A Exact Boltzmann distribution

  • Thread starter rabbed
  • Start date
232
2
I would like to see a derivation of the exact Maxwell-Boltzmann distribution shown as (16) in this document: https://www.researchgate.net/publication/222670999_Exact_Maxwell-Boltzmann_Bose-Einstein_and_Fermi-Dirac_Statistics

This is my starting point (f being the function to maximize, g and h being the constraints, a and b being the lagrange multipliers and nk being the number of particles in level/energy k):

f(n0, ..., n6) = ln(6!/(n0!*n1!*n2!*n3!*n4!*n5!*n6!))
g(n0, ..., n6) = 0*n0 + 1*n1 + 2*n2 + 3*n3 + 4*n4 + 5*n5 + 6*n6 = 6
h(n0, ..., n6) = n0 + n1 + n2 + n3 + n4 + n5 + n6 = 6
gradient(f) = a*gradient(g) + b*gradient(h)

I know ln(gamma(x))' = digamma(x)
 
232
2
I believe the most probable macrostate (for 6 particles, constant energy 6) is achieved with occupancy 3,1,1,1,0,0,0 in energy levels 0,1,2,3,4,5,6.
Is the solution to take the outputs from the following calculation (where i've picked suitable values a=-0.06 and b=1.96 to fit the solution, but maybe there are correct values?) and round them to their nearest integers?
 

Want to reply to this thread?

"Exact Boltzmann distribution" You must log in or register to reply here.

Related Threads for: Exact Boltzmann distribution

  • Posted
2
Replies
26
Views
5K
  • Posted
Replies
2
Views
2K
  • Posted
Replies
2
Views
2K
  • Posted
Replies
11
Views
2K
Replies
1
Views
492

Physics Forums Values

We Value Quality
• Topics based on mainstream science
• Proper English grammar and spelling
We Value Civility
• Positive and compassionate attitudes
• Patience while debating
We Value Productivity
• Disciplined to remain on-topic
• Recognition of own weaknesses
• Solo and co-op problem solving

Hot Threads

Top