[Goala]

Hi guys,

I have a system of particles in different energy levels. I know how many particles are in each energy level.

 I was wondering how I could find out if the distribution of these particles was in a Boltzmann distribution or not?

Thanks!

In other words, how do I determine if a distribution is a Boltzmann distribution or not?

[edgeh72]

The Boltzmann distribution for a fractional amount of particles in a set of states i possessing energy E equals [gi*e^(-Ei/kBT)]/Z(T)
where kB is the Boltzmann constant, gi is the degeneracy, and Z(T) is the partition function.

If your data fits this than you have a Boltzmann distribution.

[qw098]

Ok, interesting. All I am given as data is: That the system contains 38 particles with three equally spaced energy levels (0 J, a J, and 2a J) and I know that the population distribution is: A(18,12,8)

I am not entirely sure how to use the equation you provided me with.

[Enthalpy]

Your sample is too small to answer that.

Take the brutal fit of (18,12,8) by the exponential that passes by 18 and 8, then its expected value at the middle would be 12, the geometrical mean since the levels are equally spaced. So a Boltzmann fits.

But a (non-natural) linear distribution passing by 18 and 8 would give 13 as the expected middle value, which is very close to the observed 12. A rough estimate of the standard deviation would be sqrt(13), and the observed value would be at 0.3*sigma, excellent as well.

Even a flat distribution of 13 particles per level with SD~3.6 would be only 1.4*sigma away at both extreme levels, still good.

More complicated cases would be seriously difficult to answer, requiring harder math tests like chi-squared to give a mathematical opinion - which, in statistics, would be only one element for your human decision.

[qw098]

Take the brutal fit of (18,12,8) by the exponential that passes by 18 and 8, then its expected value at the middle would be 12, the geometrical mean since the levels are equally spaced. So a Boltzmann fits.

Thank you Enthalpy for the reply. I don't quite understand how you made this "fit" and determined that a Boltzmann fit the data. If you could explain how you did it, it would be much appreciated!

[Hello12]

Think there is an easier way than the ones people are talking about. The definition of a boltzmann distribution is it has the property that the ratio of the population of adjacent levels are CONSTANT. Therefore if you fiind the ratios of the adjacent energy levels and there are the same then it is a Boltzmann distribution. So if given N0,N1.N2. Then N1/N0 should = N2/N1

[qw098]

Very interesting Daniel!

I've never heard of this property of the Boltzmann distribution. Could you show me a reference please or where you found that?

Thanks a ton!

[qw098]

Ok, you are COMPLETELY correct Daniel! I have managed to prove it/ convince myself that because the energy separations are equal the ratios must be equal! Thank you Dan once again!!!!

[juanrga]

Hi guys,

I have a system of particles in different energy levels. I know how many particles are in each energy level.

 I was wondering how I could find out if the distribution of these particles was in a Boltzmann distribution or not?

Thanks!

In other words, how do I determine if a distribution is a Boltzmann distribution or not?

Let me emphasize that the Boltzmann distribution is an equilibrium distribution, whereas that a system with only 38 particles cannot be in equilibrium. I.e. not only the ratio n_i/n_j may verify the distribution but also be constant in time, and for 38 particles this is not the case due to fluctuations.

[Hello12]

No problem . It was in my lecture notes and we had a tutorial on it.