Consider the following stochastic dynamical system.

Fix $a > 0$, $b > 0$ and $v > 0$, and let $\mathbf{r}(t)=(x(t),y(t))$ be the position at time $t$ of a point which moves in the rectangle $R=\{ (x,y) \in \mathbb{R}^2: 0 \leq x \leq a, 0 \leq y \leq b \}$ with velocity of constant magnitude $v$ a according to the following rules:

(i) in the interior of $R$ the point is subject to no force, so that is moves with constant velocity $\mathbf{v}(t)=\frac{d\mathbf{r}}{dt}(t)$;

(ii) when the point reaches one of the vertical sides of $R$ then it is reflected elastically, that is the $y$-component of $\mathbf{v}$ is preserved, while the $x$-component of $\mathbf{v}$ changes sign;

(iii) when the point reaches one of the horizontal sides of $R$ then it is reflected diffusely, that is its velocity after the collision has always magnitude $v$, and the convex angle $\theta$ that $\mathbf{v}$ makes with the versor $\mathbf{i}=(1,0)$ takes any value in $[0,\pi]$ with equal probability (that is $\theta$ is uniformly distributed on $[0,\pi]$);

(iv) finally, if the point reaches one of the vertices of $R$, then its velocity after the collision has magnitude $v$ and the convex angle that $\mathbf{v}$ makes with the versor $\mathbf{i}$, when the point reaches one of vertices $(0,0)$ and $(0,b)$, or rispectively $-\mathbf{i}$, when the point reaches one of the vertices $(a,0)$ and $(a,b)$, assumes any value in $[0,\pi/2]$ with equal probability.

Consider a time $T> 0$, and let $N(T)$ be the number of time the point touches one of the vertical sides of $R$ (you can compute or not in $N(T)$ the times the point touches one of the vertices of $R$: it should make no essential difference for what we want to prove, I think). Let $\theta_j$ be the angle that $\mathbf{v}$ makes with $\mathbf{i}$ the $j$-th time the point touches one of the vertical sides of $R$ (or one of the vertices, if you have considered also them in the computation of $N(t)$), and form the random sum \begin{equation} \sum_{j=1}^{N(T)} |cos \theta_j|. \end{equation} I would like to prove that for any initial conditions $(\mathbf{r}(0), \mathbf{v}(0))$, the following equation holds \begin{equation} \lim_{T \rightarrow \infty} \frac{1}{T} E \left[ \sum_{j=1}^{N(T)} |cos \theta_j| \right] = \frac{v}{2 a}. \end{equation}

Any help is welcome. For now, I have no idea about a possible proof.

NOTE. This problem has been suggested to me by an interesting proof of a physical law called Wien's displacement law given by Richtmyer, Kennard and Cooper in their book "Introduction to Modern Physics", Sixth Edition, Appendix of Chapter 5 (see in particular p. 145). The equation I would like to prove should be true for compelling physical reasons of thermodynamical character. Actually it should be true more generally also if all the sides of $R$ reflected elastically and only a small segment on one or both of the horizontal sides should reflect in a diffuse way, but the proof of this last statement seems absolutely prohibitive to me. Even in the simplified case I described in my post above a proof seems not trivial at all.