Editing Brownian motion (section)

===Smoluchowski model===
{{Primary sources|date=January 2025}}
[[Marian Smoluchowski|Smoluchowski]]'s theory of Brownian motion<ref>{{cite journal | last = Smoluchowski | first = M. M. | year = 1906 | title = Sur le chemin moyen parcouru par les molécules d'un gaz et sur son rapport avec la théorie de la diffusion | language = fr | trans-title = On the average path taken by gas molecules and its relation with the theory of diffusion | url = https://archive.org/stream/bulletininternat1906pols#page/202/mode/2up | journal = [[Bulletin International de l'Académie des Sciences de Cracovie]] | page = 202 }}</ref> starts from the same premise as that of Einstein and derives the same probability distribution {{math|''ρ''(''x'', ''t'')}} for the displacement of a Brownian particle along the {{mvar|x}} in time {{mvar|t}}. He therefore gets the same expression for the mean squared displacement: {{nowrap|<math>\mathbb{E}{\left[(\Delta x)^2\right]}</math>.}} However, when he relates it to a particle of mass {{mvar|m}} moving at a velocity {{mvar|u}} which is the result of a frictional force governed by Stokes's law, he finds
<math display="block">\mathbb{E}{\left[(\Delta x)^2\right]} = 2Dt
= t \frac{32}{81} \frac{mu^2}{\pi \mu a}
= t\frac{64}{27} \frac{\frac{1}{2}mu^2}{3 \pi \mu a},</math>
where {{mvar|μ}} is the viscosity coefficient, and {{mvar|a}} is the radius of the particle. Associating the kinetic energy <math>mu^2/2</math> with the thermal energy {{math|''RT''/''N''}}, the expression for the mean squared displacement is {{math|64/27}} times that found by Einstein. The fraction 27/64 was commented on by [[Arnold Sommerfeld]] in his necrology on Smoluchowski: "The numerical coefficient of Einstein, which differs from Smoluchowski by 27/64 can only be put in doubt."<ref>See p. 535 in {{cite journal | last = Sommerfeld | first = A. | year = 1917 | title = Zum Andenken an Marian von Smoluchowski |language = de | trans-title = In Memory of Marian von Smoluchowski | journal = [[Physikalische Zeitschrift]] | volume = 18 | issue = 22 | pages = 533–539 }}</ref>

Smoluchowski<ref>{{cite journal | last = Smoluchowski | first = M. M. | year = 1906 | title = Essai d'une théorie cinétique du mouvement Brownien et des milieux troubles | url = https://archive.org/stream/bulletininternat1906pols#page/577/mode/2up | language = fr | trans-title = Test of a kinetic theory of Brownian motion and turbid media | journal = [[Bulletin International de l'Académie des Sciences de Cracovie]] | page = 577 }}</ref> attempts to answer the question of why a Brownian particle should be displaced by bombardments of smaller particles when the probabilities for striking it in the forward and rear directions are equal.
If the probability of {{mvar|m}} gains and {{math|''n'' − ''m''}} losses follows a [[binomial distribution]],
<math display="block">P_{m,n} = \binom{n}{m} 2^{-n},</math>
with equal {{em|a priori}} probabilities of 1/2, the mean total gain is
<math display="block">\mathbb{E}{\left[2m-n\right]} = \sum_{m=\frac{n}{2}}^n (2m-n)P_{m,n}=\frac{n n!}{2^{n+1} \left [ \left (\frac{n}{2} \right )! \right ]^2}.</math>

If {{mvar|n}} is large enough so that Stirling's approximation can be used in the form
<math display="block">n!\approx\left(\frac{n}{e}\right)^n\sqrt{2\pi n},</math>
then the expected total gain will be{{citation needed|date=July 2012}}
<math display="block">\mathbb{E}{\left[2m - n\right]} \approx \sqrt{\frac{n}{2\pi}},</math>
showing that it increases as the square root of the total population.

Suppose that a Brownian particle of mass {{mvar|M}} is surrounded by lighter particles of mass {{mvar|m}} which are traveling at a speed {{mvar|u}}. Then, reasons Smoluchowski, in any collision between a surrounding and Brownian particles, the velocity transmitted to the latter will be {{math|''mu''/''M''}}. This ratio is of the order of {{val|e=-7|u=cm/s}}. But we also have to take into consideration that in a gas there will be more than 10<sup>16</sup> collisions in a second, and even greater in a liquid where we expect that there will be 10<sup>20</sup> collision in one second. Some of these collisions will tend to accelerate the Brownian particle; others will tend to decelerate it. If there is a mean excess of one kind of collision or the other to be of the order of 10<sup>8</sup> to 10<sup>10</sup> collisions in one second, then velocity of the Brownian particle may be anywhere between {{val|10|-|1000|u=cm/s}}. Thus, even though there are equal probabilities for forward and backward collisions there will be a net tendency to keep the Brownian particle in motion, just as the ballot theorem predicts.

These orders of magnitude are not exact because they don't take into consideration the velocity of the Brownian particle, {{mvar|U}}, which depends on the collisions that tend to accelerate and decelerate it. The larger {{mvar|U}} is, the greater will be the collisions that will retard it so that the velocity of a Brownian particle can never increase without limit. Could such a process occur, it would be tantamount to a perpetual motion of the second type. And since equipartition of energy applies, the kinetic energy of the Brownian particle, {{nowrap|<math>MU^2/2</math>,}} will be equal, on the average, to the kinetic energy of the surrounding fluid particle, {{nowrap|<math>mu^2/2</math>.}}

In 1906 Smoluchowski published a one-dimensional model to describe a particle undergoing Brownian motion.<ref>{{cite journal | last=von Smoluchowski | first = M. | year = 1906 | title = Zur kinetischen Theorie der Brownschen Molekularbewegung und der Suspensionen | language = de | journal = [[Annalen der Physik]] | volume = 326 | issue = 14 | pages = 756–780 | bibcode = 1906AnP...326..756V | doi = 10.1002/andp.19063261405 | url = https://zenodo.org/record/1424073 }}</ref> The model assumes collisions with {{math|''M'' ≫ ''m''}} where {{mvar|M}} is the test particle's mass and {{mvar|m}} the mass of one of the individual particles composing the fluid. It is assumed that the particle collisions are confined to one dimension and that it is equally probable for the test particle to be hit from the left as from the right. It is also assumed that every collision always imparts the same magnitude of {{nowrap|Δ''V''}}. If {{math|''N''<sub>R</sub>}} is the number of collisions from the right and {{math|''N''<sub>L</sub>}} the number of collisions from the left then after {{mvar|N}} collisions the particle's velocity will have changed by {{math|Δ''V''(2''N''<sub>R</sub> − ''N'')}}. The [[multiplicity (mathematics)|multiplicity]] is then simply given by:
<math display="block"> \binom{N}{N_\text{R}} = \frac{N!}{N_\text{R}!(N - N_\text{R})!}</math>
and the total number of possible states is given by {{math|2<sup>''N''</sup>}}. Therefore, the probability of the particle being hit from the right {{math|''N''<sub>R</sub>}} times is:
<math display="block">P_N(N_\text{R}) = \frac{N!}{2^N N_\text{R}!(N-N_\text{R})!}</math>

As a result of its simplicity, Smoluchowski's 1D model can only qualitatively describe Brownian motion. For a realistic particle undergoing Brownian motion in a fluid, many of the assumptions don't apply. For example, the assumption that on average occurs an equal number of collisions from the right as from the left falls apart once the particle is in motion. Also, there would be a distribution of different possible {{math|Δ''V''}}s instead of always just one in a realistic situation.