jump stochastically obeying an exclusion rule and having a “small” drift toward regions of higher concentration. We prove convergence in the
Phase Segregation Dynamics in Particle Systems with Long Range Interactions II: Interface Motion
It is argued that the obtained interface evolution laws coincide with the ones which can be obtained in the analogous limits from the Cahn--Hilliard equation, the fourth-order PDE which is the standard macroscopic model for phase segregation with one conservation law.
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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
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
Metastability for the Exclusion Process with Mean-Field Interaction
AbstractWe consider an exclusion particle system with long-range, mean-field-type interactions at temperature 1/β. The hydrodynamic limit of such a system is given by an integrodifferential equation
Scaling Limits in Statistical Mechanics and Microstructures in Continuum Mechanics
Statistical Mechanics of Ising systems.- Thermodynamic limit in the Ising model.- The phase diagram of Ising systems.- Mean field, Kac potentials and the Lebowitz-Penrose limit.- Stochastic
Rigorous Treatment of the Van Der Waals-Maxwell Theory of the Liquid-Vapor Transition
Rigorous upper and lower bounds are obtained for the thermodynamic free‐energy density a(ρ, γ) of a classical system of particles with two‐body interaction potential q(r) + γνφ(γr) where ν is the
The low-temperature phase of Kac-Ising models
We analyze the low-temperature phase of ferromagnetic Kax-Ising models in dimensionsd≥2. We show that if the range of interactions is γ−1, then two disjoint translation-invariant Gibbs states exist
Phase segragation dynamics in particle systems with long range interactions. I
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Phase transitions in Ising systems with long but finite range interactions
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