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.