Maxwell's distribution of velocities of molecules

[Radhakrishnam]

The expression relating the mean number of molecules with velocities in the range v and v + dv and position r and r + dr is given by

where n = N/V is the number density of molecules.
My question is: Since LHS is an integer, how do we ascertain the RHS is an integer, since it involves pi and an exponential factor that are irrational numbers? How do we ascertain the product of the irrational numbers always leads to an integer?

 

[Vanadium 50]

The mean number of molecules does not have to be an integer. I have two apples. You have three apples. What is the mean number of apples we have?

 

[Radhakrishnam]

I am sorry,the mean number of molecules must be a rational number. the word 'integer' in my earlier post must be replaced by rational number. Could you please tell how to reconcile the two sides of the equation now?

 

[Philip Wood]

What an interesting question! It had never occurred to me up to now that a mean of integers must be rational. I think the reason for the discrepancy is that the equation is an approximation. One approach to deriving the equation involves the counting of evenly spaced points contained in a spherical shell, and the counting is only approximate. I recommend you look up the derivation of the equation. You'll see for yourself.

 

[Vanadium 50]

I am sorry,the mean number of molecules must be a rational number.

No, it doesn't. I have 1000 molecules in a box of length x. What is the mean number inside radius x?

 

[Radhakrishnam]

The number of molecules in the sphere of radius x must be an integer at any instant of time and the average over a number of instants must be a rational number. The number of molecules being an integer in the chosen region (sphere in the present case) at any instant, does not depend on the shape or geometry of the region chosen. Such being the case, the irrational numbers that we get by multiplying the number density by volume creates problems in the course of further developments of the subject, as for example, with regard to chemical potentials.

When we take care to see that the integral over the whole volume always gives the same value for the number of molecules ( the normalization, as it is sometimes called), should not some such condition be imposed to see that we get an integer number of molecules in any chosen region of space at any instant of time?

 

[Vanadium 50]

You didn't answer my question. I have 1000 molecules in a box of length x. What is the mean number inside radius x? You claim it is a rational number. I'm asking you to tell me what that rational number is.

 

[Arsenic&Lace]

The number of molecules in the sphere of radius x must be an integer at any instant of time and the average over a number of instants must be a rational number. The number of molecules being an integer in the chosen region (sphere in the present case) at any instant, does not depend on the shape or geometry of the region chosen. Such being the case, the irrational numbers that we get by multiplying the number density by volume creates problems in the course of further developments of the subject, as for example, with regard to chemical potentials.

Sure, the average over a finite number of instants is rational, but what happens in the limit of an infinite number of instants?

Moreover, what statistical quantity would an integral over a subvolume actually specify, and does this quantity have a meaningful interpretation even if it is not an integer?

 

[Radhakrishnam]

what happens in the limit of an infinite number of instants?

The mean settles to a constant value, in the limit of an infinite number of instants.

I don't understand the next question.

 

[Radhakrishnam]

I'm asking you to tell me what that rational number is.

The mean value of the number of molecules in the sphere is 1000.

The box gets defined only if it is a cube. So, we assume the box to be a cube. Being of smaller volume, it lies within the sphere. Therefore, the sphere contains 1000 molecules at any instant of time. Thus the mean value is 1000.

The spirit of my question is not so much on the numerical values of the numbers but on the nature (or properties) of the numbers. The LHS and RHS of Maxwell's equation must correspond to a number with certain properties, such as an integer, rational number, irrational number etc. LHS is a rational number, but is RHS a rational number? If it is rational, how do we ascertain (prove) that it is a rational number?

 

[Vanadium 50]

The problem is that you misunderstand something very important mathematically - the mean value of a function that can only take on integer values doesn't have to be an integer (as you first claimed) or a rational number (as you later claimed) - it can be any real number. That's where your misconception is, and that's what we have to address if you want to progress.

Now, the sphere is inside the cube. The sphere has volume pi/6 and the cube has volume 1. You have 1000 molecules in the cube. What is the mean number inside the sphere?

 

[Radhakrishnam]

You are trying to stress about the RHS of the equation which is a complicated mathematical function and I am not that proficient about understanding it in its elements. Therefore, I am not claiming anything about the values it can take - whether integer or rational number or any real number etc.

What I am asking is this: I know the LHS of the equation must be a rational number because it represents a mean of integers. I would like to know how to satisfy myself about the nature of the values of the function on the RHS, which for me is a complicated function. If you could guide me as to how to ascertain that RHS gives rational numbers always, lest an in consistency arise between RHS and LHS values of the equation, it would be helpful.

You are giving the volume of the sphere as pi/6, and that of the cube 1 with length of side x.

I am getting the volume of the sphere to be (4/3)pi x³ and the volume of the cube to be x³ and consequently the cube is inside the sphere, in contrast to your result that the sphere is inside the cube! Hence as I stated earlier, I am getting the result: 1000 molecules as the answer to your question.

 

[Arsenic&Lace]

Well an infinite series of rational numbers can in fact converge to an irrational number. For instance, if you take a Taylor series expansion of arctan(x) centered on 0, you will have an infinite series of rational numbers on one side, and pi on the other.

More obviously: pi= 3 + 1/10 + 4/100 +1/1000...

The second question regarded what would happen if I were to take an integral of the number density (not the MB distribution) over V'<V. This is a question about what the term number density actually means.

 

[Radhakrishnam]

Thank you, it is such results of mathematics (involving, for example infinity) that cause difficulty in understanding matters in a simple way. While we learn, on the one hand, that an irrational number is a non recurring non terminating decimal and that pi is an irrational number, and on the other hand the series on RHS is a series of rational numbers that extends to infinity, causes the uneasy feeling that the sum of rational numbers is giving an irrational number.

The above is similar to the difficulty I have with Maxwell's distribution, but with a difference. While the expansion of pi is unique, the RHS of Maxwell's equation which is a complicated function, always leads to a rational number is strange - one may say that it is a mathematical result, but if there is a proof it would be so comfortable.

The second question regarded what would happen if I were to take an integral of the number density (not the MB distribution) over V'<V.
I understand what this statement means. But reading it along with the next sentence does not make the impression that you are trying to convey, whatever that might be - perhaps because, the mind is already got fixated on what number density is.

Under equilibrium conditions, the number density has a unique value for a well defined system. the integral over a sub volume, I believe, leads to that value.
n_i/v_i = n_j/v_j = Σn_i/ Σv_i = N/V = constant. n_i is the number of molecules in sub volume v_i and n_j is the number of molecules in sub volume v_j, the two sub volumes being chosen at random. N the number of molecules in the total volume V.