Mean Free Path, Markov Processes, Diffusion, and Visibility

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 derived by considering the probability of a collision in an infinitesimal path. The classic result is:

$\ell = \frac{1}{n\,\sigma} \,,$

meaning the average distance between collisions is the inverse of the product nσ (Mean free path – Wikipedia). Intuitively, as n or σ increases (more targets or larger collision cross-section), the mean free path shortens (collisions become more frequent), and vice versa. This result can be obtained by imagining a particle moving through a slab of material: in traveling a distance L, it sweeps out a volume ~Lσ and will on average encounter ~nLσ targets; setting this ~1 gives L ≈ 1/(nσ) (Chapter3_Diffusion.dvi) (Chapter3_Diffusion.dvi).

Derivation of the Mean Free Path Equation

For a more rigorous derivation, consider a particle beam passing through a uniform medium and let $I(x)$ be the intensity (or probability of a particle surviving) after traveling distance $x$ . The decrease in intensity over an infinitesimal path $dx$ is proportional to the number of targets in that slice:

$\frac{dI}{dx} = -\,n\,\sigma \, I(x)\,,$

where $n\sigma$ is the extinction coefficient (collision probability per unit length). This differential equation integrates to the Beer–Lambert law:

$I(x) = I_0 \exp(-n\sigma\,x) = I_0 \exp\!\Big(-\frac{x}{\ell}\Big)\,,$

where $I_0$ is the initial intensity and $\ell = (n\sigma)^{-1}$ . The interpretation is that the probability a particle travels a distance $x$ without collision (i.e. survives to $x$ ) decays exponentially as $e^{-x/\ell}$ . From this, one can obtain the free path length distribution. The probability density that the particle’s next collision occurs at distance $x$ (exactly at $x$ , after no collision before) is:

$f_{L}(x) = -\frac{d}{dx}\big[e^{-x/\ell}\big] = \frac{1}{\ell}\,e^{-x/\ell}\,, \qquad x\ge0~.$

This is an exponential distribution, normalized since $\int_0^\infty (1/\ell)e^{-x/\ell}dx=1$ . The mean of this distribution is indeed $\int_0^\infty x,(1/\ell)e^{-x/\ell}dx = \ell$ , consistent with the definition of $\ell$ as the average free path. Thus, the mean free path equation $\ell = 1/(n\sigma)$ emerges naturally from the exponential attenuation law and its probability interpretation. Essentially, $n\sigma$ plays the role of a constant hazard rate for collisions, so that the “waiting distance” for the next collision is memoryless and averages to $1/(n\sigma)$ .

Markov Process and Free Path Distributions

The exponential free-path distribution reflects a Markov (memoryless) process for collisions. In a homogeneous medium, each infinitesimal segment of path presents the same collision probability, regardless of how far the particle has already traveled. Mathematically, this memoryless property means:

$P(L > x+y \mid L > x) = P(L > y)\,,$

i.e. the probability that a particle travels an additional distance $y$ without collision does not depend on the already traveled distance $x$ . This property is satisfied only by the exponential distribution. We can see this by discretizing the path: if the probability to avoid a collision in a small step $\Delta x$ is $1 - n\sigma,\Delta x$ , then avoiding collision over $N$ steps of length $\Delta x = x/N$ is latex^N[/latex]. In the limit $N\to\infty$ (continuous travel), this product converges to $\exp(-n\sigma x)$ . Thus, the survival probability is $e^{-x/\ell}$ and is multiplicative over independent segments, confirming the Markov nature. In other words, collisions occur according to a Poisson process along the path with constant rate $n\sigma$ , and the free paths between collisions are i.i.d. (independent and identically distributed) exponential random lengths.

Because each collision “resets” the particle’s direction (and potentially energy) independently, one can model the particle’s successive states as a Markov chain. For example, in Monte Carlo simulations of radiative transport, photons are propagated by sampling free path lengths from an exponential distribution and then scattering with a random new direction at each collision – a direct application of Markov process modeling. The Chapman–Kolmogorov property holds: the probability of going from one collision to the next at a certain distance does not depend on the history before that collision. This memorylessness is what allows analytical treatment of transport: it leads directly to the Beer–Lambert exponential attenuation and greatly simplifies the mathematics of multiple scattering. Any deviation from an exponential free-path distribution would indicate some form of correlation or “memory” in the collision process, breaking the simple Markov assumption.

From Random Walks to Diffusion

After many collisions, the particle’s trajectory looks like a random walk (a zig-zag path). Each segment between collisions has a random length (with mean $\ell$ ) and a random direction (assuming collisions isotropically scatter the particle). Over a large number of collisions, the cumulative displacement of the particle can be treated statistically. In fact, the Central Limit Theorem tells us that the sum of many independent random steps will approach a Gaussian distribution around the starting point. This means the particle’s motion can be approximated by a diffusion process at long times or length-scales much larger than $\ell$ .

Mean squared displacement: For an isotropic random walk in 3D, the mean squared displacement after $N$ collisions is $\langle r^2 \rangle \approx N,\langle s^2 \rangle$ , where $s$ is the step length between collisions. Since steps have mean $\ell$ and variance $\text{Var}(s)=\ell^2$ (for an exponential, $\langle s^2 \rangle = 2\ell^2$ ), we have $\langle r^2 \rangle \sim N,(2\ell^2)$ . Meanwhile, if the particle travels with speed $v$ , the average time between collisions is $\tau = \ell / v$ . In time $t$ , the expected number of collisions is about $N \approx t/\tau = v,t/\ell$ . Combining these, $\langle r^2(t)\rangle \approx 2\ell^2 \cdot (v t/\ell) = 2,v,\ell,t$ . In a truly diffusive regime, we expect $\langle r^2 \rangle = 6 D,t$ in three dimensions. Equating $6 D t \approx 2 v \ell t$ , we get an effective diffusion coefficient

$D \approx \frac{1}{3} v\,\ell \,.$

This result can be made more rigorous with transport theory. In fact, for particles moving at speed $v$ and scattering isotropically with mean free path $\ell$ , one can show $D = \frac{v,\ell}{3}$ . (More generally, if scatterings are not isotropic, the transport mean free path $\ell^<em>$ replaces $\ell$ in the formula. $D = \frac{v,\ell^</em>}{3}$ .) This diffusion constant can be understood as follows: the particle diffuses by a sequence of steps of length ~ $\ell$ at speed ~ $v$ , so in time $t$ it takes about $t/\tau$ steps (with $\tau=\ell/v$ ), and each step contributes on the order of $\ell$ to the root-mean-square displacement. In line with kinetic theory, diffusion coefficient is essentially the product of mean step length and mean speed (up to a factor 1/3 in 3D). In gases, for instance, a higher average molecular speed or a longer mean free path (e.g. lower density gas) leads to higher diffusion – consistent with $D \propto v,\ell$ .

As a concrete example, photons diffusing through a turbid medium obey this relationship. The radiation diffusion equation (an approximation of the radiative transfer equation in optically thick media) has a diffusion coefficient $D = c,\ell^*/3$ for photon speed $c$ . This is the random-walk result mentioned above, and it remains valid independent of absorption in the medium (absorption adds a finite lifetime but does not alter the spatial diffusion of surviving photons). In summary, the Markovian assumption of independent collisions allows us to treat multiple scattering as a random walk, which in the continuum limit yields Fick’s laws of diffusion. Position distributions evolve according to the diffusion equation for times much larger than the collision time (or distances $\gg \ell$ ), with the diffusion constant determined by $\ell$ and the typical particle speed.

Visibility, Optical Depth, and Mean Free Path

Visibility in a scattering medium is intimately linked to the mean free path and the resulting attenuation of light. In atmospheric science and optics, a concept called meteorological optical range (MOR) is defined as the path length over which a collimated beam of light is attenuated to a small fraction of its original intensity (such that a reference object is just barely discernible). Because light intensity decays approximately as $I(x) = I_0 e^{-x/\ell}$ in a mist or haze (where $\ell$ here is the effective attenuation mean free path of light), one can define visibility distance based on a threshold in transmitted intensity or contrast. A common criterion is a reduction to 2% of the initial intensity ( $I/I_0 = 0.02$ ) for a dark object viewed against the horizon under daylight. Plugging $e^{-x/\ell}=0.02$ into that gives $x = -\ell \ln(0.02) \approx 3.912,\ell$ . This yields the Koschmieder’s law for visual range:

$x_{V} \approx \frac{3.912}{b_{\rm ext}}\,,$

where $b_{\rm ext}=1/\ell$ is the extinction coefficient. In practical terms, visibility distance is on the order of a few times the mean free path of light in the medium. For example, if the scattering/absorbing coefficient $b_{\rm ext}$ increases (heavy fog or aerosol content), $\ell=1/b_{\rm ext}$ decreases and the visibility $x_V$ shortens proportionally. Conversely, in very clear air (small $b_{\rm ext}$ ), $\ell$ is large and one can see much farther.

It’s important to note that optical depth is defined as $\tau = x/\ell$ . When $\tau \sim 1$ , the intensity is about $e^{-1}$ . When $\tau \gg 1$ , the beam is essentially extinguished. Human vision typically requires objects to have a certain minimum contrast against the background to be detected, which corresponds to a finite optical depth (around $\tau \approx 4$ under daylight conditions). Beyond a few mean free paths, most photons from a particular object have been either scattered away or absorbed, so the object becomes indiscernible. In a highly scattering medium, photons that do reach the observer have likely been scattered multiple times, effectively coming from all directions. Thus, high optical depth leads to diffusion of light: instead of a clear line-of-sight image, one sees a uniform glow or haze. This is a direct manifestation of the diffusion regime discussed earlier – when the distance exceeds the transport mean free path by several-fold, light loses memory of its original direction (a Markovian multiple-scattering behavior) and the scene appears foggy.

In summary, visibility is directly tied to the mean free path: it quantifies how far a particle (or photon) can travel on average before a scattering/absorbing event significantly diminishes its intensity or information. A large mean free path means a transparent medium with long visibility range (approaching ballistic transport), whereas a short mean free path means a diffuse medium where light quickly loses directionality and diffusive transport dominates. The rigorous connections are made through Markov process modeling (yielding exponential free-path distributions and Beer’s law attenuation) and through the diffusion approximation (for many scatterings), with the concept of optical depth bridging mean free path to practical visibility distance (Visibility – Wikipedia). Each of these theoretical tools – from the mean free path formula to Markov processes and diffusion equations – helps explain and quantify how particles move through random media and how far we can see through such media.

References: The derivations and relationships above are grounded in standard kinetic theory and transport theory. Key results (mean free path formula, exponential path distribution, Beer–Lambert law) are found in physics textbooks and references (Mean free path – Wikipedia) (Mean free path – Wikipedia). The memoryless (Markovian) nature of collision processes is a fundamental assumption leading to the exponential law (Markov Chain Monte Carlo solutions for radiative transfer problems | Astronomy & Astrophysics (A&A)). The connection to diffusion is established in texts on stochastic processes and physical kinetics, where $D \sim \text{(speed)}\times\text{(mean free path)}/(\text{dimensional factor})$ , as confirmed for photons by transport theory (giving $D = c,\ell^*/3$ ) (Photon diffusion coefficient in scattering and absorbing media – PubMed). Finally, the role of visibility and optical depth is well described in atmospheric optics, with Koschmieder’s formula relating the extinction coefficient to visual range (Visibility – Wikipedia). The interplay of these concepts provides a coherent theoretical framework linking microscopic random motion to macroscopic observables like diffusion and visibility.

Mean Free Path, Markov Processes, Diffusion, and Visibility

Derivation of the Mean Free Path Equation

Markov Process and Free Path Distributions

From Random Walks to Diffusion

Visibility, Optical Depth, and Mean Free Path

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