Consider the following stochastic dynamical system.

Fix $ a > 0$ , $ b > 0$ , $ c>0$ and $ v > 0$ , and let $ \mathbf{r}(t)=(x(t),y(t),z(t))$ be the position at time $ t$ of a point which moves in the parallelepiped $ R=\{ (x,y,z) \in \mathbb{R}^3: 0 \leq x \leq a, 0 \leq y \leq b, 0 \leq z \leq c \}$ with velocity of constant magnitude $ v$ a according to the following rules (we shall consider the usual spherical coordinates $ (\rho, \theta, \phi)$ in $ \mathbb{R}^3$ , where $ \theta$ is the polar angel with respect to the $ z$ -axis and $ \phi$ the azimuthal angle):

(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 the face of $ R$ corresponding to $ z=0$ it is reflected diffusely, that is its velocity after the collision has always magnitude $ v$ , and the versor $ \mathbf{v} / v$ has polar coordinates $ (1,\theta, \phi)$ with $ (\theta, \phi)$ distributed over $ [0,\pi/2] \times [0,2\pi)$ with probability density law (Knudsen law) $ f(\theta, \phi)= \frac{1}{\pi} \cos \theta d \Omega= \frac{1}{\pi} \cos \theta \sin \theta d \theta d \phi$ ;

(iii) if the point reaches any other side of $ R$ , then it is reflected elastically: e.g. in the case of the face corresponding to $ z=a$ the $ x$ and $ y$ components of $ \mathbf{v}$ are preserved, while the $ z$ -component of $ \mathbf{v}$ changes sign;

(iv) finally, if the point reaches an edge (or a vertex) of $ R$ , then the reflection must be thought as the combination of reflections of the two (respectively three) faces involved; so if the point hits the edge $ \{ (x,0,0): 0 < x < a \}$ it is reflected diffusely, that is its velocity after the collision has always magnitude $ v$ , and the versor $ \mathbf{v} / v$ has polar coordinates $ (1,\theta, \phi)$ with $ (\theta, \phi)$ distributed over $ [0,\pi/2] \times [0, pi]$ with probability density law $ g(\theta, \phi)= \frac{2}{\pi} \cos \theta d \Omega= \frac{2}{\pi} \cos \theta \sin \theta d \theta d \phi$ ; if instead e.g. the point hits the edge $ \{ (x,0,a): 0 < x < a \}$ then is reflected elastically, that is the $ x$ component is preserved, while the $ y$ and $ z$ components change sign.

For any time $ T> 0$ , let $ N(T)$ be the number of times the point hits the face $ z=a$ (included its edges and vertices) before or at time $ T$ . Let $ (1,\theta_j,\phi_j)$ be the polar coordinates of the versor $ \mathbf{v} / v$ just before the $ j$ -th collision 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}{3 a}. \end{equation}

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, and see also ter Haar and Wergeland, Elements of Thermodynamics, $ \S 5.3.3$ ). I formulated a first model of dynamic stochastic billiard in my post Stochastic Dynamic Billiard. This model was too complicated to be carefully analyzed, so I simplified it to get a much simpler version of it in a second post A Simple Stochastic Dynamic Billiard. The answer by Iosif Pinelis to this last post shows that my conjecture was trivially wrong. Anyway, as suggested by Benoît Kloeckner in his insightful comments, this was due to the fact I had assumed random uniform reflection on $ z=0$ in my two previous posts. Replacing the random uniform reflection with the Knudsen law should finally give the right answer.