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  1. Available Volume: Extensive vs. Discrete at Different Densities Claim: “Available volume is an extensive quantity at low densities but transitions to a discrete space at higher densities.” Fact-check: This claim is supported by studies on hard-sphere fluids. At low particle densities, the “free” or available volume per particle scales extensively with system size…

Below is an explanation of the table you provided, describing different chord distributions in a sphere under various sampling models (ways of selecting chords). These chord‐selection schemes are classic examples in geometric probability (especially for spheres in 3D). The table shows: The model or method for choosing chords. The mean chord length (or some normalized…

Mean Free Path (MFP) is the average distance a particle travels before a collision (or any event that significantly changes its direction/energy) (Mean free path – Wikipedia) (Mean free path – Wikipedia). In a dilute homogeneous medium with number density n of target particles and a collision cross-section σ, the mean free path can be…

1. Memoryless (Markov) Property A fundamental assumption in deriving the mean free path is the memoryless property. Specifically, let be the random variable for the distance traveled by a particle before it collides. The Markov (or memoryless) property states:   P(X>x+Δx ∣ X>x)  =  P(X>Δx),∀ x, Δx ≥0.P\bigl(X > x + \Delta x \,\big\vert\, X > x\bigr) \;=\; P\bigl(X > \Delta x\bigr),…

Conceptual Relationship Mean free path (MFP) is the average distance a particle travels between successive collisions (Chapter3_Diffusion.dvi). In a gas or any collection of moving particles, each particle’s trajectory becomes a zig-zag as it collides and changes direction. The sequence of straight-line “flights” between collisions can be thought of as diffusion steps in a random…

Mean Free Path and Diffusion Steps Conceptual Relationship Mean free path (MFP) is the average distance a particle travels between successive collisions (Chapter3_Diffusion.dvi). In a gas or any collection of moving particles, each particle’s trajectory becomes a zig-zag as it collides and changes direction. The sequence of straight-line “flights” between collisions can be thought of…

Below is a more detailed (yet still relatively accessible) derivation of the 3D Cauchy formula (also called the mean chord length formula) in integral geometry, which states: (L)∘  =  4 VA,(L)_\circ \;=\; \frac{4\,V}{A}, where (L)∘(L)_\circ is the mean chord length of a convex body K⊂R3K \subset \mathbb{R}^3; VV is the volume of KK; AA is the surface area…

In a finite, convex region containing a uniform, isotropic medium with a constant collision rate (or equivalently a mean free path ), there is a competition between: 1.Geometric Size of : captured by the mean chord length . 2.Physical Mean Free Path : the distance a particle typically travels before colliding. When a particle starts…

When we say a particle has a low escape probability from a region , we mean there is a small chance it will traverse 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…

In stochastic geometry and related fields, the term stochastic visibility generally refers to the probability (or statistical characterization) that a given point or region is visible—i.e., can be “seen” or reached along an unobstructed line of sight—from another point or region, when the scene contains random obstacles or is otherwise subject to random processes. Below…