Low escape probability on the context of mean free path

When we say a particle has a low escape probability from a region $K$ , we mean there is a small chance it will traverse $K$ and exit through the boundary without undergoing any collisions (i.e., interactions like scattering or absorption). In the context of mean free path, this happens when the particle’s typical collision-free travel distance is much smaller than the region’s typical linear extent.

1. Mean Free Path vs. Mean Chord Length

1. Mean Free Path $\lambda$

•Physical property of the medium.

• $\lambda = \tfrac{1}{\Sigma}$ , where $\Sigma$ is the collision rate (collisions per unit length).

•The particle, on average, travels a distance $\lambda$ before colliding.

2. Mean Chord Length $\langle \ell \rangle = \tfrac{4V}{A}$

•Geometric property of the convex region $K$ .

•If a particle moves in a straight line (no collisions), the average distance from its starting point to an exit on the boundary is $\tfrac{4V}{A}$ .

When $\lambda \ll \tfrac{4V}{A}$ , the region is large compared to the particle’s typical collision-free path. The particle will likely undergo a collision before it can cross the region—hence, it has a low probability of escaping collision-free.

2. Why a Small Mean Free Path Leads to a Low Escape Probability

2.1 Exponential Attenuation

In a homogeneous medium with collision rate $\Sigma$ , the probability that a particle does not collide over a straight-line distance $\ell$ is

P(\text{no collision up to distance } \ell)

;=;

e^{-\Sigma ,\ell}.

•If $\ell$ is larger than $\frac{1}{\Sigma} = \lambda$ , then $\Sigma,\ell \gg 1$ and $e^{-\Sigma,\ell}$ is very small.

•This means the chance of traveling a distance $\ell$ without collision is negligible when $\ell \gg \lambda$ .

2.2 Chord Lengths vs. $\lambda$

In a finite convex body $K$ , the distance to the boundary (the chord length) varies for different starting points and directions, but on average it is $\langle \ell \rangle = \tfrac{4V}{A}$ . If $\tfrac{4V}{A}$ is much bigger than $\lambda$ , then

\Sigma ,\bigl(\tfrac{4V}{A}\bigr)

;=;

\frac{\tfrac{4V}{A}}{\lambda}

;\gg;1,

\quad

\text{so }

e^{-\Sigma ,\langle \ell \rangle}

;\ll;1.

Thus, only a small fraction of particles can traverse the average chord without colliding—hence low escape probability.

3. “Optically Thick” / “Opaque” Medium

In radiative transfer, neutron transport, or similar fields, when $\lambda \ll \tfrac{4V}{A}$ , one often says the medium is “optically thick” or “opaque.” This terminology reflects that:

•The region’s typical size (as a path the particle must travel) is large.

•The typical collision-free distance is small.

So collisions happen frequently relative to how far the particle needs to go to exit.

4. Escape Probability (Heuristic)

A rough—though not exact—estimate of the escape probability $P_{\text{escape}}$ (i.e., the probability that a particle, created randomly inside $K$ with isotropic direction, exits without any collision) uses the mean chord length:

P_{\text{escape}}

;\approx;

\exp!\Bigl(-\Sigma,\langle \ell \rangle\Bigr)

;=;

\exp!\Bigl(-\tfrac{\langle \ell \rangle}{\lambda}\Bigr).

•When $\langle \ell \rangle \gg \lambda$ , then $\tfrac{\langle \ell \rangle}{\lambda} \gg 1$ and $P_{\text{escape}} \approx e^{-\text{(large number)}} \ll 1$ .

•In reality, chord lengths vary (some shorter, some longer than $\tfrac{4V}{A}$ ), so the exact calculation requires integrating over the chord-length distribution. But this exponential factor conveys the qualitative result: if the typical path is much longer than $\lambda$ , a collision almost surely happens first.

5. Physical Interpretation: Low Escape Probability

1. Frequent Collisions

Because $\lambda$ is small, the particle seldom travels far without an interaction.

2. Small Chance to Traverse $K$

Since the region’s average chord length is much larger than $\lambda$ , most paths are “cut short” by collisions.

3. High Likelihood of Absorption or Scattering

Once a collision occurs, the particle may be absorbed (disappears) or scattered (changes direction, possibly losing energy). Either way, it is less likely to emerge from $K$ in the original, collision-free manner.

Hence, low escape probability in the context of mean free path means that the internal collisions dominate before boundary escape can occur, making the region effectively “opaque” to the traveling particles.

Low escape probability on the context of mean free path

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