Why is air more dense closer to the Earth's surface? - particle's view

[Martian2020]

TL;DR Summary

From particles moving in gravitational field view, explain emerging phenomena of air more dense closer to gravitational attraction center.

Could not find reason on molecular level. E.g. here some explanation
"There are two reasons: at higher altitudes, there is less air pushing down from above, and gravity is weaker farther from Earth's center."

However, when I start to imagine particles, when there are only a few I get opposite result: as particle moves faster closer to Earth (like a ball bouncing off the floor), it spends less time there, hence average number of particles is less closer to ground. I doubt horizontal speeds make a difference.

Looks to me we have emerging phenomena when number of particles increase over some boundary, the result reverses.
What is that boundary and mechanism that causes such reverse?

 

[Dale]

Summary:: From particles moving in gravitational field view, explain emerging phenomena of air more dense closer to gravitational attraction center.

I doubt horizontal speeds make a difference.

The horizontal speeds do make a difference in the sense that on average only 1/3 of the momentum will be directed vertically.

How much time does a low-energy molecule spend near the surface compared to a high-energy molecule?

 

[Martian2020]

How much time does a low-energy molecule spend near the surface compared to a high-energy molecule?

More. You hint that energy levels of particles are generally different.
Properties_to_Kinetic_Theory_of_Gase
"When examining the gas molecules individually, we see that not all of the molecules of a particular gas at a given temperature move at exactly the same speed. This means that each molecule of a gas have slightly different kinetic energy."
Only slightly? or distribution of energies of particles of ideal gas is from near 0 to max and e.g. like normal...?

 

[Nugatory]

However, when I start to imagine particles, when there are only a few I get opposite result: as particle moves faster closer to Earth (like a ball bouncing off the floor), it spends less time there, hence average number of particles is less closer to ground.

We aren't dropping molecules from the top of the atmosphere to the ground, so that analogy doesn't apply. Instead, each individual molecule generally remains at more or less the same average height, just randomly bouncing back and forth in a narrow range around that average as it is knocked about by its neighbors.

So here is a very hand-waving sort of explanation - and I cannot stress enough this is handwaving, an "OK I see how it might be" sort of thing that is no substitute for a proper derivation of the ideal gas laws using statistical mechanics. (Statistical mechanics is the application of Newton's laws and basic probability and statistics to get quantitative precision behind the phrases "more or less", "generally", "over time", "more often", "average", that I'm sloppily tossing around below):
The molecule is subject to a downwards gravitational force, yet it remains at more or less the same height. That can only happen if there is a compensating upwards force, meaning that the molecule is, over time, hit from below and knocked upwards more often than it is hit from above and knocked downwards, which implies that the density below is greater than the density above. If the density below wasn't enough greater to keep the molecules above more or less at the same height, then those molecules would drift downwards under the influence of gravity, thereby increasing the density until everything balances again.

So at every height, the density below is slightly greater than the density above. Integrate this small gradient from the bottom of the atmosphere to the top and you get the observed result.

 

[russ_watters]

Why is air more dense closer to the Earth's surface?

Could not find reason on molecular level.

I don't see that a molecular-level analysis tells us anything at all about atmospheric pressure. Given that the individual molecules are more or less bouncing-around in place, there's no way to tell whether a small group of molecules is outside in the atmosphere or in an airtight box. It doesn't "know" anything about what's going on beyond it or why.

 

[sophiecentaur]

Summary:: From particles moving in gravitational field view, explain emerging phenomena of air more dense closer to gravitational attraction center.

gravity is weaker farther from Earth's center.

Using the well known inverse square law, you can see that the gravitational force at 100km ( above any significant atmosphere) is hardly any different from that at sea level. The ratio is

(Earth Radius / (Earth Radius + 100))²
which equals (6371/6471)²
or 0.97 which is virtually the same. And that goes for satellites in low Earth orbit, too.

as particle moves faster closer to Earth

Reason? The mean speed of particles mainly depends on the Temperature (not density). Temperature drops by about 6.5C per km of altitude so you could say that is why they are faster lower down.

The causes of the temperature distribution with altitude are complicated, to do with EM energy being absorbed as it passes through the atmosphere from the Sun and from the warm Earth. Then there's convection and wind that stirs it all up and condensation of water vapour etc. etc... A simple model of a vertical tube going up from the surface into space is not enough.

The molecule is subject to a downwards gravitational force, yet it remains at more or less the same height.

Simple kinetic theory of gases implies that molecules hit each other and exchange momentum and, in a box you can think in terms of a single particle bouncing from wall to wall (that's the overall effect). In a tall column of air, although individual molecules don't actually get far up or down, the effect is similar to a set of individual trajectories going from sea level up to whatever height their initial velocity would take them (like an artillery shell). Their initial kinetic energy is transferred to gravitational energy due to their altitude. So that will lower the temperature. But that's an oversimplification as the temperature lapse rate doesn't carry on above about 11km.

PS There are many people who get pretty cross when you try to say that the pressure down here is due to the weight of all the air above. Afaics, it's not a bad starting description, though.

 

[hilbert2]

The Boltzmann distribution predicts that the likelihood of a particle to have kinetic energy ##K## and potential energy ##V## is proportional to ##\exp (-(K+V)/k_B T)##. So, when the potential energy ##V## increases when going upwards (almost linearly when not too far from Earth's surface), the minimum value of ##K+V## also increases and the probability of finding that kind of particle decreases exponentially like the observed air pressure does at increasing altitude.

This is not anywhere near a rigorous justification, as the Earth's atmosphere is not a system in thermal equilibrium, but the correct idea can be seen from it.

 

[russ_watters]

PS There are many people who get pretty cross when you try to say that the pressure down here is due to the weight of all the air above. Afaics, it's not a bad starting description, though.

They shouldn't; it's true. I don't see it as merely a starting assumption at all. What is it missing/can't it explain?

 

[Dale]

PS There are many people who get pretty cross when you try to say that the pressure down here is due to the weight of all the air above. Afaics, it's not a bad starting description, though.

They shouldn't; it's true. I don't see it as merely a starting assumption at all. What is it missing/can't it explain?

The correct phrasing is that the weight of the column of hydrostatic fluid above some spot gives the change in pressure. When you are dealing with a column that goes up to space the pressure is equal to the change in pressure. But in other cases the distinction is important and can lead to confusion as in this thread:

https://www.physicsforums.com/threads/pressure-on-a-sample-of-fluid-at-rest.979560/#post-6253067

 

[russ_watters]

The correct phrasing is that the weight of the column of hydrostatic fluid above some spot gives the change in pressure. When you are dealing with a column that goes up to space the pressure is equal to the change in pressure.

K. Different situations are different. Agreed. I thought you were going in a different direction with that.
[edit; oops, different person answered.]

 

[Dale]

or distribution of energies of particles of ideal gas is from near 0 to max and e.g. like normal...?

It is given by the Boltzmann distribution. This is a very important statistical distribution for kinetic theory of gasses. It is what @hilbert2 was talking about in his excellent answer.

 

[sophiecentaur]

What is it missing/can't it explain?

I think that they want to include all the other factors in an actual weather system. (But experts can get very excited about details) Everything is dynamic and the velocity of air above and the temperatures in the vicinity will affect the answer. The 'weight' of the upper parts will take a time to affect the pressure on the ground; I don't think it's just the speed of sound that causes the delay.

 

[russ_watters]

I think that they want to include all the other factors in an actual weather system. (But experts can get very excited about details) Everything is dynamic and the velocity of air above and the temperatures in the vicinity will affect the answer. The 'weight' of the upper parts will take a time to affect the pressure on the ground; I don't think it's just the speed of sound that causes the delay.

IMO it is a mistake to focus on little intricacies until the primary issue has been addressed.

 

[Martian2020]

Reason? The mean speed of particles mainly depends on the Temperature (not density). Temperature drops by about 6.5C per km of altitude so you could say that is why they are faster lower down.

Because they gain speed in a gravitational field toward the ground

The Boltzmann distribution predicts that the likelihood of a particle to have kinetic energy ##K## and potential energy ##V## is proportional to ##\exp (-(K+V)/k_B T)##. So, when the potential energy ##V## increases when going upwards (almost linearly when not too far from Earth's surface), the minimum value of ##K+V## also increases and the probability of finding that kind of particle decreases exponentially like the observed air pressure does at increasing altitude.

that could be result I want to understand from analyzing individual particles.
P.S. interestingly I see you formulas as formulas in a quote above, but in original post I see ordinary symbols bracketed with ##...

We aren't dropping molecules from the top of the atmosphere to the ground, so that analogy doesn't apply. Instead, each individual molecule generally remains at more or less the same average height, just randomly bouncing back and forth in a narrow range around that average as it is knocked about by its neighbors.

In a tall column of air, although individual molecules don't actually get far up or down, the effect is similar to a set of individual trajectories going from sea level up to whatever height their initial velocity would take them

The two above seems at first glance to contradict one another. My reasoning was that due to acceleration individual particles spend less time near the ground than up above. If "the effect is similar to a set of individual trajectories going from sea level up to whatever height their initial velocity would take them", what is incorrect about my initial reasoning?

 

[snorkack]

Earth atmosphere is dense and molecules do collide between ground and outer space.
But imagine an atmosphere which is less dense like that of Moon.
IF the molecules were not subject to Maxwell distribution, but all had equal vertical speed, then indeed they would all fall back from equal height, density would decrease up, reach maximum at the top height and fall to nothing above it.
But they are subject to Maxwell distribution. As you rise up, the slowest molecules fall back, the faster ones slow down. The result is a new Maxwell distribution with the same temperature but lower density.

 

[mpresic3]

The link in your post refers to a article for Earth Science for Middle Schools. These students (really kids) are typically 12-14 years old. Intricacies as to how gravity grows weaker with altitude and Boltzmann's distribution is likely going to be far removed from the average middle school science student. The explanation listed is unsatisfactory at the college level. Your professor/instructor might fail you if you expressed this answer in a physics course, especially if the Boltzmann distribution was introduced in class.

Now the equation derived in many freshman physics textbooks is density = exp ( -m g h / k T ). The simplifying assumptions can almost be read off from the equation. m is the mass of the molecule, and k is boltzmann's constant. We can assume these are constant as m is for the "molecule" of oxygen/nitrogen/trace gases mixture (known as "air" ). the letter h is the altitude, the T is the temperature, and g is the local gravity value at the Earth surface, ( typically 9.8 m / s squared ). The idea that the freshman physics student will use this equation suggests that they would evaluate the values at a temperature T of interest, and a g value, that does not vary with height.

The application of this equation suggests it can be applied to an isothermal atmosphere (T = constant), and the gravitational acceleration does not vary with height (g = constant). Consequently, (if this equation is correct, and to large measure it is applicable) density in this equation diminishes in altitude, even if the temperature is constant and the gravity does not vary with altitude.

The answer for middle schoolers is inadequate for at least two reasons. First, the fact that there is "less air above the molecule" is saying the pressure is lower at higher altitudes. If you are trying to explain why density is lower at higher altitudes with the pressure is lower at higher altitudes, doesn't this beg the question why is the pressure lower at higher altitude?
Second, the equation above (at the college level 7-8 years beyond middle school), demonstrates even assuming g is constant, and does not vary with height, the density decreases with increasing height or altitude.

It might seem I think middle school students should be given perfectly accurate information. I never taught middle school, but I think it is OK to simplify concepts to introduce some main points. (I do think the explanation offered in the link is inadequate and misleading, more than I would like. That could be the subject of a future thread)

 

[willem2]

First, the fact that there is "less air above the molecule" is saying the pressure is lower at higher altitudes. If you are trying to explain why density is lower at higher altitudes with the pressure is lower at higher altitudes, doesn't this beg the question why is the pressure lower at higher altitude?

No it doesn't. If you are higher up, there will be less air above you, this will be true for any pressure distribution.

 

[sophiecentaur]

Intricacies as to how gravity grows weaker with altitude and Boltzmann's distribution is likely going to be far removed from the average middle school science student.

A good post, basically and you demonstrate that it's actually a very complicated topic with (literally) lots of balls in the air at the same time.
But is it really an "intricacy" for them to be disabused of the idea that there's no gravity when you're up in orbit? The small change in gravity (1/r²) is highly relevant to the Potential Energy of molecules high in the atmosphere. But, more to the point, they get told, all the time, a false notion about the 'no gravity in space' thing. On one hand, gravity is the same and on the other hand (at the same scale of 100km -ish) gravity goes to nothing. Any bright student would spot that inconsistency and point it out or (worse) go away without asking about it.

Forget the details of Boltzmann. You don't need Maths to talk about a range of speeds and the (loose) definition of temperature a gas. I favour the approach which starts with the (measured) temperature lapse rate. Not fundamental but it is consistent with the more advanced approaches which (some) students may graduate to.

 

[Martian2020]

Earth atmosphere is dense and molecules do collide between ground and outer space.
But imagine an atmosphere which is less dense like that of Moon.
IF the molecules were not subject to Maxwell distribution, but all had equal vertical speed, then indeed they would all fall back from equal height, density would decrease up, reach maximum at the top height and fall to nothing above it.
But they are subject to Maxwell distribution. As you rise up, the slowest molecules fall back, the faster ones slow down. The result is a new Maxwell distribution with the same temperature but lower density.

This seems like useful explanation. Could you please clarify:
1) at what density (start from 0 and increase) distribution becomes so "Maxwell like" that density starts to increase downward?
2) explain in more detail mechanism of the process highlighted in bold?

 

[snorkack]

This seems like useful explanation. Could you please clarify:
1) at what density (start from 0 and increase) distribution becomes so "Maxwell like" that density starts to increase downward?

1) The distribution is "Maxwell like" all the time, even at negligible density. The reason being that gas molecules also acquire Maxwell distribution when they collide with ground rather than each other. So the density will decrease upwards no matter whether it is a tenuous atmosphere like that of Moon where molecules only collide with ground, or a thick one like Earth where molecules collide with each other.

2) explain in more detail mechanism of the process highlighted in bold?

Maxwell distribution is an exponential distribution by energy.
So, if on sea level, 1/2 of air molecules are fast enough to rise to 5 km, 1/4 fast enough to rise to 10 km, 1/8 fast enough to rise to 15 km... then if you look from a mountaintop 5 km above sea level, those 1/2 of the molecules too slow on sea level never get to that altitude, but the 1/4 that were fast enough to rise to 10 km have slowed down... to rise further 5 km above the summit. So of the 1/2 air molecules that reach the summit, there is again 1/2 fast enough to rise to 5 km, 1/4 fast enough to rise to 10 km... Maxwell distribution of the same temperature, just lower density.

 

[sophiecentaur]

explain in more detail mechanism of the process highlighted in bold?

There won't be any satisfactory explanation until the Energy Flow is considered. The atmosphere is not just a vertical tube, full of gas. Radiation comes in from the Sun at a steady rate (same rate for millions of years) and that energy is absorbed on the way down and by the ground. The resulting ground temperature will produce IR which will heat the atmosphere on the way out. At different levels there will be different amounts of incoming and outgoing heat supplied and that will drive the ground level pressure as much as anything else.

Whatever the composition of the surface and the atmosphere, there will be an equivalent equilibrium temperature of around 300K, looking from outside. There is a steady 1kW per metre squared, whatever happens to the atmosphere. Any simplified explanation (for education purposes) needs to have accompanying caveats and relevant hints about the Energy situation.

 

[Dale]

Forget the details of Boltzmann. You don't need Maths to talk about a range of speeds and the (loose) definition of temperature a gas. I favour the approach which starts with the (measured) temperature lapse rate.

I don't know. If you do look at Boltzmann or Maxwell then you get a real molecular explanation which gives the right density for the right reason even in a uniform gravitational field with a constant temperature.

If you want a simpler explanation then I would not do a "molecular" explanation at all.

I am not sure that the temperature lapse rate works since the temperature is colder higher up which would imply more density. Am I missing something about that explanation?

 

[sophiecentaur]

even in a uniform gravitational field with a constant temperature.

Uniform gravity is fine but the temperature is definitely not uniform. Like I said before, we're not dealing with a simple vertical tube. The Energy input to the system (highly relevant ) is not uniform because it depends on where the various wavelengths are absorbed and emitted.

I am not sure that the temperature lapse rate works since the temperature is colder higher up

But that only applies up to about 11km of altitude. Thereafter the temperature increases again. (Although the weight of gas at that height and above won't be great).
It's hard stuff.

 

[Dale]

But that only applies up to about 11km of altitude. Thereafter the temperature increases again. (Although the weight of gas at that height and above won't be great).
It's hard stuff

I think that is definitely an intricacy. It is an explanation that can explain why the details change at 11 km, but by itself it gives the wrong "first approximation" overall.

When I am looking for a non-intricacy explanation I want an explanation that gives the right first-approximation behavior with the most simplified scenario. For me the most simplified scenario is a stationary column of air under constant gravity and constant temperature. Yes, this doesn't represent the real atmosphere, but it is a physically reasonable idealized scenario. The explanation for this scenario is the "first approximation" to more realistic scenarios.

 

[sophiecentaur]

Yes, this doesn't represent the real atmosphere, but it is a physically reasonable idealized scenario.

Yes it's very idealised but it hasn't anything to say how the situation comes about. That's my problem. It's a model of a type that proves 1=1. The Boltzmann distribution relates to the temperature and vice versa but there's no actual cause stated. I think that any claimed understanding with it is more of an illusion than anything.

Bearing in mind that a planet without a nearby star would have no atmosphere at all then I think you have to consider the Sun from the very start. Everything would be solid or liquid without it and you have to ask how the arriving solar energy causes the atmosphere to be the way it is.

If you took a tall tube in a water bath which kept the walls at the same temperature as our atmosphere all the way up then I guess you might have the 'correct' density profile with altitude but our atmosphere doesn't get its energy that way. It gets energy from a warm ground and from some direct incoming EM. It's all about heat transfer and not a static situation.

This makes the problem an awful lot harder. But you can't expect that vertical tube to be a good model for a spherical planet which is heated from one direction, which spins and which has a significant amount of water at around its triple point. Close but no cigar, imo.

 

[Dale]

Yes it's very idealised but it hasn't anything to say how the situation comes about. That's my problem.

Ok, but explaining how the situation comes about is going even beyond an intricacy of the current question to the intricacies of a different question. That certainly isn’t my approach to teaching, but I guess it would be satisfying to some students. I just think that it would bewilder the rest.

Of course, I think that the whole idea of a molecular explanation for something like this is bad teaching until they have already studied some thermodynamics. I would honestly probably answer the question “you will cover that in thermodynamics”.

 

[sophiecentaur]

Of course, I think that the whole idea of a molecular explanation for something like this is bad teaching

Agreed. There's a popular misconception that discussing Physics in terms of particles (Photons, Electrons etc etc) is somehow a higher level. Our world is largely experienced on a macroscopic level so why not describe it that way when you can?

 

[sophiecentaur]

Summary:: From particles moving in gravitational field view, explain emerging phenomena of air more dense closer to gravitational attraction center.

Could not find reason on molecular level.

I re-read this first post. I'm not surprised that you can't find it because it's just not worth trying. It's needlessly complex - as the thread demonstrates. The intermediate stage of description involves the macroscopic behaviour of gases (gas laws) and provides a pretty good model (to whatever level you want). These gas laws can be derived from the molecular level but you don't need to bother with the extra complexity of most real-life situations.
This is how the rest of Science works. You study it at various levels, that are valid in themselves and then use the results from one level to work on another level. It's the same with Engineering , which uses the results from Materials Science and that works very well.

 

[mpresic3]

Summary:: From particles moving in gravitational field view, explain emerging phenomena of air more dense closer to gravitational attraction center.

However, when I start to imagine particles, when there are only a few I get opposite result: as particle moves faster closer to Earth (like a ball bouncing off the floor), it spends less time there, hence average number of particles is less closer to ground.

Has anyone addressed this part of the thread by martian2020. It seems to be true for a falling object that the object spends less time in the vicinity of the floor than higher up. I noted Dale has mentioned the horizontal velocities also are present, but I think you can explain the density decrease with height without invoking the horizontal velocities. I can add my thoughts but I would like to sample some of the other thoughts first.

 

[mpresic3]

No it doesn't. If you are higher up, there will be less air above you, this will be true for any pressure distribution.

True. I did not think about this aspect correctly.

Martian2020 had me think about his thought experiment regarding dropped bouncing balls. I figured I would review my favorite paper, Stochastic problems in Physics and Astronomy by S Chandrasekhar. Chandrasekhar approaches this problem from the particle viewpoint. He shows a solution to the problem as a first order stochastic differential equation where the energy kT drives the equation for the velocity. In his solution leading to the exponential distribution, he uses Stokes law for a falling sphere, and defines a "dynamical friction". What I believe he is getting at is the air molecules cannot be thought as accelerating in a gravitational field.

The many collisions with other molecules quickly impose dominating forces, so the air molecules quickly approach what might be considered a terminal (i.e constant) velocity. Don Lemons in a small textbook mentions in the preface this review article by Chandrasekhar is beyond first time students in the subject and has written, An introduction to Stochastic Problems in Physics. I agree the Chandrasekhar paper is difficult, but it is well-worth examining.

Of course, this is many years beyond middle school. Although I was introduced to Chanrasekhar's paper in my late 20's, I did not realize it's significance until I was in my late 40's.

 

[sophiecentaur]

What I believe he is getting at is the air molecules cannot be thought as accelerating in a gravitational field.

That's not to be confused with their weight, of course. But the 'acceleration' (actual increase in velocity) still applies, in the same sense that Newton's Cradle operates. Each inner ball in the cradle doesn't move far but momentum is transferred from end to end from the fast arriving ball to the fast departing ball. Each ball (and, by extension, each air molecule) shares momentum with another molecule, either below or above. In the brief time between collisions, there is acceleration. So you can think in terms of one 'virtual' molecule making the whole journey - and accelerating downwards. (Electron Holes in semiconductors are a sort of parallel.)
That simple example has to be extended for 'glancing' collisions between molecules, which share the KE over all three degrees of freedom.

 

[snorkack]

In atmosphere where gravity can be regarded as constant, pressure at ground must equal the weight of air column.
But this says nothing about density. In a lake of fresh water 10 m deep, the water also exerts 1 bar pressure on bottom, yet the density just below surface is practically the same as at bottom, while density just above surface is practically nothing.

 

[mpresic3]

I reread the Chandrasekhar article and now should write some clarification to my earlier ideas.
Chandrasekhar derives the Barometric equation by assuming a stochastic differential equation of the form:

dv/dt = - bv + A(t). where b is a constant and A consists of two parts, A1, and A2(t)

A1 is a constant part (g), and A2(t) is a part involving many fluctuations caused by collisions.

So my earlier statement was incorrect. The acceleration is present, (otherwise g would not show up in the final equation at all). The equation by Chandrasekhar does still show a damping term, so that the characteristics of the solution are closer to a first order differential equation with a damped exponential in v(t), rather than a linearly decreasing or increasing v(t).

The beauty in the Chandrasekhar paper is that he demonstrates not only the solution in the form of the barometric equation, but exactly how the solution is reached as a function of time from (arbitrary) initial
conditions. He uses delta functions as initial conditions.

Chandrasekhar uses several methods to do this in his review article. Chandra notes the some solutions are not his, but he cites Smoluchowski, Langevin, Einstein, Guoy, and others. Smoluchowski (1908) shows graphs on how the barometric equation is approached as time goes by starting with delta function initial conditions. Today, this would be done by computer. in 1908, this probably required some effort.

Through libraries and some effort, I have the original papers by Smoluchowski, and some translations from German. Unfortunately, not all the papers are translated to English. This led me to online language translators and I have had some success in interpreting these papers too.

 

[sophiecentaur]

pressure at ground must equal the weight of air column.

. . . even if the planet is so cold that the air becomes liquid!

 

[vanhees71]

I reread the Chandrasekhar article and now should write some clarification to my earlier ideas.
Chandrasekhar derives the Barometric equation by assuming a stochastic differential equation of the form:

dv/dt = - bv + A(t). where b is a constant and A consists of two parts, A1, and A2(t)

A1 is a constant part (g), and A2(t) is a part involving many fluctuations caused by collisions.

So my earlier statement was incorrect. The acceleration is present, (otherwise g would not show up in the final equation at all). The equation by Chandrasekhar does still show a damping term, so that the characteristics of the solution are closer to a first order differential equation with a damped exponential in v(t), rather than a linearly decreasing or increasing v(t).

The beauty in the Chandrasekhar paper is that he demonstrates not only the solution in the form of the barometric equation, but exactly how the solution is reached as a function of time from (arbitrary) initial
conditions. He uses delta functions as initial conditions.

Chandrasekhar uses several methods to do this in his review article. Chandra notes the some solutions are not his, but he cites Smoluchowski, Langevin, Einstein, Guoy, and others. Smoluchowski (1908) shows graphs on how the barometric equation is approached as time goes by starting with delta function initial conditions. Today, this would be done by computer. in 1908, this probably required some effort.

Through libraries and some effort, I have the original papers by Smoluchowski, and some translations from German. Unfortunately, not all the papers are translated to English. This led me to online language translators and I have had some success in interpreting these papers too.

Here's the link to Chandrasekhar's great article:

S. Chandrasekhar, Stochastic problems in physics and
astronomy, Rev. Mod. Phys. 15, 1 (1943),
https://dx.doi.org/10.1103/RevModPhys.15.1