*Moderator: *Anna De Masi (L'Aquila)

Cédric Bernardin (Nice),
Gioia Carinci (Delft),
Bernard Derrida (Paris),
Laurent Desvillettes (Paris),
Cristian Giardinà (Modena),
Rajamani Krishna (Amsterdam),
Dimitrios Tsagkarogiannis (Brighton).

Macroscopic fractional diffusion appears in various situations. Lévy flights or Lévy walks are standard microscopic models to describe such phenomena. Interesting problems appear when we consider them evolving in bounded domains. In this talk I will discuss simple microscopic models of interacting Lévy flights (exclusion process with long jumps) giving rise to some fractional reaction-diffusion equation with some boundary conditions.

**
Gioia Carinci** (Delft):
*Particle Systems and Free Boundary Problems.*

There are several systems of interest in physics and biology which are related to free boundary problems in PDE's. In particular I will refer to the Brunet-Derrida models for biological selection mechanisms and the Fourier law for particles systems in domains which change in time due to reservoirs which fix the current (while usually it is the spatial domain and the density at the boundaries which are fixed). In both cases the hydrodynamic (or continuum) limit is described by parabolic equations with free boundaries of Stefan-type. I will give in this talk a short survey of the methods used to study such particle systems and the free boundary problems for the corresponding PDE's. Well known theorems on the Stefan problem yield a local existence for "classical initial data". I will introduce a new notion of relaxed solution for which global existence and uniqueness is proven for general initial data.

**
Bernard Derrida** (Paris):
*The Fisher KPP equation: some exactly soluble versions.*

After a short review of the known properties of the solution Fisher-KPP
equation, in
particular on how front positions depend on initial condition, an
exactly soluble case will be presented, where the problem is reduced to a
question of complex analysis.

E. Brunet, B. Derrida, An Exactly Solvable Travelling Wave Equation in the
Fisher KPP Class J. Stat. Phys. 161, 801-820 (2015). arXiv:1506.06559

J. Berestycki, E. Brunet, B. Derrida, Exact solution and precise
asymptotics of a Fisher-KPP type front. arXiv:1705.08416

**
Laurent Desvillettes** (Paris):
*A microscopic approach to cross diffusion equations in population
dynamics.*

Cross diffusion equations have been used in population dynamics at least since the 70s, in order to model the segregation of population in competition. One of the emblematic models in this field is the "SKT"-model, proposed by Shigesada, Kawasaki and Teramato. A microscopic version of this model, using a relaxation mechanism on a small time scale, was later introduced, and the convergence towards the original model when the relaxation time tends to 0, was conjectured and checked numerically by IIda, Mimura, and Ninomiya. Recent progresses in the theory of reaction-diffusion equations, obtained by M. Breden, F. Conforto, C. Soresina, A. Trescases and myself, now enable to show rigorously this convergence. We present those advances and discuss their possible use in other applicative contexts.

**
Cristian Giardinà** (Modena):
*Non-equilibrium 2D Ising model with stationary uphill diffusion.*
pdf

The phenomenon of "uphill diffusion" (a current which goes up the gradient) has been observed in several systems, in particular in multi-component systems. Recently, uphill diffusion has also been observed, by means of numerical simulations, in the low-temperature stationary state of the non-equilibrium 2D Ising model in contact with magnetization reservoirs, as well as in 1D spin systems with Kac potential. This opens up a novel perspective on uphill diffusion, that is due to the existence of a phase transition and the work performed by the reservoirs. We shall present some conjectures that explain the observed phenomenology by a stability analysis of interfaces.

**
Rajamani Krishna** (Amsterdam):
*Uphill Diffusion.*

Molecular diffusion is an omnipresent phenomena that is important in a wide variety of contexts in chemical, physical, and biological processes. In the majority of cases, the diffusion process can be adequately described by Fick's law that postulates a linear relation between the flux of any species and its own concentration gradient. Most commonly, a component diffuses down the concentration gradient. The major objective of my presentation is to highlight a very wide variety of situations that cause uphill transport of one constituent in the mixture.

Uphill diffusion may occur in multicomponent mixtures in which the diffusion flux of any species is strongly coupled to that of partner species. Such coupling effects often arise from strong thermodynamic non-idealities; for a quantitative description we need to use chemical potential gradients as driving forces. The transport of ionic species in aqueous solutions is coupled with its partner ions because of the electro-neutrality constraints; such constraints may accelerate or decelerate a specific ion. When uphill diffusion occurs, we observe transient overshoots during equilibration; the equilibration process follows serpentine trajectories in composition space. For mixtures of liquids, alloys, ceramics and glasses the serpentine trajectories could cause entry into meta-stable composition zones; such entry could result in phenomena such as spinodal decomposition, spontaneous emulsification, and the Ouzo effect.

For distillation of multicomponent mixtures that form azeotropes, uphill diffusion may allow crossing of distillation boundaries that are normally forbidden. For mixture separations with microporous adsorbents, uphill diffusion can cause supra-equilibrium loadings to be achieved during transient uptake within crystals; this allows the possibility of over-riding adsorption equilibrium for achieving difficult separations.

The theoretical background for the material that I will present is contained in my paper: R. Krishna, Uphill Diffusion in Multicomponent Mixtures, Chem. Soc. Rev. 44 (2015) 2812-2836.

**
Dimitrios Tsagkarogiannis** (Brighton):
*Fourier law, phase transitions and the stationary Stefan problem.*

We construct stationary solutions for the one-dimensional integro-differential equation describing the limit of Kawasaki dynamics with Kac potential. We prove the validity of the Fourier law in the thermodynamic limit showing that below the critical temperature the limit equilibrium profile has a discontinuity (which defines the position of the interface) and satisfies a stationary free boundary Stefan problem. If metastable values are imposed at the boundaries then the mesoscopic stationary profile is no longer monotone and therefore the Fourier law is not satisfied. It regains its validity however in the thermodynamic limit where the limit profile is again monotone away from the interface. This is joint work with Anna De Masi and Errico Presutti.