This is the collection of summaries in the winter school in Rome in 2025.

Speaker: Alessia Nota

Title On the Smoluchowski coagulation equation for aggregation phenomena

Day 1: Well-posedness, properties, physical relevant cogulation kernels, Dynamical scaling solutions

Mass/ moment conservation

Day2: space-dependent particle model convergence to Smoluchowski equation. Physical kernels: $K=l/d$, $l$ mean free path, $d$ particle density,
$K \to 0$ continuum regime, Brownian kernel, $K(v,w)\approx (v^{1/3}+ w^{1/3})$
$K=O(1)$ transition regime,
$K \to \infty$ Free molecular regime (Ballistic kernel) $K(v,w) \approx (1/v +1/w)^{1/2} (v^{1/3}+w^{1/3})^2 $

Vaildation of Smoluchowski equation: from Marcus-Lushnikov process Large system limit and identification for gelation Propagation of chaos for a coalescing particle system with constant kernel

  1. Derivation from mechanical particle system with spatial inhomogeneous system

Day3: With source term, stationary injection solution

Bogoliubov theory in many-body quantum mechanics

Speaker: Benjamin Schlein

Main content: Introduction of second quantization formalism
Idea of Bogolouiov approximation -> Approximation of constant $C$ in $N\hat{V}(0)/2 -c \le E_N \le (N+1)/2 \hat{V}(0)$ gives information of BEC. Because order $N$ particles on ground state, $O(1)$ particles in excited states.

Gross-Pitaevskii equation, mean field equation,

Two-temperature fluid models for a polyatomic gas derived from kinetic theory

Speaker: Kazuo Aoki

keywords: BGK model, ES model for Boltzmann equation