I What does equiprobable mean in the context of thermal motion?

[Mike_bb]

Hello!

I read that the probability of a molecule moving with a certain velocity component in the x-direction is equal to the probability of it moving with the same velocity component in the y or z direction.

I can't fully understand what does this mean and why are the square of the components of velocity equal?

Could anyone explain it?
Thanks.

 

[.Scott]

Hello!

I read that the probability of a molecule moving with a certain velocity component in the x-direction is equal to the probability of it moving with the same velocity component in the y or z direction.

I can't fully understand what does this mean and why are the square of the components of velocity equal?

Could anyone explain it?
Thanks.

The first part is easy. If you have a volume of gas at a particular temperature and pressure, you might ask how many of those molecules have approximately a specific vertical velocity - say 100Kmps +/- 1Kmps. Whatever that number is, it would be correct even if the question had been about the x, y, z, or any other axis (for example -0.8x, 0y, 0.6z).

I'm having a problem with "the square of the components of velocity". The sum of the squares of the components might be equal to something - but that word "sum" is important. Could you give us a direct quote on the statement you don't understand?

 

[sbrothy]

Sorry for this stupid follow-up question... Is this what is meant by the "Ergodic Hypothesis"?

"[...] i.e., that all accessible microstates are equiprobable over a long period of time? [...]" ---- Wikipedia

 

[Mike_bb]

The first part is easy. If you have a volume of gas at a particular temperature and pressure, you might ask how many of those molecules have approximately a specific vertical velocity - say 100Kmps +/- 1Kmps. Whatever that number is, it would be correct even if the question had been about the x, y, z, or any other axis (for example -0.8x, 0y, 0.6z).

I'm having a problem with "the square of the components of velocity". The sum of the squares of the components might be equal to something - but that word "sum" is important. Could you give us a direct quote on the statement you don't understand?

If we consider example with coin then we'll obtain that probability of one of two side is 1/2. But experiment show that we can obtain one of two side of coin more than another.

 

[renormalize]

But experiment show that we can obtain one of two side of coin more than another.

Not in the limit as the number of flips approaches infinity, assuming it's a fair (perfectly symmetrical) coin.

 

[Dale]

what does this mean and why are the square of the components of velocity equal?

This is a statement of the isotropy of physics. There is no specific direction in the universe where things naturally align. The molecules in the pot of water boiling on the stove are not particularly pulled in the direction of NGC 4041.

If we consider example with coin then we'll obtain that probability of one of two side is 1/2. But experiment show that we can obtain one of two side of coin more than another.

The statement that a coin is fair doesn't mean that you will obtain the same number of heads as tails for any finite sample.

 

[Herman Trivilino]

If we consider example with coin then we'll obtain that probability of one of two side is 1/2. But experiment show that we can obtain one of two side of coin more than another.

The difference in sample sizes is highly significant. You have Avagadro's Number of molecules, say, in a liter of gas. I can't imagine flipping a coin that many times. If you did one flip per second it would take about 10¹⁶ years. The universe is only about 10¹⁰ years old, so that's about a million times longer than the age of the universe.

 

[Andy Resnick]

Sorry for this stupid follow-up question... Is this what is meant by the "Ergodic Hypothesis"?

"[...] i.e., that all accessible microstates are equiprobable over a long period of time? [...]" ---- Wikipedia

I don't think so. I explain the Ergodic hypothesis as the claim that 1 system repeatedly subjected to the same experiment a million times will give the same results as a million identical systems each experimented on 1 time.

 

[Mike_bb]

Could anyone explain which component Vy (Y-axis) will have velocity vector with Vx component (X-axis)? Is it true that if velocity vector has Vx component in X-axis direction and Vy component in Y-axis direction then there is velocity vector with Vy component in X-axis direction and Vx component in Y-axis direction?

 

[Herman Trivilino]

Could anyone explain which component Vy (Y-axis) will have velocity vector with Vx component (X-axis)?

If I understand your question, the answer would be any vector not parallel to the y-axis will have a nonzero x-component.

Is it true that if velocity vector has Vx component in X-axis direction and Vy component in Y-axis direction then there is velocity vector with Vy component in X-axis direction and Vx component in Y-axis direction?

Not sure what you're asking here. A y-component never has a direction parallel to the x-axis, and a x-component never has a direction parallel to the y-axis.

Perhaps you could explain why you're asking these questions. They are worded strangely.

 

[Mike_bb]

If I understand your question, the answer would be any vector not parallel to the y-axis will have a nonzero x-component.

Not sure what you're asking here. A y-component never has a direction parallel to the x-axis, and a x-component never has a direction parallel to the y-axis.

Perhaps you could explain why you're asking these questions. They are worded strangely.

Sorry, but I try to ask correctly. If we have velocity vector V1(V1x, V1y) then do we have velocity vector V2(V1y,V1x) (if there is no specific direction)?

 

[Herman Trivilino]

Sorry, but I try to ask correctly. If we have velocity vector V1(V1x, V1y) then do we have velocity vector V2(V1y,V1x) (if there is no specific direction)?

Are you talking about the velocities of two different molecules? And those are two molecules in a macroscopic sample of gas? What do you mean by no specific direction?

I'm thinking there's a language barrier in this discussion.

 

[A.T.]

If we have velocity vector V1(V1x, V1y) then do we have velocity vector V2(V1y,V1x) (if there is no specific direction)?

By swapping the components you are not canceling the momentum, you have to negate the components for that, if that was your idea here. If not, then explain what you are trying to achieve here. Your question makes little sense.

And it's not like in an ensemble of particles there are particle pairs with opposite velocities. You can cancel momentum with an odd number of particles too.

 

[Dale]

If we have velocity vector V1(V1x, V1y) then do we have velocity vector V2(V1y,V1x) (if there is no specific direction)?

If I understand your question then the answer is no. Instead, what you can say is that you have equal probability for V1 and V2.

Again, a fair coin does not mean that you will have as many heads as tails. It means they have the same probability.

 

[Mike_bb]

If I understand your question then the answer is no. Instead, what you can say is that you have equal probability for V1 and V2.

Again, a fair coin does not mean that you will have as many heads as tails. It means they have the same probability.

I read in one source that molecules of gas are moving in two directions (X-axis and Y-axis):

Are these images correctly? And how to understand what does this mean?

 

[Herman Trivilino]

I read in one source

What source? We need to know.

that molecules of gas are moving in two directions (X-axis and Y-axis):

View attachment 364522View attachment 364523

Are these images correctly? And how to understand what does this mean?

Without context I can only guess, and chances are that would just take you down the wrong rode.

Tell us the source so we have some context.

 

[Mike_bb]

What source? We need to know.

It's russian book.

 

[A.T.]

It's russian book.

Make a picture of the page and put it into Google Lens or similar to translate it. We have the year 2025.

 

[Herman Trivilino]

It's russian book.

We need more than that. A proper reference gives us the book's title, author, publisher, etc. And the page number.

 

[Mike_bb]

We need more than that. A proper reference gives us the book's title, author, publisher, etc. And the page number.

https://scask.ru/e_book_jphis.php?id=31

 

[Dale]

Are these images correctly?

I don’t think those images are good. They give the impression that every molecule is going outward from a single point. They also give the impression that for every molecule going one way there is another with equal and opposite velocity. They also give the impression that all of the motion is in a 2D plane. None of those are correct impressions.

 

[Mike_bb]

I don’t think those images are good. They give the impression that every molecule is going outward from a single point. They also give the impression that for every molecule going one way there is another with equal and opposite velocity. They also give the impression that all of the motion is in a 2D plane. None of those are correct impressions.

Now I understand you. In this book it's written that one part of molecules (N/2) move in positive direction and another part of molecules(N/2) move in negative direction.

It seems logically to me because average of components of velocity vectors (axis X) is 0.

 

[Herman Trivilino]

Are these images correctly? And how to understand what does this mean?

What I can glean from the translation is that you imagine drawing an arrow representing the velocity vector for each molecule, and then translating each vector so that the tails of the arrows are all located at point A. The figures show a few of those arrows.

Is your native language Russian or English? What level of physics education have you had?

 

[Mike_bb]

Is your native language Russian or English? What level of physics education have you had?

Russian. School level.

 

[Mike_bb]

What I can glean from the translation is that you imagine drawing an arrow representing the velocity vector for each molecule, and then translating each vector so that the tails of the arrows are all located at point A. The figures show a few of those arrows.

I understand it. But how is it possible that for Y-axis we have the same figure with arrows?

 

[Dale]

In this book it's written that one part of molecules (N/2) move in positive direction and another part of molecules(N/2) move in negative direction.

On average that is true. So the numbers 0.188, 0.268, and -0.456 are randomly distributed from a standard normal distribution, but none of them are equal and opposite to each other. It is not necessary for every value to be matched with its opposite for the average to be zero.

 

[Mike_bb]

On average that is true. So the numbers 0.188, 0.268, and -0.456 are randomly distributed from a standard normal distribution, but none of them are equal and opposite to each other. It is not necessary for every value to be matched with its opposite for the average to be zero.

Ok. But in this book it's written that every value of X-component of velocity vectors match with its opposite. Is it wrong?

 

[Mike_bb]

On average that is true. So the numbers 0.188, 0.268, and -0.456 are randomly distributed from a standard normal distribution, but none of them are equal and opposite to each other. It is not necessary for every value to be matched with its opposite for the average to be zero.

This book says that each of vectors has opposite vector because both direction (negative and positive) are equiprobable. It's true, yes?

 

[Dale]

I read in one source that molecules of gas are moving in two directions (X-axis and Y-axis):

View attachment 364522View attachment 364523

Are these images correctly? And how to understand what does this mean?

Here is a more realistic image:

The main thing wrong with this image is that it is 2D and this kind of motion happens in 3D, and also the arrowheads are a fixed size so on the shorter arrows you cannot see the tail. But this accurately represents a collection of 200 gas molecules that are randomly distributed (uniform distribution) in a box with randomly distributed (standard normal distribution on each axis) velocities.

 

[Herman Trivilino]

I understand it. But how is it possible that for Y-axis we have the same figure with arrows?

Because the x-direction is chosen randomly. It could have been the y-direction that was chosen instead, or indeed any direction.

 

[Herman Trivilino]

This book says that each of vectors has opposite vector because both direction (negative and positive) are equiprobable. It's true, yes?

Yes, with the qualifications stated below.

Ok. But in this book it's written that every value of X-component of velocity vectors match with its opposite. Is it wrong?

Not if you have a tremendously large number of molecules, which you do.

So the chances of these statements being true is so incredibly large that they can be considered true.

 

[Herman Trivilino]

Russian. School level.

The book you're reading is written for physics majors. Probably at the graduate level, so it's meant for students who've had previous courses in thermodynamics, statistics, and differential equations.

What is it you're trying to learn from this book? Or to put it another way, why are you reading it?

 

[Mike_bb]

What is it you're trying to learn from this book? Or to put it another way, why are you reading it?

I found this book because I need in more detail explanation. (In my old school book there was brief description only: "x,y,z are equiprobable directions and square of velocities components are the same".

 

[Mike_bb]

Dale, Herman Trivilino,
I'm so grateful for your help!! I understand now how it works!

PS. I found on SE alternative explanation but I can't understand it. Could you explain in more detail? Big thanks!

 

[weirdoguy]

but I can't understand it

What exactly you don't understand? Which sentence? I think this description is really straightforward...

 

[Mike_bb]

What exactly you don't understand? Which sentence? I think this description is really straightforward...

"... it would mean that on average..." this сonfused me. What does it mean? Average of components?

 

[Herman Trivilino]

"... it would mean that on average..." this сonfused me. What does it mean? Average of components?

It means that you take the average. That's what the bracket notation <...> means. So the average value of $v_x^2$ is $<v_x^2>$.

And yes, it's average of components. Because $v_x$ is the x-component of $\vec{v}$.

 

[Herman Trivilino]

Dale, Herman Trivilino,
I'm so grateful for your help!! I understand now how it works.

You're welcome.

PS. I found on SE alternative explanation but I can't understand it.

I don't know what SE means.

The part of the book you were reading appears to be a derivation of the Boltzmann distribution. I don't know if that's what you were interested in, or if you have the background for a book that advanced. I know you said "school level" but I don't know what that's equivalent to in the American education system

Also, I had trouble with the translate function in my Chrome browser. It would translate the table of contents, but not the pages of the book. Except one time when I clicked on the chapter title in the table of contents, and I was able to read that one chapter section, but I couldn't duplicate it, so I was unable to reread it, and unable to read any other chapters or sections.

 

[Mike_bb]

I don't know what SE means.

StackExchange.

I know you said "school level" but I don't know what that's equivalent to in the American education system

I said so because I have knowledge gaps. (In reality, I have incomplete course of Physics in university)
Physics is difficult for me but interesting.

 

[Herman Trivilino]

(In reality, I have incomplete course of Physics in university

Ahhh... At that point in my education I wouldn't have been able to understand that book, either.

I suggest you start with a college-level introductory physics textbook and review the chapters on vectors, and then the chapters on thermodynamics. The section where the ideal gas law is derived would address all the questions you've asked in this thread.

From there you could move up to a textbook on introductory thermodynamics or statistical physics if you want.

 

[Gavran]

I said so because I have knowledge gaps. (In reality, I have incomplete course of Physics in university)
Physics is difficult for me but interesting.

There is an interesting text which can help you for better understanding.
Maxwell–Boltzmann distribution

 

[Mike_bb]

Hi all!

In the book there was following expression:

$N\frac{mc_x^2}{2} = \frac{N}{3} \frac{mc^2}{2}$
$c^2 = 3с_x^2$


Explanation: 1/3 of all particles moves along one axis (in both directions).

I depicted 12 particles and their velocities. But I don't understand which 4 molecules move along one axis.

Could anyone explain it?
Thanks.

 

[Dale]

What are the variables?

I depicted 12 particles and their velocities.

Again, this is not a good depiction, in my opinion. See above.

 

[Mike_bb]

What are the variables?

Again, this is not a good depiction, in my opinion. See above.

$c^2=<3V_x^2>$

 

[A.T.]

Hi all!

In the book there was following expression:

$N\frac{mc_x^2}{2} = \frac{N}{3} \frac{mc^2}{2}$
$c^2 = 3с_x^2$


Explanation: 1/3 of all particles moves along one axis (in both directions).

I depicted 12 particles and their velocities. But I don't understand which 4 molecules move along one axis.

View attachment 364673

Could anyone explain it?

Looks like you have depicted something else than the book describes.

 

[Mike_bb]

Looks like you have depicted something else than the book describes.

Each velocity vector has opposite velocity vector because gas is isotropic.

 

[Dale]

Each velocity vector has opposite velocity vector because gas is isotropic.

That isn’t correct, as we already explained above.

 

[Mike_bb]

That isn’t correct, as we already explained above.

If that isn't correct then there is specific direction but it's wrong.

 

[Dale]

$c^2=<3V_x^2>$

This is not a very good explanation of what the variables mean. It is helpful if you write the expression that you want help with, as well as a list of what each variable in that expression means and as much background about the expression as possible. I can guess, but if you want accurate help then you should eliminate guessing.

 

[Dale]

If that isn't correct then there is specific direction but it's wrong.

Really? What is the specific direction in the figure in post 29?