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PrefaceThe idea of writing tbis book came after a course given in Boulder, Colorado, in 1988

This paper studies the hydrodynamic limit of a stochastic process describing the time evolution of a system with N neurons with mean-field interactions produced both by chemical and by electrical synapses. This system can be informally described as follows. Each neuron spikes randomly following a point process with rate depending on its membrane potential. At its spiking time, the membrane potential of the spiking neuron is reset to the value 0 and, simultaneously, the membrane potentials of the other neurons are increased by an amount of potential 1 N . This mimics the effect of chemical synapses. Additionally, the effect of electrical synapses is represented by a deterministic drift of all the membrane potentials towards the average value of the system.We show that, as the system size N diverges, the distribution of membrane potentials becomes deterministic and is described by a limit density which obeys a non linear PDE which is a conservation law of hyperbolic type.

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We consider solutions of the Boltzmann equation, in a d-dimensional torus, d = 2,3, for macroscopic times T = t / e N , e a: 1, t 2 0, when the space variations are on a macroscopic scale x = eN-'r, N 2 2, x in the unit torus. Let u ( x , r ) be, for t to, a smooth solution of the incompressible Navier Stokes equations (INS) for N = 2 and of the Incompressible Euler equation (IE) for N > 2.We prove that (*) has solutions for r 6 to which are close, to O(e2) in a suitable norm, to the local Maxwellian [6/(2~T)~/']exp( -[ ueu(x, t)]'/2T} with constant density 6 and temperature 7. This is a particular case, defined by the choice of initial values of the macroscopic variables, of a class of such solutions in which the macroscopic variables satisfy more general hydrodynamical equations. For N 2 3 these equations correspond to variable density IE while for N = 2 they involve higher-order derivatives of the density.It is now possible to make various formal expansions of f, in powers of E about some local or global Maxwellian. In particular the well-known Hilbert-Chapman-Enskog method [l] leads to a sequence of equations for the hydrodynamical variables corresponding, respectively, to the (non-dissipative) Euler equations, the Navier-Stokes equations, etc. From now on we always consider a collision operator Q corresponding, a la Grad [2], to cut-off hard potentials. An important step in the mathematical development of the theory was taken by Caflisch [7] (and by Caflisch and Papanicolau [8] for a discrete velocities model) who proved the following: let p, u, T be smooth solutions of Euler equations in a torus for r E [0, to]. Then there exists a solution f E ( r , T ; u ) of the

This is the fin1 of three papers on the Glauber evolution of king spin systems with Kac potentials. We begin with the analysis of the mesoscopic limit, where space scales like the diverging range. y-', of the interaction while time is kepi finite: ye prove that in this limit the magnetization density converges to the solution of a deterministic, nonlinear, nonlocal evolution equation. We also show that the long time behaviour of this equation describes correctly the evolution of the spin system till times which diverge as y + 0 but are small in units logy-'. In this time regime we can give a "ry precise description of the evolution and a sharp characterhion of the spin trajectories. As an application of the general theory, we then prove that for ferromagnetic interactions, in the absence of extema! magnetic fields and below the critical temperature, on a suitable macroscopic limit an interface between two stable phases moves by mean curvature. All ihe proofs are consequek of sharp estimates on special correlation functions, the v-functions, whose analysis is reminiscent of the cluster expansion in equilibrium statistical mechanics.

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