{"id":89,"date":"2025-03-17T00:00:22","date_gmt":"2025-03-17T00:00:22","guid":{"rendered":"https:\/\/freepath.info\/?p=89"},"modified":"2025-03-17T00:53:50","modified_gmt":"2025-03-17T00:53:50","slug":"mean-free-path-and-diffusion-steps","status":"publish","type":"post","link":"https:\/\/freepath.info\/?p=89","title":{"rendered":"Mean Free Path and Diffusion Steps"},"content":{"rendered":"<h2>Conceptual Relationship<\/h2>\n<p><strong>Mean free path (MFP)<\/strong> is the average distance a particle travels between successive collisions (<a href=\"https:\/\/ps.uci.edu\/~cyu\/p51A\/LectureNotes\/Chapter3\/Chapter3_Diffusion.pdf#:~:text=it%20presents%20as%20it%20moves,1\">Chapter3_Diffusion.dvi<\/a>). In a gas or any collection of moving particles, each particle\u2019s trajectory becomes a zig-zag as it collides and changes direction. The sequence of straight-line \u201cflights\u201d between collisions can be thought of as <strong>diffusion steps<\/strong> in a random walk. Each step has some length (distance traveled before a collision) and a random direction. The mean free path is essentially the average step length in this random walk.<\/p>\n<p>Because collision events redirect the particle randomly, the particle\u2019s net displacement grows <strong>sub-diffusively<\/strong>: in fact, the root-mean-square distance from the starting point increases proportional to the square root of time (<a href=\"https:\/\/ps.uci.edu\/~cyu\/p51A\/LectureNotes\/Chapter3\/Chapter3_Diffusion.pdf#:~:text=the%20average%20magnitude%20of%20the,to%20the%20time%20of%20travel\">Chapter3_Diffusion.dvi<\/a>). This is a hallmark of diffusive motion (random walk), as opposed to ballistic motion (straight-line travel with no collisions) where distance grows linearly with time (<a href=\"https:\/\/ps.uci.edu\/~cyu\/p51A\/LectureNotes\/Chapter3\/Chapter3_Diffusion.pdf#:~:text=the%20average%20magnitude%20of%20the,to%20the%20time%20of%20travel\">Chapter3_Diffusion.dvi<\/a>). In other words:<\/p>\n<ul>\n<li><strong>Ballistic motion (no collisions):<\/strong> <img decoding=\"async\" src=\"https:\/\/s0.wp.com\/latex.php?latex=distance+%5Cpropto+t&#038;bg=ffffff&#038;fg=000&#038;s=0&#038;c=20201002\" alt=\"distance &#92;propto t\" class=\"latex\" \/> (travel is unimpeded at constant velocity).<\/li>\n<li><strong>Diffusive motion (many random collisions):<\/strong> <img decoding=\"async\" src=\"https:\/\/s0.wp.com\/latex.php?latex=distance+%5Cpropto+%5Csqrt%7Bt%7D&#038;bg=ffffff&#038;fg=000&#038;s=0&#038;c=20201002\" alt=\"distance &#92;propto &#92;sqrt{t}\" class=\"latex\" \/> (many small random steps tend to cancel out on average, leading to slower spread).<\/li>\n<\/ul>\n<p><strong>Intuitively<\/strong>, a shorter mean free path (due to frequent collisions) means each individual step is short, so the particle\u2019s direction is randomized often. The particle will take many small zig-zag steps to cover a given distance, resulting in slow diffusion. Conversely, a longer mean free path means the particle travels farther before changing direction, tending toward faster spread. Thus, mean free path is a key link between microscopic collision processes and macroscopic diffusion behavior.<\/p>\n<h2>Mathematical Derivation<\/h2>\n<h3>Mean Free Path in a Uniform Medium<\/h3>\n<p>Consider a particle traveling through a uniform medium with number density <img decoding=\"async\" src=\"https:\/\/s0.wp.com\/latex.php?latex=n&#038;bg=ffffff&#038;fg=000&#038;s=0&#038;c=20201002\" alt=\"n\" class=\"latex\" \/> (particles per unit volume) and an effective collision cross-section <img decoding=\"async\" src=\"https:\/\/s0.wp.com\/latex.php?latex=%5Csigma&#038;bg=ffffff&#038;fg=000&#038;s=0&#038;c=20201002\" alt=\"&#92;sigma\" class=\"latex\" \/> (interaction area). We can derive the classic <strong>mean free path formula<\/strong> using a probabilistic (Markovian) argument:<\/p>\n<ol>\n<li><strong>Small-step probability:<\/strong> In an infinitesimal distance <img decoding=\"async\" src=\"https:\/\/s0.wp.com\/latex.php?latex=dx&#038;bg=ffffff&#038;fg=000&#038;s=0&#038;c=20201002\" alt=\"dx\" class=\"latex\" \/>, the probability of a collision is approximately <img decoding=\"async\" src=\"https:\/\/s0.wp.com\/latex.php?latex=n+%5Csigma+%2C+dx&#038;bg=ffffff&#038;fg=000&#038;s=0&#038;c=20201002\" alt=\"n &#92;sigma , dx\" class=\"latex\" \/>. This is because the particle sweeps out a volume in which the expected number of target particles is <img decoding=\"async\" src=\"https:\/\/s0.wp.com\/latex.php?latex=n%2Cdx&#038;bg=ffffff&#038;fg=000&#038;s=0&#038;c=20201002\" alt=\"n,dx\" class=\"latex\" \/> times the cross-sectional area <img decoding=\"async\" src=\"https:\/\/s0.wp.com\/latex.php?latex=%5Csigma&#038;bg=ffffff&#038;fg=000&#038;s=0&#038;c=20201002\" alt=\"&#92;sigma\" class=\"latex\" \/>, leading to a collision probability ~<img decoding=\"async\" src=\"https:\/\/s0.wp.com\/latex.php?latex=n%5Csigma+%2C+dx&#038;bg=ffffff&#038;fg=000&#038;s=0&#038;c=20201002\" alt=\"n&#92;sigma , dx\" class=\"latex\" \/> (<a href=\"https:\/\/en.wikipedia.org\/wiki\/Mean_free_path#:~:text=Image%3A%20,2%7D%7D%7D%3Dn%5Csigma%20%5C%2Cdx\">Mean free path &#8211; Wikipedia<\/a>). Equivalently, the probability of <strong>no<\/strong> collision in <img decoding=\"async\" src=\"https:\/\/s0.wp.com\/latex.php?latex=dx&#038;bg=ffffff&#038;fg=000&#038;s=0&#038;c=20201002\" alt=\"dx\" class=\"latex\" \/> is <img decoding=\"async\" src=\"https:\/\/s0.wp.com\/latex.php?latex=%5Cbigl%281+-+n%5Csigma+%2C+dx%5Cbigr%29&#038;bg=ffffff&#038;fg=000&#038;s=0&#038;c=20201002\" alt=\"&#92;bigl(1 - n&#92;sigma , dx&#92;bigr)\" class=\"latex\" \/>, assuming collisions are memoryless (independent in each segment).<\/li>\n<li><strong>Exponential survival:<\/strong> Let <img decoding=\"async\" src=\"https:\/\/s0.wp.com\/latex.php?latex=P%28%5Ctext%7Bno+collision+up+to+distance+%7D+x%29&#038;bg=ffffff&#038;fg=000&#038;s=0&#038;c=20201002\" alt=\"P(&#92;text{no collision up to distance } x)\" class=\"latex\" \/> be <img decoding=\"async\" src=\"https:\/\/s0.wp.com\/latex.php?latex=S%28x%29&#038;bg=ffffff&#038;fg=000&#038;s=0&#038;c=20201002\" alt=\"S(x)\" class=\"latex\" \/>. Over a small step, <img decoding=\"async\" src=\"https:\/\/s0.wp.com\/latex.php?latex=S%28x+%2B+dx%29+%3D+S%28x%29%2C%5Cbigl%281+-+n%5Csigma%2Cdx%5Cbigr%29.&#038;bg=ffffff&#038;fg=000&#038;s=0&#038;c=20201002\" alt=\"S(x + dx) = S(x),&#92;bigl(1 - n&#92;sigma,dx&#92;bigr).\" class=\"latex\" \/><br \/>\nThis leads to the differential equation<br \/>\n<img decoding=\"async\" src=\"https:\/\/s0.wp.com\/latex.php?latex=%5Cfrac%7BdS%7D%7Bdx%7D+%3D+-%2Cn%5Csigma%2CS%2C&#038;bg=ffffff&#038;fg=000&#038;s=0&#038;c=20201002\" alt=\"&#92;frac{dS}{dx} = -,n&#92;sigma,S,\" class=\"latex\" \/><br \/>\nwhose solution is<br \/>\n<img decoding=\"async\" src=\"https:\/\/s0.wp.com\/latex.php?latex=S%28x%29+%3D+%5Cexp%5Cbigl%28-n%5Csigma%2Cx%5Cbigr%29.&#038;bg=ffffff&#038;fg=000&#038;s=0&#038;c=20201002\" alt=\"S(x) = &#92;exp&#92;bigl(-n&#92;sigma,x&#92;bigr).\" class=\"latex\" \/><br \/>\nThis is the Beer\u2013Lambert law for attenuation and indicates an <em>exponential<\/em> distribution of free-path lengths. The probability density that a collision occurs at distance <img decoding=\"async\" src=\"https:\/\/s0.wp.com\/latex.php?latex=x&#038;bg=ffffff&#038;fg=000&#038;s=0&#038;c=20201002\" alt=\"x\" class=\"latex\" \/> is<br \/>\n<img decoding=\"async\" src=\"https:\/\/s0.wp.com\/latex.php?latex=f%28x%29+%3D+-%5Cfrac%7BdS%7D%7Bdx%7D+%3D+n%5Csigma%2Ce%5E%7B-%2Cn%5Csigma%2Cx%7D.&#038;bg=ffffff&#038;fg=000&#038;s=0&#038;c=20201002\" alt=\"f(x) = -&#92;frac{dS}{dx} = n&#92;sigma,e^{-,n&#92;sigma,x}.\" class=\"latex\" \/><\/li>\n<li><strong>Mean free path:<\/strong> The mean of this exponential distribution is calculated as<br \/>\n<span class=\"katex\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mo stretchy=\"false\">\u27e8<\/mo><mi>x<\/mi><mo stretchy=\"false\">\u27e9<\/mo><mtext>\u2005\u200a<\/mtext><mo>=<\/mo><mtext>\u2005\u200a<\/mtext><msubsup><mo>\u222b<\/mo><mn>0<\/mn><mi mathvariant=\"normal\">\u221e<\/mi><\/msubsup><mi>x<\/mi><mtext>\u2009<\/mtext><mi>f<\/mi><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo stretchy=\"false\">)<\/mo><mtext>\u2009<\/mtext><mi>d<\/mi><mi>x<\/mi><mtext>\u2005\u200a<\/mtext><mo>=<\/mo><mtext>\u2005\u200a<\/mtext><msubsup><mo>\u222b<\/mo><mn>0<\/mn><mi mathvariant=\"normal\">\u221e<\/mi><\/msubsup><mi>x<\/mi><mtext>\u2009<\/mtext><mo stretchy=\"false\">(<\/mo><mi>n<\/mi><mi>\u03c3<\/mi><mo stretchy=\"false\">)<\/mo><mtext>\u2009<\/mtext><msup><mi>e<\/mi><mrow><mo>\u2212<\/mo><mi>n<\/mi><mi>\u03c3<\/mi><mi>x<\/mi><\/mrow><\/msup><mtext>\u2009<\/mtext><mi>d<\/mi><mi>x<\/mi><mtext>\u2005\u200a<\/mtext><mo>=<\/mo><mtext>\u2005\u200a<\/mtext><mfrac><mn>1<\/mn><mrow><mi>n<\/mi><mi>\u03c3<\/mi><\/mrow><\/mfrac><mtext>\u2009<\/mtext><mi mathvariant=\"normal\">.<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\langle x \\rangle \\;=\\; \\int_0^{\\infty} x\\,f(x)\\,dx \\;=\\; \\int_0^{\\infty} x\\,(n\\sigma)\\,e^{-n\\sigma x}\\,dx \\;=\\; \\frac{1}{n\\sigma}\\,.<\/annotation><\/semantics><\/math><\/span> (Recognizing the mean of an Exp(<img decoding=\"async\" src=\"https:\/\/s0.wp.com\/latex.php?latex=%5Clambda&#038;bg=ffffff&#038;fg=000&#038;s=0&#038;c=20201002\" alt=\"&#92;lambda\" class=\"latex\" \/>) distribution is <img decoding=\"async\" src=\"https:\/\/s0.wp.com\/latex.php?latex=1%2F%5Clambda&#038;bg=ffffff&#038;fg=000&#038;s=0&#038;c=20201002\" alt=\"1\/&#92;lambda\" class=\"latex\" \/>.) Thus we obtain:<br \/>\n<strong><span class=\"katex\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>M<\/mi><mi>e<\/mi><mi>a<\/mi><mi>n<\/mi><mi>f<\/mi><mi>r<\/mi><mi>e<\/mi><mi>e<\/mi><mi>p<\/mi><mi>a<\/mi><mi>t<\/mi><mi>h<\/mi><mo>:<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">Mean free path:<\/annotation><\/semantics><\/math><\/span><\/strong> <span class=\"katex\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>\u03bb<\/mi><mtext>\u2005\u200a<\/mtext><mo>=<\/mo><mtext>\u2005\u200a<\/mtext><mfrac><mn>1<\/mn><mrow><mi>n<\/mi><mtext>\u2009<\/mtext><mi>\u03c3<\/mi><\/mrow><\/mfrac><mtext>\u2009<\/mtext><mi mathvariant=\"normal\">.<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\lambda \\;=\\; \\frac{1}{n\\,\\sigma}\\,.<\/annotation><\/semantics><\/math><\/span><\/li>\n<\/ol>\n<p>This result <img decoding=\"async\" src=\"https:\/\/s0.wp.com\/latex.php?latex=%5Clambda+%5Capprox+%5Cfrac%7B1%7D%7Bn%5Csigma%7D&#038;bg=ffffff&#038;fg=000&#038;s=0&#038;c=20201002\" alt=\"&#92;lambda &#92;approx &#92;frac{1}{n&#92;sigma}\" class=\"latex\" \/> matches the intuitive expectation that higher density <img decoding=\"async\" src=\"https:\/\/s0.wp.com\/latex.php?latex=n&#038;bg=ffffff&#038;fg=000&#038;s=0&#038;c=20201002\" alt=\"n\" class=\"latex\" \/> or larger cross-section <img decoding=\"async\" src=\"https:\/\/s0.wp.com\/latex.php?latex=%5Csigma&#038;bg=ffffff&#038;fg=000&#038;s=0&#038;c=20201002\" alt=\"&#92;sigma\" class=\"latex\" \/> shorten the mean free path (<a href=\"https:\/\/ps.uci.edu\/~cyu\/p51A\/LectureNotes\/Chapter3\/Chapter3_Diffusion.pdf#:~:text=it%20presents%20as%20it%20moves,1\">Chapter3_Diffusion.dvi<\/a>). The derivation assumes a <strong>Markov (memoryless) process<\/strong> for collisions \u2013 i.e. the probability of collision in the next small interval does not depend on how far the particle has already traveled without colliding. This memoryless property is what yields the exponential distribution of path lengths and the above formula for <img decoding=\"async\" src=\"https:\/\/s0.wp.com\/latex.php?latex=%5Clambda&#038;bg=ffffff&#038;fg=000&#038;s=0&#038;c=20201002\" alt=\"&#92;lambda\" class=\"latex\" \/>, rigorously confirming the relationship.<\/p>\n<h3>Random Walk and Diffusive Spread<\/h3>\n<p>Now consider a particle undergoing a sequence of collisions (a random walk in continuous space). If <strong><img decoding=\"async\" src=\"https:\/\/s0.wp.com\/latex.php?latex=%5Clambda&#038;bg=ffffff&#038;fg=000&#038;s=0&#038;c=20201002\" alt=\"&#92;lambda\" class=\"latex\" \/><\/strong> is the mean free path (average step length) and <strong><img decoding=\"async\" src=\"https:\/\/s0.wp.com\/latex.php?latex=%5Ctau&#038;bg=ffffff&#038;fg=000&#038;s=0&#038;c=20201002\" alt=\"&#92;tau\" class=\"latex\" \/><\/strong> is the mean time between collisions, we can quantify the connection between these microscopic parameters and diffusion. After <img decoding=\"async\" src=\"https:\/\/s0.wp.com\/latex.php?latex=N&#038;bg=ffffff&#038;fg=000&#038;s=0&#038;c=20201002\" alt=\"N\" class=\"latex\" \/> random steps (collisions), the net displacement <img decoding=\"async\" src=\"https:\/\/s0.wp.com\/latex.php?latex=%5Cmathbf%7BR%7D&#038;bg=ffffff&#038;fg=000&#038;s=0&#038;c=20201002\" alt=\"&#92;mathbf{R}\" class=\"latex\" \/> is the vector sum <img decoding=\"async\" src=\"https:\/\/s0.wp.com\/latex.php?latex=%5Cmathbf%7Bd%7D_1+%2B+%5Cmathbf%7Bd%7D_2+%2B+%5Ccdots+%2B+%5Cmathbf%7Bd%7D_N&#038;bg=ffffff&#038;fg=000&#038;s=0&#038;c=20201002\" alt=\"&#92;mathbf{d}_1 + &#92;mathbf{d}_2 + &#92;cdots + &#92;mathbf{d}_N\" class=\"latex\" \/>. Because the direction of each step is random, the <em>average<\/em> displacement <img decoding=\"async\" src=\"https:\/\/s0.wp.com\/latex.php?latex=%5Clangle+%5Cmathbf%7BR%7D+%5Crangle&#038;bg=ffffff&#038;fg=000&#038;s=0&#038;c=20201002\" alt=\"&#92;langle &#92;mathbf{R} &#92;rangle\" class=\"latex\" \/> is zero \u2013 the motion is unbiased. However, the mean-square displacement <img decoding=\"async\" src=\"https:\/\/s0.wp.com\/latex.php?latex=%5Clangle+R%5E2%5Crangle&#038;bg=ffffff&#038;fg=000&#038;s=0&#038;c=20201002\" alt=\"&#92;langle R^2&#92;rangle\" class=\"latex\" \/> is not zero. Summing the contributions of independent steps:<\/p>\n<ul>\n<li>Each step has length <img decoding=\"async\" src=\"https:\/\/s0.wp.com\/latex.php?latex=d+%5Capprox+%5Clambda&#038;bg=ffffff&#038;fg=000&#038;s=0&#038;c=20201002\" alt=\"d &#92;approx &#92;lambda\" class=\"latex\" \/>. One can show that the <strong>mean squared displacement<\/strong> after <img decoding=\"async\" src=\"https:\/\/s0.wp.com\/latex.php?latex=N&#038;bg=ffffff&#038;fg=000&#038;s=0&#038;c=20201002\" alt=\"N\" class=\"latex\" \/> steps is <img decoding=\"async\" src=\"https:\/\/s0.wp.com\/latex.php?latex=%5Clangle+R%5E2+%5Crangle+%3D+N%2C%5Clambda%5E2&#038;bg=ffffff&#038;fg=000&#038;s=0&#038;c=20201002\" alt=\"&#92;langle R^2 &#92;rangle = N,&#92;lambda^2\" class=\"latex\" \/>. (In an ensemble average, cross terms cancel out, leaving <img decoding=\"async\" src=\"https:\/\/s0.wp.com\/latex.php?latex=N&#038;bg=ffffff&#038;fg=000&#038;s=0&#038;c=20201002\" alt=\"N\" class=\"latex\" \/> times the mean squared step length <img decoding=\"async\" src=\"https:\/\/s0.wp.com\/latex.php?latex=%5Clambda%5E2&#038;bg=ffffff&#038;fg=000&#038;s=0&#038;c=20201002\" alt=\"&#92;lambda^2\" class=\"latex\" \/>.)<\/li>\n<li>The <strong>root-mean-square (RMS)<\/strong> distance from the origin is then <img decoding=\"async\" src=\"https:\/\/s0.wp.com\/latex.php?latex=R_%7B%5Ctext%7Brms%7D%7D+%3D+%5Csqrt%7BN%7D%2C%5Clambda&#038;bg=ffffff&#038;fg=000&#038;s=0&#038;c=20201002\" alt=\"R_{&#92;text{rms}} = &#92;sqrt{N},&#92;lambda\" class=\"latex\" \/>. This shows <img decoding=\"async\" src=\"https:\/\/s0.wp.com\/latex.php?latex=R_%7B%5Ctext%7Brms%7D%7D&#038;bg=ffffff&#038;fg=000&#038;s=0&#038;c=20201002\" alt=\"R_{&#92;text{rms}}\" class=\"latex\" \/> grows with the square root of the number of steps, consistent with diffusive behavior.<\/li>\n<\/ul>\n<p>If the particle moves with average speed <img decoding=\"async\" src=\"https:\/\/s0.wp.com\/latex.php?latex=v&#038;bg=ffffff&#038;fg=000&#038;s=0&#038;c=20201002\" alt=\"v\" class=\"latex\" \/> (or more precisely <img decoding=\"async\" src=\"https:\/\/s0.wp.com\/latex.php?latex=v_%7B%5Crm+rms%7D&#038;bg=ffffff&#038;fg=000&#038;s=0&#038;c=20201002\" alt=\"v_{&#92;rm rms}\" class=\"latex\" \/> for a thermal particle), the mean time between collisions is <img decoding=\"async\" src=\"https:\/\/s0.wp.com\/latex.php?latex=%5Ctau+%3D+%5Cfrac%7B%5Clambda%7D%7Bv%7D&#038;bg=ffffff&#038;fg=000&#038;s=0&#038;c=20201002\" alt=\"&#92;tau = &#92;frac{&#92;lambda}{v}\" class=\"latex\" \/>. In a total time <img decoding=\"async\" src=\"https:\/\/s0.wp.com\/latex.php?latex=t&#038;bg=ffffff&#038;fg=000&#038;s=0&#038;c=20201002\" alt=\"t\" class=\"latex\" \/>, the expected number of steps is <img decoding=\"async\" src=\"https:\/\/s0.wp.com\/latex.php?latex=N+%3D+%5Cfrac%7Bt%7D%7B%5Ctau%7D&#038;bg=ffffff&#038;fg=000&#038;s=0&#038;c=20201002\" alt=\"N = &#92;frac{t}{&#92;tau}\" class=\"latex\" \/>. Substituting, we get the RMS distance in time <img decoding=\"async\" src=\"https:\/\/s0.wp.com\/latex.php?latex=t&#038;bg=ffffff&#038;fg=000&#038;s=0&#038;c=20201002\" alt=\"t\" class=\"latex\" \/>:<\/p>\n<p><span class=\"katex\"><math display=\"block\" xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msub><mi>R<\/mi><mtext>rms<\/mtext><\/msub><mo>=<\/mo><msqrt><mfrac><mi>t<\/mi><mi>\u03c4<\/mi><\/mfrac><\/msqrt><mtext>\u2005\u200a<\/mtext><mi>\u03bb<\/mi><mtext>\u2005\u200a<\/mtext><mo>=<\/mo><mtext>\u2005\u200a<\/mtext><msqrt><mfrac><mrow><mi>t<\/mi><mtext>\u2009<\/mtext><mi>v<\/mi><\/mrow><mi>\u03bb<\/mi><\/mfrac><\/msqrt><mtext>\u2005\u200a<\/mtext><mi>\u03bb<\/mi><mtext>\u2005\u200a<\/mtext><mo>=<\/mo><mtext>\u2005\u200a<\/mtext><msqrt><mrow><mtext>\u2009<\/mtext><mi>v<\/mi><mtext>\u2009<\/mtext><mi>\u03bb<\/mi><mtext>\u2009<\/mtext><mi>t<\/mi><mtext>\u2009<\/mtext><\/mrow><\/msqrt><mi mathvariant=\"normal\">.<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">R_{\\text{rms}} = \\sqrt{\\frac{t}{\\tau}}\\;\\lambda \\;=\\; \\sqrt{\\frac{t\\,v}{\\lambda}}\\;\\lambda \\;=\\; \\sqrt{\\,v\\,\\lambda\\,t\\,}.<\/annotation><\/semantics><\/math><\/span><\/p>\n<p>Thus <img decoding=\"async\" src=\"https:\/\/s0.wp.com\/latex.php?latex=R_%7B%5Ctext%7Brms%7D%7D+%5Cpropto+%5Csqrt%7Bt%7D&#038;bg=ffffff&#038;fg=000&#038;s=0&#038;c=20201002\" alt=\"R_{&#92;text{rms}} &#92;propto &#92;sqrt{t}\" class=\"latex\" \/>, as noted earlier. This random-walk result can be connected to the <strong>diffusion coefficient <img decoding=\"async\" src=\"https:\/\/s0.wp.com\/latex.php?latex=D&#038;bg=ffffff&#038;fg=000&#038;s=0&#038;c=20201002\" alt=\"D\" class=\"latex\" \/><\/strong>. In three dimensions, one can show that <img decoding=\"async\" src=\"https:\/\/s0.wp.com\/latex.php?latex=%5Clangle+R%5E2+%5Crangle+%3D+6%2CD%2Ct&#038;bg=ffffff&#038;fg=000&#038;s=0&#038;c=20201002\" alt=\"&#92;langle R^2 &#92;rangle = 6,D,t\" class=\"latex\" \/> for long-time diffusive motion. Comparing with <img decoding=\"async\" src=\"https:\/\/s0.wp.com\/latex.php?latex=%5Clangle+R%5E2%5Crangle+%3D+v%2C%5Clambda%2Ct&#038;bg=ffffff&#038;fg=000&#038;s=0&#038;c=20201002\" alt=\"&#92;langle R^2&#92;rangle = v,&#92;lambda,t\" class=\"latex\" \/> from above, we identify:<\/p>\n<img decoding=\"async\" src=\"https:\/\/s0.wp.com\/latex.php?latex=D+%3D+%5Cfrac%7Bv%2C%5Clambda%7D%7B6%7D%2C.&#038;bg=ffffff&#038;fg=000&#038;s=0&#038;c=20201002\" alt=\"D = &#92;frac{v,&#92;lambda}{6},.\" class=\"latex\" \/>\n<p>Using <img decoding=\"async\" src=\"https:\/\/s0.wp.com\/latex.php?latex=v_%7B%5Crm+rms%7D&#038;bg=ffffff&#038;fg=000&#038;s=0&#038;c=20201002\" alt=\"v_{&#92;rm rms}\" class=\"latex\" \/> (root-mean-square speed) instead of a single <img decoding=\"async\" src=\"https:\/\/s0.wp.com\/latex.php?latex=v&#038;bg=ffffff&#038;fg=000&#038;s=0&#038;c=20201002\" alt=\"v\" class=\"latex\" \/>, this is often written <img decoding=\"async\" src=\"https:\/\/s0.wp.com\/latex.php?latex=D+%3D+%5Ctfrac%7B1%7D%7B3%7D%2Cv_%7B%5Crm+rms%7D%2C%5Clambda&#038;bg=ffffff&#038;fg=000&#038;s=0&#038;c=20201002\" alt=\"D = &#92;tfrac{1}{3},v_{&#92;rm rms},&#92;lambda\" class=\"latex\" \/>. In fact, kinetic theory of gases yields <strong><img decoding=\"async\" src=\"https:\/\/s0.wp.com\/latex.php?latex=D+%3D+%5Ctfrac%7B1%7D%7B3%7D%2C%5Cbar%7Bv%7D%2C%5Clambda&#038;bg=ffffff&#038;fg=000&#038;s=0&#038;c=20201002\" alt=\"D = &#92;tfrac{1}{3},&#92;bar{v},&#92;lambda\" class=\"latex\" \/><\/strong> (with <img decoding=\"async\" src=\"https:\/\/s0.wp.com\/latex.php?latex=%5Cbar%7Bv%7D&#038;bg=ffffff&#038;fg=000&#038;s=0&#038;c=20201002\" alt=\"&#92;bar{v}\" class=\"latex\" \/> the mean molecular speed) as a standard result. This shows the diffusion coefficient is directly proportional to the mean free path: a longer <img decoding=\"async\" src=\"https:\/\/s0.wp.com\/latex.php?latex=%5Clambda&#038;bg=ffffff&#038;fg=000&#038;s=0&#038;c=20201002\" alt=\"&#92;lambda\" class=\"latex\" \/> leads to a larger <img decoding=\"async\" src=\"https:\/\/s0.wp.com\/latex.php?latex=D&#038;bg=ffffff&#038;fg=000&#038;s=0&#038;c=20201002\" alt=\"D\" class=\"latex\" \/>, all else being equal.<\/p>\n<blockquote><p><strong>Key relationships:<\/strong><\/p>\n<ul>\n<li><strong>Mean free path:<\/strong> <img decoding=\"async\" src=\"https:\/\/s0.wp.com\/latex.php?latex=%5Clambda+%3D+%5Cfrac%7B1%7D%7Bn%2C%5Csigma%7D&#038;bg=ffffff&#038;fg=000&#038;s=0&#038;c=20201002\" alt=\"&#92;lambda = &#92;frac{1}{n,&#92;sigma}\" class=\"latex\" \/> (for a uniform medium of density <img decoding=\"async\" src=\"https:\/\/s0.wp.com\/latex.php?latex=n&#038;bg=ffffff&#038;fg=000&#038;s=0&#038;c=20201002\" alt=\"n\" class=\"latex\" \/> and cross-section <img decoding=\"async\" src=\"https:\/\/s0.wp.com\/latex.php?latex=%5Csigma&#038;bg=ffffff&#038;fg=000&#038;s=0&#038;c=20201002\" alt=\"&#92;sigma\" class=\"latex\" \/>).<\/li>\n<li><strong>Random-walk displacement:<\/strong> <img decoding=\"async\" src=\"https:\/\/s0.wp.com\/latex.php?latex=%5Clangle+R%5E2+%5Crangle+%3D+N%2C%5Clambda%5E2&#038;bg=ffffff&#038;fg=000&#038;s=0&#038;c=20201002\" alt=\"&#92;langle R^2 &#92;rangle = N,&#92;lambda^2\" class=\"latex\" \/> \u21d2 <img decoding=\"async\" src=\"https:\/\/s0.wp.com\/latex.php?latex=R_%7B%5Ctext%7Brms%7D%7D+%3D+%5Csqrt%7BN%7D%2C%5Clambda&#038;bg=ffffff&#038;fg=000&#038;s=0&#038;c=20201002\" alt=\"R_{&#92;text{rms}} = &#92;sqrt{N},&#92;lambda\" class=\"latex\" \/>.<\/li>\n<li><strong>Collision rate:<\/strong> <img decoding=\"async\" src=\"https:\/\/s0.wp.com\/latex.php?latex=%5Ctau+%3D+%5Cfrac%7B%5Clambda%7D%7Bv_%7B%5Crm+rms%7D%7D&#038;bg=ffffff&#038;fg=000&#038;s=0&#038;c=20201002\" alt=\"&#92;tau = &#92;frac{&#92;lambda}{v_{&#92;rm rms}}\" class=\"latex\" \/> (mean time per step), so steps per unit time = <img decoding=\"async\" src=\"https:\/\/s0.wp.com\/latex.php?latex=%5Cfrac%7B1%7D%7B%5Ctau%7D+%3D+%5Cfrac%7Bv_%7B%5Crm+rms%7D%7D%7B%5Clambda%7D&#038;bg=ffffff&#038;fg=000&#038;s=0&#038;c=20201002\" alt=\"&#92;frac{1}{&#92;tau} = &#92;frac{v_{&#92;rm rms}}{&#92;lambda}\" class=\"latex\" \/>.<\/li>\n<li><strong>Diffusion coefficient:<\/strong> <img decoding=\"async\" src=\"https:\/\/s0.wp.com\/latex.php?latex=D+%3D+%5Cfrac%7B1%7D%7B3%7D%2Cv_%7B%5Crm+rms%7D%2C%5Clambda&#038;bg=ffffff&#038;fg=000&#038;s=0&#038;c=20201002\" alt=\"D = &#92;frac{1}{3},v_{&#92;rm rms},&#92;lambda\" class=\"latex\" \/> (in 3D, for an ideal gas or similar isotropic random flight).<\/li>\n<li><em>(In 1D, one finds <img decoding=\"async\" src=\"https:\/\/s0.wp.com\/latex.php?latex=D+%3D+%5Ctfrac%7B1%7D%7B2%7D%2Cv_%7B%5Crm+rms%7D%2C%5Clambda&#038;bg=ffffff&#038;fg=000&#038;s=0&#038;c=20201002\" alt=\"D = &#92;tfrac{1}{2},v_{&#92;rm rms},&#92;lambda\" class=\"latex\" \/>; in 2D, <img decoding=\"async\" src=\"https:\/\/s0.wp.com\/latex.php?latex=D+%3D+%5Ctfrac%7B1%7D%7B4%7D%2Cv_%7B%5Crm+rms%7D%2C%5Clambda&#038;bg=ffffff&#038;fg=000&#038;s=0&#038;c=20201002\" alt=\"D = &#92;tfrac{1}{4},v_{&#92;rm rms},&#92;lambda\" class=\"latex\" \/>. The <img decoding=\"async\" src=\"https:\/\/s0.wp.com\/latex.php?latex=%5Ctfrac%7B1%7D%7B3%7D&#038;bg=ffffff&#038;fg=000&#038;s=0&#038;c=20201002\" alt=\"&#92;tfrac{1}{3}\" class=\"latex\" \/> is the 3D-specific factor reflecting three degrees of freedom.)<\/em><\/li>\n<\/ul>\n<\/blockquote>\n<h2>Markov Chain Perspective<\/h2>\n<p>The above derivations can be framed in terms of <strong>Markov chains<\/strong> or Markov processes. A Markov chain is a stochastic process with no memory beyond the current state. The random collision process <strong>is memoryless<\/strong>: once a collision happens, the particle \u201cforgets\u201d its previous direction and starts a new step fresh. Similarly, the distance traveled without collision has no memory \u2013 the chance of colliding in the next <img decoding=\"async\" src=\"https:\/\/s0.wp.com\/latex.php?latex=%5Cdelta+x&#038;bg=ffffff&#038;fg=000&#038;s=0&#038;c=20201002\" alt=\"&#92;delta x\" class=\"latex\" \/> is the same regardless of how long it has been since the last collision. This <strong>Markov property<\/strong> underpins the exponential distribution of free path lengths derived earlier. Formally, one can say the <em>distance-to-collision<\/em> <img decoding=\"async\" src=\"https:\/\/s0.wp.com\/latex.php?latex=X&#038;bg=ffffff&#038;fg=000&#038;s=0&#038;c=20201002\" alt=\"X\" class=\"latex\" \/> satisfies:<\/p>\n<img decoding=\"async\" src=\"https:\/\/s0.wp.com\/latex.php?latex=P%5Cbigl%28X+%3E+x+%2B+%5CDelta+x+%3B%5Cbigm%7C%3B+X+%3E+x%5Cbigr%29+%3D+P%5Cbigl%28X+%3E+%5CDelta+x%5Cbigr%29&#038;bg=ffffff&#038;fg=000&#038;s=0&#038;c=20201002\" alt=\"P&#92;bigl(X &gt; x + &#92;Delta x ;&#92;bigm|; X &gt; x&#92;bigr) = P&#92;bigl(X &gt; &#92;Delta x&#92;bigr)\" class=\"latex\" \/>\n<p>for all <img decoding=\"async\" src=\"https:\/\/s0.wp.com\/latex.php?latex=x%2C+%5CDelta+x+%5Cge+0&#038;bg=ffffff&#038;fg=000&#038;s=0&#038;c=20201002\" alt=\"x, &#92;Delta x &#92;ge 0\" class=\"latex\" \/>, which is the defining property of the exponential distribution. Solving this functional equation yields<\/p>\n<img decoding=\"async\" src=\"https:\/\/s0.wp.com\/latex.php?latex=P%5Cbigl%28X+%3E+x%5Cbigr%29+%3D+e%5E%7B-%2Cx%2F%5Clambda%7D%2C&#038;bg=ffffff&#038;fg=000&#038;s=0&#038;c=20201002\" alt=\"P&#92;bigl(X &gt; x&#92;bigr) = e^{-,x\/&#92;lambda},\" class=\"latex\" \/>\n<p>with <img decoding=\"async\" src=\"https:\/\/s0.wp.com\/latex.php?latex=%5Clambda+%3D+E%5BX%5D&#038;bg=ffffff&#038;fg=000&#038;s=0&#038;c=20201002\" alt=\"&#92;lambda = E[X]\" class=\"latex\" \/>.<\/p>\n<p>We can model the particle\u2019s trajectory as a <strong>Markov chain in discrete steps<\/strong>: each step ends in a collision (state transition), and the direction of the next step is chosen randomly (independent of previous directions). In this chain, the <em>step length<\/em> is a random variable with mean <img decoding=\"async\" src=\"https:\/\/s0.wp.com\/latex.php?latex=%5Clambda&#038;bg=ffffff&#038;fg=000&#038;s=0&#038;c=20201002\" alt=\"&#92;lambda\" class=\"latex\" \/> (and exponential distribution, in a uniform medium), and the <em>transition probability<\/em> for turning into any direction is uniform (isotropic scattering). Using Markov chain theory, one can rigorously compute quantities like the expected number of steps or the distribution of displacements. For example:<\/p>\n<ul>\n<li>The <em>expected number of collisions<\/em> in a path of length <img decoding=\"async\" src=\"https:\/\/s0.wp.com\/latex.php?latex=L&#038;bg=ffffff&#038;fg=000&#038;s=0&#038;c=20201002\" alt=\"L\" class=\"latex\" \/> is <img decoding=\"async\" src=\"https:\/\/s0.wp.com\/latex.php?latex=N+%3D+%5Ctfrac%7BL%7D%7B%5Clambda%7D&#038;bg=ffffff&#038;fg=000&#038;s=0&#038;c=20201002\" alt=\"N = &#92;tfrac{L}{&#92;lambda}\" class=\"latex\" \/>. This aligns with treating collisions as a Poisson process (which is a continuous-time Markov process) with rate <img decoding=\"async\" src=\"https:\/\/s0.wp.com\/latex.php?latex=%5Ctfrac%7B1%7D%7B%5Clambda%7D&#038;bg=ffffff&#038;fg=000&#038;s=0&#038;c=20201002\" alt=\"&#92;tfrac{1}{&#92;lambda}\" class=\"latex\" \/> per unit length.<\/li>\n<li>The <em>transition kernel<\/em> of the Markov chain for direction ensures that after many steps, the probability distribution of the particle\u2019s displacement approaches a Gaussian (by the Central Limit Theorem), satisfying the diffusion equation. In fact, one can derive the <strong>diffusion equation<\/strong> (Fick\u2019s second law) from the master equation of this Markov chain. For a one-dimensional random walk with step length <img decoding=\"async\" src=\"https:\/\/s0.wp.com\/latex.php?latex=%5Clambda&#038;bg=ffffff&#038;fg=000&#038;s=0&#038;c=20201002\" alt=\"&#92;lambda\" class=\"latex\" \/> and step time <img decoding=\"async\" src=\"https:\/\/s0.wp.com\/latex.php?latex=%5Ctau&#038;bg=ffffff&#038;fg=000&#038;s=0&#038;c=20201002\" alt=\"&#92;tau\" class=\"latex\" \/>, the chain\u2019s evolution for the probability <img decoding=\"async\" src=\"https:\/\/s0.wp.com\/latex.php?latex=P%28x%2Ct%29&#038;bg=ffffff&#038;fg=000&#038;s=0&#038;c=20201002\" alt=\"P(x,t)\" class=\"latex\" \/> satisfies <img decoding=\"async\" src=\"https:\/\/s0.wp.com\/latex.php?latex=P%28x%2Ct%2B%5Ctau%29+%5Capprox+%5Ctfrac12%2CP%28x-%5Clambda%2Ct%29+%2B+%5Ctfrac12%2CP%28x%2B%5Clambda%2Ct%29&#038;bg=ffffff&#038;fg=000&#038;s=0&#038;c=20201002\" alt=\"P(x,t+&#92;tau) &#92;approx &#92;tfrac12,P(x-&#92;lambda,t) + &#92;tfrac12,P(x+&#92;lambda,t)\" class=\"latex\" \/>. In the limit <img decoding=\"async\" src=\"https:\/\/s0.wp.com\/latex.php?latex=%5Clambda+%5Cto+0&#038;bg=ffffff&#038;fg=000&#038;s=0&#038;c=20201002\" alt=\"&#92;lambda &#92;to 0\" class=\"latex\" \/>, <img decoding=\"async\" src=\"https:\/\/s0.wp.com\/latex.php?latex=%5Ctau+%5Cto+0&#038;bg=ffffff&#038;fg=000&#038;s=0&#038;c=20201002\" alt=\"&#92;tau &#92;to 0\" class=\"latex\" \/> with <img decoding=\"async\" src=\"https:\/\/s0.wp.com\/latex.php?latex=D%3D%5Ctfrac%7B%5Clambda%5E2%7D%7B2%5Ctau%7D&#038;bg=ffffff&#038;fg=000&#038;s=0&#038;c=20201002\" alt=\"D=&#92;tfrac{&#92;lambda^2}{2&#92;tau}\" class=\"latex\" \/> fixed, this leads to <img decoding=\"async\" src=\"https:\/\/s0.wp.com\/latex.php?latex=%5Cfrac%7B%5Cpartial+P%7D%7B%5Cpartial+t%7D+%3D+D%2C%5Cfrac%7B%5Cpartial%5E2+P%7D%7B%5Cpartial+x%5E2%7D&#038;bg=ffffff&#038;fg=000&#038;s=0&#038;c=20201002\" alt=\"&#92;frac{&#92;partial P}{&#92;partial t} = D,&#92;frac{&#92;partial^2 P}{&#92;partial x^2}\" class=\"latex\" \/>, the diffusion equation. Markov chain analysis thus provides a <strong>rigorous bridge<\/strong> from microscopic jumps to macroscopic diffusion laws.<\/li>\n<\/ul>\n<p>In summary, the Markov chain perspective treats each collision (or each step) as one step of a stochastic process with a simple memoryless transition rule. This approach not only reproduces the mean free path formula, but also gives a systematic way to derive diffusion properties. Indeed, researchers have used Markov chain techniques to analyze complex transport processes. For example, the <strong>transport mean free path<\/strong> and diffusion length in neutron scattering can be derived via Markov chain statistics (<a href=\"https:\/\/www.osti.gov\/biblio\/4788156#:~:text=A%20derivation%20is%20given%20of,R\">TRANSPORT MEAN FREE PATH AND DIFFUSION LENGTH OF NEUTRONS CALCULATED FROM MARKOV CHAIN STATISTICS (Journal Article) | OSTI.GOV<\/a>), offering insights beyond a basic random-walk argument. The power of the Markov chain approach is that it can handle arbitrary scattering phase functions or absorption probabilities by appropriate transition probabilities, and still yield exact or approximate analytical results for measures like mean distance traveled, variance, or absorption probability.<\/p>\n<h2>Applications and Examples<\/h2>\n<p>The relationship between mean free path and diffusion steps (random walks) is widely used in physics and engineering to model transport processes:<\/p>\n<ul>\n<li><strong>Gas Diffusion (Kinetic Theory):<\/strong> Gas molecules undergo random collisions with each other. Using the mean free path (<img decoding=\"async\" src=\"https:\/\/s0.wp.com\/latex.php?latex=%5Clambda+%3D+1%2F%28n%5Csigma%29&#038;bg=ffffff&#038;fg=000&#038;s=0&#038;c=20201002\" alt=\"&#92;lambda = 1\/(n&#92;sigma)\" class=\"latex\" \/>) and average molecular speed, one can predict self-diffusion coefficients. For instance, for air molecules at atmospheric pressure, <img decoding=\"async\" src=\"https:\/\/s0.wp.com\/latex.php?latex=%5Clambda&#038;bg=ffffff&#038;fg=000&#038;s=0&#038;c=20201002\" alt=\"&#92;lambda\" class=\"latex\" \/> is on the order of 100\u00a0nm, and plugging into <img decoding=\"async\" src=\"https:\/\/s0.wp.com\/latex.php?latex=D%3D%5Ctfrac%7B1%7D%7B3%7D%2C%5Cbar%7Bv%7D%2C%5Clambda&#038;bg=ffffff&#038;fg=000&#038;s=0&#038;c=20201002\" alt=\"D=&#92;tfrac{1}{3},&#92;bar{v},&#92;lambda\" class=\"latex\" \/> correctly estimates the diffusion coefficient in air. The same microscopic picture explains viscosity and thermal conductivity of gases (molecules carry momentum or energy <img decoding=\"async\" src=\"https:\/\/s0.wp.com\/latex.php?latex=%5Clambda&#038;bg=ffffff&#038;fg=000&#038;s=0&#038;c=20201002\" alt=\"&#92;lambda\" class=\"latex\" \/> distance on average before exchanging it), with similar <img decoding=\"async\" src=\"https:\/\/s0.wp.com\/latex.php?latex=%5Ctfrac%7B1%7D%7B3%7D&#038;bg=ffffff&#038;fg=000&#038;s=0&#038;c=20201002\" alt=\"&#92;tfrac{1}{3}\" class=\"latex\" \/> factors appearing in those transport coefficients.<\/li>\n<li><strong>Brownian Motion:<\/strong> In a liquid, a small particle (like a pollen grain in water) is bombarded by fluid molecules. Although the collisions are more frequent and chaotic, one can define an effective mean free path and step time for the particle\u2019s jittery motion. The Markov chain\/random walk model (as first quantitatively explained by Einstein) leads to the diffusion constant for Brownian motion. This explains, for example, how particle size and fluid viscosity affect the diffusion rate of colloidal particles.<\/li>\n<li><strong>Radiative Transfer (Photon Diffusion):<\/strong> Photons traveling through a scattering medium (like light in fog, or gamma rays in the atmosphere) undergo random scatters. The mean free path might be the distance between photon interactions. Modeling the photon\u2019s path as a Markov chain (often via Monte Carlo simulation) is common practice. It yields the exponential attenuation law (Beer\u2019s law) for intensity and diffusion-like spread of radiation in thick materials. In stars, for example, a photon\u2019s random walk from the core to the surface can be analyzed by treating each scatter as a step; the <strong>random walk length<\/strong> with a given <img decoding=\"async\" src=\"https:\/\/s0.wp.com\/latex.php?latex=%5Clambda&#038;bg=ffffff&#038;fg=000&#038;s=0&#038;c=20201002\" alt=\"&#92;lambda\" class=\"latex\" \/> can be millions of steps.<\/li>\n<li><strong>Neutron Diffusion in Reactors:<\/strong> Neutrons born from fission collide with nuclei in a reactor. Their paths form a random walk until absorption. The <strong>transport mean free path<\/strong> (which accounts for forward-scattering effects) can be derived with Markov chain models. This is used to calculate the diffusion length and moderation length of neutrons in reactor physics. Markov chain approaches offer a concise way to derive these quantities, clarifying the random-walk mechanism of neutron thermalization.<\/li>\n<\/ul>\n<p>In all these examples, treating the particle\u2019s trajectory as a Markovian random walk provides a powerful framework. It connects <strong>microscopic parameters<\/strong> (like cross-sections, densities, or scattering probabilities) to <strong>macroscopic behavior<\/strong> (like diffusion rates, attenuation lengths, and spatial spread). The mean free path emerges as a fundamental bridge between the two scales \u2013 essentially setting the \u201cstep size\u201d of the diffusion process. By leveraging Markov chain theory, one can rigorously derive the equations governing mean free path and diffusion, and confidently apply them in diverse fields ranging from gas kinetics to astrophysics.<\/p>\n<p><strong>Sources:<\/strong> The relationship <img decoding=\"async\" src=\"https:\/\/s0.wp.com\/latex.php?latex=%5Clambda+%3D+1%2F%28n%5Csigma%29&#038;bg=ffffff&#038;fg=000&#038;s=0&#038;c=20201002\" alt=\"&#92;lambda = 1\/(n&#92;sigma)\" class=\"latex\" \/> and its derivation are discussed in kinetic theory texts (<a href=\"https:\/\/ps.uci.edu\/~cyu\/p51A\/LectureNotes\/Chapter3\/Chapter3_Diffusion.pdf#:~:text=it%20presents%20as%20it%20moves,1\">Chapter3_Diffusion.dvi<\/a>) (<a href=\"https:\/\/en.wikipedia.org\/wiki\/Mean_free_path#:~:text=Image%3A%20%7B%5Cdisplaystyle%20dI%3D\">Mean free path &#8211; Wikipedia<\/a>). Random-walk derivations of diffusion (showing <img decoding=\"async\" src=\"https:\/\/s0.wp.com\/latex.php?latex=%5Clangle+R%5E2%5Crangle+%5Cpropto+t&#038;bg=ffffff&#038;fg=000&#038;s=0&#038;c=20201002\" alt=\"&#92;langle R^2&#92;rangle &#92;propto t\" class=\"latex\" \/> and <img decoding=\"async\" src=\"https:\/\/s0.wp.com\/latex.php?latex=D%3D%5Ctfrac%7B1%7D%7B3%7Dv%5Clambda&#038;bg=ffffff&#038;fg=000&#038;s=0&#038;c=20201002\" alt=\"D=&#92;tfrac{1}{3}v&#92;lambda\" class=\"latex\" \/> in 3D) appear in physics literature (<a href=\"https:\/\/ps.uci.edu\/~cyu\/p51A\/LectureNotes\/Chapter3\/Chapter3_Diffusion.pdf#:~:text=The%20root,d%2C%20then%20Drms%20%3D%20%E2%88%9A\">Chapter3_Diffusion.dvi<\/a>) and are also found in resources like the Wolfram Formula Repository. Markov chain methods have been used in neutron transport and radiative transfer to derive mean free paths and diffusion lengths with greater generality (<a href=\"https:\/\/www.osti.gov\/biblio\/4788156#:~:text=A%20derivation%20is%20given%20of,R\">TRANSPORT MEAN FREE PATH AND DIFFUSION LENGTH OF NEUTRONS CALCULATED FROM MARKOV CHAIN STATISTICS (Journal Article) | OSTI.GOV<\/a>). These approaches illustrate the deep connection between simple probabilistic steps and the emergent behavior of diffusion.<\/p>\n<p>&nbsp;<\/p>\n","protected":false},"excerpt":{"rendered":"<p>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\u2019s trajectory becomes a zig-zag as it collides and changes direction. The sequence of straight-line \u201cflights\u201d between collisions can be thought of as diffusion steps in a random [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_crdt_document":"","_jetpack_memberships_contains_paid_content":false,"footnotes":""},"categories":[1],"tags":[],"class_list":["post-89","post","type-post","status-publish","format-standard","hentry","category-uncategorized"],"jetpack_featured_media_url":"","jetpack_sharing_enabled":true,"brizy_media":[],"_links":{"self":[{"href":"https:\/\/freepath.info\/index.php?rest_route=\/wp\/v2\/posts\/89","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/freepath.info\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/freepath.info\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/freepath.info\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/freepath.info\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=89"}],"version-history":[{"count":3,"href":"https:\/\/freepath.info\/index.php?rest_route=\/wp\/v2\/posts\/89\/revisions"}],"predecessor-version":[{"id":102,"href":"https:\/\/freepath.info\/index.php?rest_route=\/wp\/v2\/posts\/89\/revisions\/102"}],"wp:attachment":[{"href":"https:\/\/freepath.info\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=89"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/freepath.info\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=89"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/freepath.info\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=89"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}