What does equiprobable mean in the context of thermal motion?

[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?

 

[Mike_bb]

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

Does $<V_x>=<V_y>=<V_z>=0$ hold in your case in post 29?

 

[Dale]

Does $<V_x>=<V_y>=<V_z>=0$ hold in your case in post 29?

Yes. The velocity in each direction is given by a standard normal distribution, so it has mean = 0 and standard deviation = 1.

 

[weirdoguy]

I depicted 12 particles and their velocities.

Since when is 12 "extremely large"? You are not listening to what people are saying. All the book is saying is true:
- on average
- when we have around $10^{23}$ molecules

12 molecules is not a case for statistical physics.

 

[Mike_bb]

Yes. The velocity in each direction is given by a standard normal distribution, so it has mean = 0 and standard deviation = 1.

Thanks! I understand how it works.

But why does my book say another thing?

 

[Dale]

But why does my book say another thing?

From what I can tell it is not a very high quality book on this specific topic. I think it might be suitable as a reference for someone who already understands the material, but it doesn’t seem like a good source to learn it.

 

[Mike_bb]

Dale, could you explain what is idea of this proof? I can't understand it. Thanks.

 

[Herman Trivilino]

In the book

Which book? Are you still reading that advanced 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).

Your explanation isn't correct. What they're saying is that the square of the magnitude of $\vec{c}$ equals three times the square of the x-component of $\vec{c}$.

The Pythagorean theorem tells us that $c^2=c_x^2+c_y^2+c_z^2$. And since $c_x^2=c_y^2=c_z^2$, we have $c^2=3c_x^2$.

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?

Yes, your conclusion is based on an incorrect assumption.

As I've tried to tell you before, if I had barely completed the university-level introductory course, I would not be able to understand a book written for either upper-level undergraduate or graduate-level physics majors. You need the understanding of vector calculus, differential equations, thermodynamics, and statistics that can only be gained by taking university-level courses in those topics.

Again, as I told you before, get a college-level introductory physics textbook and read the section where the ideal gas law is derived. A review of vectors in the earlier chapters would also be helpful.

 

[Herman Trivilino]

could you explain what is idea of this proof?

It's the first sentence in the passage you quoted:

"The main assumption here is that the direction of the velocity is distributed uniformly."

 

[weirdoguy]

You need the understanding of vector calculus, differential equations, thermodynamics, and statistics that can only be gained by taking university-level courses in those topics.

I think OP has even more basic problem - understanding what this:

"The main assumption here is that the direction of the velocity is distributed uniformly."

means. And thinking in terms of mean behavior of large amount of particles. I think so especially after the 12-particle drawing...

 

[jbriggs444]

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

Is this explanation verbatim from the book? Or is it your own interpretation? Either way, It is utterly incorrect.

If you total ${v_x}^2$ across all 12 particles you will get a result.
If you total ${v_y}^2$ across all 12 particles you will get another result.
If you total ${v_z}^2$ across all 12 particles you will get a third result.

Obviously, the sum of the three sub-totals is exactly equal to the total squared velocity across the 12 particles.

For a sample this small, the three sub-totals will be slightly different. Before looking at the velocities, there is no reason to expect any particular axis to have a greater sub-total than any other. On average (over a large number of 12 particle samples) we expect the total squared velocity to be spread evenly across the three sub-totals. Accordingly, we expect that each sub-total will be 1/3 of the overall total. On average.

For larger sample sizes, the percentage variation between the three sub-totals will become smaller and smaller. [The expected percentage variation will scale approximately with the inverse square root of the sample size.] For a sample size of $10^{23}$, we would expect variation on the rough order of $10^{-10}$ percent.

There is no particular axis along which 1/3 of the particle trajectories. will align.