Boltzmann's Legacy 150 Years After His Birth

approximation asymptotic atoms average behavior Bolgiano Boltzmann equation boundary conditions CERCIGNANI classical coefficient collision Colloquio configurations consider Convegno Internazionale convergence cross section defined degrees of freedom denote density diffusion dimensional dimensions discrete dynamics eigenvalues entropy equilibrium Ergodic Ergodic Hypothesis evolution exponential exponents fermions finite flow fluctuations formula gases Gibbs Gibbsian Giornata Lincea H-theorem Hamiltonian hydrodynamic initial integral intensive quantities interaction Internazionale invariant kinetic theory Kolmogorov lattice limit linear Lyapunov exponents macroscopic maggio Math mathematical measure method molecules motion Navier-Stokes equations neutron transport obtained parameters particle systems perturbation phase space Phys physical possible potential probability problem properties quantum rarefied gas reactor Roma scaling scattering simulation solution spin glasses statistical mechanics symmetry temperature theorem thermal thermodynamic tion trajectory transport equation turbulence v₁ variables velocity zero