Live-blogging from STATPHYS26 /3

After the first two days, it is remarkable to notice that the nonequilibrium parallel session is by far the most populated. That’s encouraging…

Fluid models as scaling limits of systems of particles, L. Saint-Raymond. From microscopic Newton’s equations, to mesoscopic description in terms of Boltzmann’s kinetic equation, and to a macroscopic description as a continuous fluid equation of hydrodynamics. Boltzmann equation: $f(t,x,v)$ fraction of particles of position x and velocity v at time t. Particles are only transported by velocity in vacuum, and the velocity is only changed by collisions. Of course at some point a dissipative step (in the eye of the beholder) will have to be attained. So one obtains

$\partial_t f + v \nabla_x f = \alpha \int \int [f(t,x,v') f(t,x,v'_2) - f(t,x,v)f(t,x,v_2) ]|(v-v_2)\cdot \nu_2| dv dv_2 d\nu_2$

Mass, momentum and kinetic energy are collision invariants [Q: is angular momentum as well?]. E.g. mass:

$\partial_t \int f dv + \nabla_x \int d v dv$

is we have a well-prescribed profile for $f(x,v,t)$ we would obtain from this an hydrodynamic equation. Lyapunov functional of the Boltzmann equation:

$S(t) = - \int f \log f dx dv$

is an increasing function of time. That this is the case for the Boltzmann equation it’s fine. But that this should be the case for any system, it’s a crazy idea and lots of people are obsessed with this. The maximum of the entropy, constrained on conserved mass, energy and momentum is

$\log f(v) = \gamma + u\cdot v + \beta v^2$

This is true in the limit where the collision process is much faster than transport, in the long time. By plugging this equation in the conservation laws one obtains the Navier-Stokes equations, which is then the quasi-classical mean-field sort of limit of the Boltzmann equation. Notice that from Newton equation to Boltzmann equation some irreversibility entered (because the entropy is a constant of motion of Newtonian mechanics), somehow embedded in the collision integral: but it’s not clear where such dissipation is. Maybe in the angular momentum which might not be conserved? [Gianmaria says that also angular momentum is conserved. So what is not conserved?]

Statistical point of view: The starting point is the Liouville equation of the $N$ particles, and one is interested in the first marginal with respect to one particle (that’s where the dissipation will come about, in this N to 1 process). At some point one resorts to the BBGKY hierarchy, where the molecular dyamics can be analyzed in terms of all possible histories, which give rise to collision trees, from the final time to the initial (you want to retrace the “parents” in the tree; the difference between the Boltzmann hierarchy and the BBGKY hierarchy is that in one of them you cannot have recollisions, i.e. the meeting of two particles that already met and hence are already correlated. There exists a theorem that tell you convergence after just a few collisions, but not in large times where one recovers the fluid dynamics.

[An idea: the BBGKY-kind of methods truncate the equations for the moments of some variable; but what if one considers another variable, or a coordinate transformation of a variable, which in some way is “optimal”?]

Human rights session. Apparently it’s a tradition instituted by J. Lebowitz at the time of the cold war, when it was difficult to have Russian scientists travelling through the iron curtain. A resolution of the United Nations for over-concern with security measures made so that five North-Corean scientists had to be expelled from SISSA and other institutions, to avoid (I guess) to provide them with some sort of strategic military information (they were working on string theory and DNA…). [By the way I was somehow relieved that the first mention of violation of human rights is a charge on the UN, since I have the eerie impression that human rights talk often serves as a vacuous propaganda tool when it comes from the side of this sort of international institutions]. Stefano Ruffo presents the case of Omid Kokabe, who was arrested in Teheran possibly because he refused to work on military projects.