Boltzmann

 

(08 May 2019)

Hard Sphere Dynamics

  1. (Alexander) For almost every initial data the dynamics are globally well defined (i.e. nothing but simultaneous binary collisions)

  2. Qualitative behavior of the system:

    1. Reversibiliy

    2. Poincaré recurrence theorem

    3. If $\varepsilon«1$ stability deteriorates

We consider $N$ particles of size $\varepsilon$, with $N\to \infty$ and $\varepsilon \to 0$.

If $\varepsilon = 0$ and we take $N\to \infty$ or $\varepsilon\to 0$ much faster compared to $N\to\infty$ then particles never or rarely collide. In the limit we end up with the equation for free transport

\[\partial_t f + v\cdot \nabla_x f = 0\]

We are interested in a regime where there are not too many collisions but not so few that they become meaningless in the limit. The specific regime we will study is what is known as the low density regime.

Maxwell did a computation regarding this regime: you consider a $N$ particles layed out on a lattice (and we see them as fixed) and we have another particle which is freely moving initial with velocity $v$

This leads to the relation

\[N\varepsilon^{d-1} \approx 1\]

In particular, $N\varepsilon^d \to 0$ as $N\to \infty$, so the particle density is small in the limit. This is what is known as the low density regime.

\[f_N^0(x_1,v_1,\ldots,x_N,v_N) \leftarrow \text{Initial distribution of states}\] \[f_N(t,x_1,v_1,\ldots,x_N,v_N) \leftarrow \text{Distribution of states at time } t\]

The Liouville equation

\[\partial_t f_N + \sum \limits_{i=1}^{N} v_i \cdot \nabla_{x_i} f_N = 0 \text{ in } (0,\infty)\times \mathcal{D}_N\]

where

\[\mathcal{D}_N = \{ (X_N,V_N) \mid |x_i-x_j| > 2\varepsilon \;\forall\;i\neq j\}\]

the boundary conditions (#what was the name of this, specular reflection?)

\[{f_N}_{\mid \Sigma_{ij}^-} = R_{ij} {f_N}_{\mid \Sigma_{ij}^+}\]

We now work instead in the grand canonical setting (explain what this means)

Boltzmann’s equation

  1. Exchangeability (we don’t care about labels on the particles)

(this translates to symmetry of $f_N$ with respect to the $(x_i,v_i)$)

  1. Averaging \(f(t,x,v) \leftarrow \text{probability density for a single particule}\)

(this means among other things that all observables shouldbe in terms of this)

Moreover, $f(t,v,x)$ can be interpreted as a marginal distribution

\[f(t,v,x) = \int f_N(t,x,v,x_2,v_2,\ldots,x_N,v_N)\;dx_2dv_2 \ldots dx_Ndv_N\]

What about the equation for $f$? We can advance that

\[\partial_tf +v\cdot \nabla_x f = \left ( \begin{array}{l} \text{boundary terms} \\ \text{in Green's formula} \end{array}\right )\] \[\sum \limits_{i=2}^N \int_{\{|x_i-x_1|=\varepsilon\}} f_N(t,X_N,V_N)(v_i-v_1) \cdot n_{i1}dv??? dv_1dx_2\ldots dx_{n}dv_n \\ = (n-1)\varepsilon^{d-1}\int f_n^{(2)}(t,x,v,x_2,v_2)(v_2-v_1)\cdot n_{12}dn_{12}dv_2\]

Here is we bring in the Boltzmann-Grad scaling.

Then,

\[\begin{array}{rl} \partial_t f + v\cdot \nabla_x f \sim & \int f_N^{(2)}(t,x_1,v_1',x_1+\varepsilon??,v_2')((v_2-v_1)\cdot n_{12})_+\\ & -\int f_N^{(2)}(t,x_1,v_1,x_1+\varepsilon??,v_2')((v_2-v_1)\cdot n_{12})_+\end{array}\]

This system is not closed!! To compute the dynamics for $f = f^{(1)}_N$ we need to know $f^{(2)}_N$ $\ldots$

Here comes a fundamental reduction in the model: Boltzmann’s chaos assumption.

\[(*) \;\;\;\;\; f_N^{(2)}(t,x_1,v_1,x_2,v_2) \approx f_N^{(1)}(t,x_1,v_1)f_N^{(1)}(t,x_2,v_2)\]

(note this states an independence relation between particles)

Then, the equation becomes

\[\partial_tf + v\cdot \nabla_x f = C(f,f)\]

Notes:

  1. Even at $t=0$, assumption $(*)$ cannot be correct.

  2. For $t>0$, there is some chance that $(*)$ is approximately true.

  3. This last model, unlike the original equation for $f_N$, is not reversible in time. So, understanding the validity of Boltzmann’s chaos assumption requires understanding the loss of time reversibility in the approximation.

Theorem (Lanford)

Consider a system of $N$ hard spheresof size $\varepsilon$ which are initially independent and identically distributed, i.e.

\[f_N^0 = \frac{1}{Z_N}(f^0)^{\otimes N} \chi_{\mathcal{D}_N}\]

Then, in the low density limit as $N\to \infty$ we have

\[\frac{1}{N}\sum \limits_{i=1}^N \delta_{(x_i(t),v_i(t))} \to f \text{ (almost surely)}\]

where $f$ is a solution of the Boltzmann equation on a time interval $[0,T_0]$. Moreover, we have the convergence of the marginals

\[f^{(1)}_N \to f,\; f^{(2)}_N \to f^{\otimes 2}\]

Note: this last limit indicates the validity of the Boltzmann chaos assumption in the limit.

Strategy of the proof

\[\partial_t f_N^{(1)} + v_1 \cdot \nabla_{x_1} f_{N}^{(1)} = C(f_N^{(2)})\] \[\partial_t f_N^{(2)} + (v_1,v_2)\cdot \nabla_{(x_1,x_2)} f_{N}^{(2)} = C_{2,3} (f_N^{(3)})\]

$(\ldots)$

and that is how we arrive at what is known as the BBGKY hierarchy.

\[\partial_t f_N^{(k)} + \sum \limits_{i=1}^k v_i \cdot \nabla_{x_i} f_N^{(k)} = C_{k,k+1}(f_N^{(k+1)})\]

Combinatorics of collisions (collision diagram)

$(\ldots)$

We will take this combinatorial data and create a series expansion

\[\sum\limits_k \sum \limits_{A_k} \int dt_{i}dx_idv_i S_1(t-t_2)C_{1,2}(S_2(t_2-t_3))\ldots\]

The proof boils down to showing the “bad trees” (those corresponding to “recollisions”) do not occur very often.

Step 1 Convergence of the series

Step 2 Controlling contribution of recollisions (i.e. show they contribute zero in the limit)

Step 3 If you are not in the bad set of parameters then the small shifts are controlled.

The reason you cannot go back in time is that you are losing information (the bad trajectories) when averaging. This information is being carried by a set of vanishing measure (this encodes the deviation of $f_N^{(2)}$ away from $f^{\otimes (2)}$).