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

[Nugatory]

(Google AI says it's true)

There is a reason why Google AI and other LLMs are not acceptable sources here. Stop quoting them, stop wasting your time trying to understand physics from what they say.

 

[jbriggs444]

Your statement "Yes, any large sample from any distribution which is symmetric about zero is overwhelmingly likely to have both positive and negative elements." contradicts with your following statements:

"The probability that all elements share the same sign is one in $2^{N−1}$"

I see no contradiction. I see consistency.

The probability that $10^{23}$ independent samples drawn from a normal distribution with mean zero will all share the same sign is $\frac{1}{2^{10^{23}-1}}$. This is effectively impossible. And agrees with what I said.

The probability that $10^{23}$ independent samples drawn from a normal distribution with mean zero will not all share the same sign is $1 - \frac{1}{2^{10^{23}-1}}$. This is virtually certain. And agrees with what I said.

It is possible that we have a language issue here. So I am giving you the benefit of the doubt in this case.

 

[Herman Trivilino]

It is possible that we have a language issue here. So I am giving you the benefit of the doubt in this case.

Yes. Along with a background that doesn't include completion of the university-level introductory physics course.

Over 100 posts in this thread attempting to explain the relationship between probability and sample size.

It's going in circles. The OP just keeps finding other sources to help him understand, but all they do is cause tangential confusions.

 

[Mike_bb]

Yes. Along with a background that doesn't include completion of the university-level introductory physics course.

Over 100 posts in this thread attempting to explain the relationship between probability and sample size.

It's going in circles. The OP just keeps finding other sources to help him understand, but all they do is cause tangential confusions.

I have university-level introductory physics course but after long break I don't remember some things.

 

[jbriggs444]

I have university-level introductory physics course but after long break I don't remember some things.

“It ain’t what you don’t know that gets you into trouble. It’s what you know for sure that just ain’t so. “ – Mark Twain

 

[Herman Trivilino]

I have university-level introductory physics course but after long break I don't remember some things.

Well then, I don't know what you meant by what you wrote in Post #39.

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

You either completed the course or you didn't.

 

[Mike_bb]

Well then, I don't know what you meant by what you wrote in Post #39.



You either completed the course or you didn't.

Yes. In Russia in most universities incomplete course is introductory physics course. (base course)

 

[Herman Trivilino]

I have university-level introductory physics course but after long break I don't remember some things.

That's why I keep telling you to start with the textbook for that course.

 

[Herman Trivilino]

Yes. In Russia in most universities incomplete course is introductory physics course. (base course)

No it's not. Nowhere is an introductory course an incomplete course. Either you complete the introductory course or you don't.

 

[Mike_bb]

No it's not. Nowhere is an introductory course an incomplete course. Either you complete the introductory course or you don't.

Complete course = base course + advanced course

Complete is equivalent for "full course" in russian

 

[Charles Link]

I didn't read every post, but Reif's book Statistical and Thermal Physics would answer just about anything you would want to know on how the Maxwell-Boltzmann distribution works, and how the states are counted.

 

[Herman Trivilino]

Complete course = base course + advanced course

Complete is equivalent for "full course" in russian

Interesting. How much time would this take?

 

[Mike_bb]

Interesting. How much time would this take?

Incomplete (base course) - 4 semesters.
Complete - 6-8 semesters

 

[Herman Trivilino]

Incomplete (base course) - 4 semesters.

In the US that would be called the introductory sequence of courses. It varies but I had 3 semesters of introductory physics (introduction to newtonian physics) plus one semester of modern physics. One could complete or not complete any one of the courses, with or without completing the entire sequence. That is what confused me when you said you had not completed the course.

Complete - 6-8 semesters

8 semesters would be equivalent to a bachelors degree here. After the 4-semester sequence one would complete the degree by taking 4 more semesters that would include courses in thermodynamics and statistical physics, mechanics, electromagnetism, and quantum mechanics.

This is also referred to as the undergraduate course of study. In there one might encounter that book you're reading, or it might be later as part of an advanced degree such as a masters degree or a Ph.D.

Along with these physics courses one would also be taking math courses in calculus, differential equations, linear algebra, statistics, etc.

So again I ask you, what is you're trying to get out of this book? Because from the very beginning of this thread you have been asking questions about the distributions of velocity components of gas molecules. We answer, and then you ask questions that address those same issues. Repeatedly. We don't seem to be able to get through to you.

I would urge you to first read and review the material in those introductory courses on vectors and the ideal gas law. It would make for a much more productive discussion.

 

[Mike_bb]

In the US that would be called the introductory sequence of courses. It varies but I had 3 semesters of introductory physics (introduction to newtonian physics) plus one semester of modern physics. One could complete or not complete any one of the courses, with or without completing the entire sequence. That is what confused me when you said you had not completed the course.

I had 4 semesters. Mechanics and molecular physics, electromagnetism and waves, optics and atomic physics, quantum mechanics.

8 semesters would be equivalent to a bachelors degree here. After the 4-semester sequence one would complete the degree by taking 4 more semesters that would include courses in thermodynamics and statistical physics, mechanics, electromagnetism, and quantum mechanics.

Yes, 8 semesters in Russian universities would be equivalent to a bachelors as in USA.

Along with these physics courses one would also be taking math courses in calculus, differential equations, linear algebra, statistics, etc.

I had 4 semesters. Linear algebra, analytic geometry, vector calculus/vector algebra, differential calculus, integral calculus and series, multivariable calculus, differential equations, probability theory and statistics, theory of functions of a complex variable.

So again I ask you, what is you're trying to get out of this book? Because from the very beginning of this thread you have been asking questions about the distributions of velocity components of gas molecules. We answer, and then you ask questions that address those same issues. Repeatedly. We don't seem to be able to get through to you.

I would urge you to first read and review the material in those introductory courses on vectors and the ideal gas law. It would make for a much more productive discussion.

I expected that you ask me about it. Physics was very difficult to me in university and as I mentioned above, after long break I don't remember many things. Not long ago I took my old school book to repeat and it was written there that $<V_x^2> = <V_y^2> = <V_z^2>$ because X-axis, Y-axis and Z-axis are equiprobable.
But I wanted to know in more details about this fact. That's all.

P.S. Now I fully (as it seems to me) understand that it's written in that book. I would like to thank you all for help, patience and feedback!

 

[jbriggs444]

Not long ago I took my old school book to repeat and it was written there that $<V_x^2> = <V_y^2> = <V_z^2>$ because X-axis, Y-axis and Z-axis are equiprobable.

It may be a language issue, but "equiprobable" is not the correct term to use.

If $V_x$ is a random variable then we could speak of the probability that $V_x$ falls within a particular range. We could speak of the probability that $V_y$ falls within the same range. Or $V_z$.

If the coordinate velocities are independent and identically distributed then those three events (that the random variables fall into a particular range) would be equiprobable. The converse does not hold.

The premise that the book should be using is that $V_x$, $V_y$ and $V_z$ are "independent and identically distributed". Not that they are "equiprobable". From that starting point, it is trivially true that any statistical measure (such as mean square) of one distribution will be identical to the same statistical measure of any other identical distribution.

This is all before we get into a discussion about the distinction between the expected value for a distribution and the mean of a particular sample.

 

[Herman Trivilino]

The premise that the book should be using is that Vx, Vy and Vz are "independent and identically distributed". Not that they are "equiprobable".

But the book is written in Russian, so it seems it hasn't been translated correctly?

Likely the translation algorithm is for everyday terminology, not technical jargon. Who knows?