Mathematical Methods for Hydrodynamic Limits

Entropy inequalities, correlation functions, couplings

between stochastic processes are powerful techniques which

have been extensively used to give arigorous foundation to

the theory of complex, many component systems and to its

many applications in a variety of fields as physics,

biology, population dynamics, economics, ...

The purpose of the book is to make theseand other

mathematical methods accessible to readers with a limited

background in probability and physics by examining in detail

a few models where the techniques emerge clearly, while

extra difficulties arekept to a minimum.

Lanford's method and its extension to the hierarchy of

equations for the truncated correlation functions, the

v-functions, are presented and applied to prove the validity

of macroscopic equations forstochastic particle systems

which are perturbations of the independent and of the

symmetric simple exclusion processes. Entropy inequalities

are discussed in the frame of the Guo-Papanicolaou-Varadhan

technique and of theKipnis-Olla-Varadhan super exponential

estimates, with reference to zero-range models. Discrete

velocity Boltzmann equations, reaction diffusion

equations and non linear parabolic equations are considered,

as limits of particles models. Phase separation phenomena

are discussed in the context of Glauber+Kawasaki evolutions

and reaction diffusion equations. Although the emphasis is

onthe mathematical aspects, the physical motivations are

explained through theanalysis of the single models, without

attempting, however to survey the entire subject of

hydrodynamical limits.