Hard sphere model (1) : Introduction

The hard sphere model starts from the very basic assumption of classical mechanics and attempts to prove the convergence from microscopic model to macroscopic model.

Notation and basic assumption

We starts from $N$ particles following the rules of classical mechanics and each one can be designed $z_i = (x_i, v_i) \in \mathbb{T}^d \times \mathbb{R}^d, i = 1, 2, 3, \cdots N$. After one collision, the change of speed is expressed as
$$
\begin{eqnarray*}
v' &=& v - ((v - v_1)  \cdot r) r \\
v'_1 &=& v_1 - ((v_1 - v) \cdot  r) r
\end{eqnarray*}
$$

The mechanics can be reversed once we change the direction of time and speed, in another word, this system is reversible. To avoid the  abuse of notation, we always define the evolution of system in a suitable subset of all configuration like
$$
D^N_{\epsilon} = \{(z_1, z_2, \cdots z_N), \forall i \neq j, |x_i - x_j| < \epsilon \}
$$
We define the density of the configuration in a produit probability space on $\mathbb{T}^d \times \mathbb{R}^d$. Some analysis can deduce that : We can define all the configuration on the probability space except a null set.

M_{\beta}^{\otimes N}(V_N)

Boltzmann-Grad scaling and Boltzmann equation

For a single particle, its density has an invariant law following the Liouville equation $$ \partial_t f(t,x) + v \cdot \partial_x f(t,x) = 0 $$ 

We generalize this equation to $N$ hard sphere model, then the Liouville equation is 

$$ \partial_t f_N(t,Z_N) + \sum_{i = 1}^N v_i \cdot \partial_{x_i} f_N(t,Z_N) = 0 $$ 

Since we would like to study the particle chaotic phenomena, a nature idea is to calculate its invariant measure - we know the limit measure is always the invariant measure. In this case, the invariant measure is called Boltzmann measure which is 

$$M_{N, \beta} = \frac{1}{\mathcal{Z}_N} \mathbb{1}_{D^{N}_{\epsilon}}(Z_N) M_{\beta}^{\otimes N}(V_N)$$ 

this one is not exactly a Maxwell-Boltzmann distribution in cause of the exclusion condition. But some detailed analysis tells us that we can do "almost" factorization of the measure to the product of the probability.  

$$|(M_{N, \beta}^{(s)} - M_{\beta}^{\otimes s}) \mathbb{1}_{D^s_{\epsilon}}| \leq C^s \rho M_{\beta}^{\otimes s}$$

In the further blog, we will study the convergence in different context precisely. We will show that, the distribution is very near to the distribution of the solution of linear Boltzmann equation $\partial_t f + v \cdot \nabla_x f = \alpha Q(f,f)$.

Entropy paradox

Entropy introduced by Shannon and Boltzmann is a genius idea to describe the disorder degree as $H(t) = \int_{D^N_{\epsilon}}  - f_N(t, Z_N) \log{f(t,Z_N)} dZ_N$. According to this definition, the entropy of the solution of hard sphere model is constant, but that of the solution of Boltzmann equation will increase. 

In our course, we list two models to explain this paradox. The two baby model like Kac ring model and Ehrenfest model are all reversible, so they have a period or Poincare recurrence result. However, the time to appear the chaos is faster than the period. That is also the case in hard sphere model : we will see the recurrence perhaps, but it takes long long long time and during this time, the chaos repeat again and again. So how do you regard it? A deterministic system or a random system? You got the answer.