renamed inference module (still in prep)

Author: drannarosen

…inference-and-machine-learning/.DS_Store …inference-and-machine-learning/.DS_Store05-statistical-thinking/05-inference-and-machine-learning/.DS_Store renamed to 05-statistical-thinking/module-5-inference-and-machine-learning/.DS_Store 05-statistical-thinking/05-inference-and-machine-learning/.DS_Store renamed to 05-statistical-thinking/module-5-inference-and-machine-learning/.DS_Store

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+# Cell Boundary Crossing Mathematics for MCRT
+
+## Ray-Grid Intersection in 3D Cartesian Coordinates
+
+### Problem Statement
+
+Given a photon packet at position $\vec{r}_0 = (x_0, y_0, z_0)$ traveling in direction $\hat{n} = (n_x, n_y, n_z)$ through a regular 3D grid, determine:
+1. The distance to the next cell boundary
+2. Which cell face will be crossed
+3. The position where optical depth accumulation should continue
+
+---
+
+## Mathematical Formulation
+
+### 1. Distance to Next Boundary in Each Dimension
+
+For each spatial dimension, calculate the distance to the nearest boundary in the direction of travel:
+
+#### X-dimension:
+$$d_x = \begin{cases}
+\dfrac{x_{\text{right}} - x_0}{n_x} & \text{if } n_x > 0 \quad \text{(moving toward +x)} \\[1em]
+\dfrac{x_{\text{left}} - x_0}{n_x} & \text{if } n_x < 0 \quad \text{(moving toward -x)} \\[1em]
+\infty & \text{if } n_x = 0 \quad \text{(parallel to x-planes)}
+\end{cases}$$
+
+#### Y-dimension:
+$$d_y = \begin{cases}
+\dfrac{y_{\text{top}} - y_0}{n_y} & \text{if } n_y > 0 \quad \text{(moving toward +y)} \\[1em]
+\dfrac{y_{\text{bottom}} - y_0}{n_y} & \text{if } n_y < 0 \quad \text{(moving toward -y)} \\[1em]
+\infty & \text{if } n_y = 0 \quad \text{(parallel to y-planes)}
+\end{cases}$$
+
+#### Z-dimension:
+$$d_z = \begin{cases}
+\dfrac{z_{\text{back}} - z_0}{n_z} & \text{if } n_z > 0 \quad \text{(moving toward +z)} \\[1em]
+\dfrac{z_{\text{front}} - z_0}{n_z} & \text{if } n_z < 0 \quad \text{(moving toward -z)} \\[1em]
+\infty & \text{if } n_z = 0 \quad \text{(parallel to z-planes)}
+\end{cases}$$
+
+---
+
+## 2. Term Definitions
+
+### Photon State Variables
+- **$\vec{r}_0 = (x_0, y_0, z_0)$**: Current photon position [cm]
+- **$\hat{n} = (n_x, n_y, n_z)$**: Unit propagation direction vector, where $|\hat{n}| = 1$
+
+### Grid Parameters
+- **$L$**: Total box size = $3.086 \times 10^{18}$ cm (1 parsec)
+- **$N_{\text{grid}}$**: Number of cells per dimension = 128
+- **$\Delta x = \Delta y = \Delta z$**: Cell size = $L/N_{\text{grid}}$ [cm]
+
+### Current Cell Indices
+For a position $(x_0, y_0, z_0)$, the cell indices are:
+$$i_x = \left\lfloor \frac{x_0 + L/2}{\Delta x} \right\rfloor, \quad i_y = \left\lfloor \frac{y_0 + L/2}{\Delta y} \right\rfloor, \quad i_z = \left\lfloor \frac{z_0 + L/2}{\Delta z} \right\rfloor$$
+
+### Cell Boundary Positions
+For the current cell with indices $(i_x, i_y, i_z)$:
+
+**X-boundaries:**
+- $x_{\text{left}} = i_x \cdot \Delta x - L/2$ (left face)
+- $x_{\text{right}} = (i_x + 1) \cdot \Delta x - L/2$ (right face)
+
+**Y-boundaries:**
+- $y_{\text{bottom}} = i_y \cdot \Delta y - L/2$ (bottom face)
+- $y_{\text{top}} = (i_y + 1) \cdot \Delta y - L/2$ (top face)
+
+**Z-boundaries:**
+- $z_{\text{front}} = i_z \cdot \Delta z - L/2$ (front face)
+- $z_{\text{back}} = (i_z + 1) \cdot \Delta z - L/2$ (back face)
+
+---
+
+## 3. Next Boundary Selection
+
+The distance to the next cell boundary crossing is the minimum of the three calculated distances:
+
+$$\boxed{d_{\text{next}} = \min(d_x, d_y, d_z)}$$
+
+The crossed boundary type is determined by which distance equals $d_{\text{next}}$:
+
+$$\text{Boundary crossed} = \begin{cases}
+\text{x-face at } x_{\text{right/left}} & \text{if } d_{\text{next}} = d_x \\[0.5em]
+\text{y-face at } y_{\text{top/bottom}} & \text{if } d_{\text{next}} = d_y \\[0.5em]
+\text{z-face at } z_{\text{back/front}} & \text{if } d_{\text{next}} = d_z
+\end{cases}$$
+
+---
+
+## 4. Optical Depth Accumulation
+
+As the photon traverses each cell, accumulate optical depth:
+
+$$\tau_{\text{accumulated}} = \tau_{\text{accumulated}} + \Delta\tau_{\text{cell}}$$
+
+where the optical depth through the current cell (or portion thereof) is:
+
+$$\Delta\tau_{\text{cell}} = \kappa_{\text{band}} \cdot \rho_{\text{dust}} \cdot d_{\text{step}}$$
+
+### Terms:
+- **$\kappa_{\text{band}}$**: Band-averaged mass absorption coefficient [cm²/g dust]
+- **$\rho_{\text{dust}}$**: Dust density in current cell [g/cm³]
+- **$d_{\text{step}}$**: Distance traveled in cell [cm]
+
+---
+
+## 5. Interaction Position Determination
+
+The photon interacts when $\tau_{\text{accumulated}} \geq \tau_{\text{target}}$.
+
+### Case 1: Interaction Within Current Cell
+If $\tau_{\text{accumulated}} + \Delta\tau_{\text{cell}} \geq \tau_{\text{target}}$:
+
+The interaction occurs at fractional distance through the cell:
+$$f = \frac{\tau_{\text{target}} - \tau_{\text{accumulated}}}{\kappa_{\text{band}} \cdot \rho_{\text{dust}} \cdot d_{\text{next}}}$$
+
+where $0 \leq f < 1$.
+
+The interaction position is:
+$$\vec{r}_{\text{interaction}} = \vec{r}_0 + f \cdot d_{\text{next}} \cdot \hat{n}$$
+
+### Case 2: Continue to Next Cell
+If $\tau_{\text{accumulated}} + \Delta\tau_{\text{cell}} < \tau_{\text{target}}$:
+
+Update position to the cell boundary:
+$$\vec{r}_{\text{new}} = \vec{r}_0 + (d_{\text{next}} + \epsilon) \cdot \hat{n}$$
+
+where $\epsilon \sim 10^{-10}$ cm ensures numerical stability at boundaries.
+
+---
+
+## 6. Physical Interpretation
+
+### Why Minimum Distance?
+The photon travels in a straight line and encounters whichever boundary is geometrically closest along its path. The other boundaries are not reached during this step.
+
+### Sign Convention
+When a direction component is negative (e.g., $n_x < 0$), both the numerator and denominator in the distance calculation are negative, yielding a positive distance. This ensures all distances represent forward propagation along the ray.
+
+### Parallel Motion
+When $n_i = 0$ for any dimension $i$, the photon travels parallel to that set of boundaries and never crosses them, hence $d_i = \infty$.
+
+### Numerical Precision
+The epsilon adjustment ($\sim 10^{-10}$ cm) after boundary crossing prevents photons from becoming numerically "stuck" at cell interfaces due to floating-point precision limits.
+
+---
+
+## 7. Validation Tests
+
+### Test 1: Axis-Aligned Ray
+For $\hat{n} = (1, 0, 0)$:
+- $d_x = \Delta x$ (regular spacing)
+- $d_y = d_z = \infty$ (never crossed)
+
+### Test 2: Diagonal Ray
+For $\hat{n} = \frac{1}{\sqrt{3}}(1, 1, 1)$:
+- Distances depend on position within starting cell
+- Pattern of boundary crossings alternates between x, y, and z faces
+
+### Test 3: Energy Conservation
+$$\left|L_{\text{in}} - (L_{\text{absorbed}} + L_{\text{escaped}})\right| < 0.001 \cdot L_{\text{in}}$$
+
+This must hold for all packet counts and grid resolutions.