The Boltzmann Distribution

1. The Boltzmann Distribution Impact of Temperature on the State of the System

2. Motivation Individual molecules in a system will not all have the same energy. Generally there will be some distribution around an average. Statistical mechanics will allow us to calculate the most probable (i.e. equilibrium) distribution.

3. Outline Definitions. Total Energy and Population Numbers. Most probable distribution given constraint of constant total energy. Lagrange method of undetermined multipliers.

4. Definition Consider a set of N identical molecules. Each molecule has a set of available energy levels {?i}. If N1 of the molecules have energy, ?1 , N2 have energy, ?2 , etc., the total energy, E is given by E=? Ni?i

5. Definitions The set of numbers, {Ni} are referred to a population numbers and the total number of molecules, N is given by, N=? Ni Defining the population numbers for a state is equivalent to defining the �distribution�.

6. To determine the most probable distribution (the equilibrium distribution) we must be able to calculate the number of configurations that lead to each of the possible distributions.

7. Justification of formula for W

8. Thus,

9. How do we find the max in a multi-variable function? Consider f(x,y). At the (unconstrained) maximum of this function,

10. How do we find the max in f(x,y) subject to a constraint on possible values of x and y? The constraint might be expressed as a function, e.g. g(x,y) = 0. Consider the similarity: We want a max in W(N1,N2,�) subject to the constraint g(N1,N2,�) = E - ?Ni?i = 0

11. We can still use�.

13. therefore

14. Likewise,

16. So our new criteria are:

17. With two constraints, g=0 and h=0,

18. Returning to the Boltzmann Distribution We wish to maximize f({Ni}) = ln W subject to the constraints g ({Ni}) = NT-? Ni=0 and h ({Ni}) = ET-? Ni?i=0

19. This gives the criterion

20. Thus, This is a little different from Noggle�s treatment and results in a different value for A. Problems arise because N in the definition of W is not a constant but instead N= sum_Ni. and the dN/dNi =1 rather than 0. Whereas N in the constraint is a constant so dN/d(anything) = 0. Poor choice of symbols. This is a little different from Noggle�s treatment and results in a different value for A. Problems arise because N in the definition of W is not a constant but instead N= sum_Ni. and the dN/dNi =1 rather than 0. Whereas N in the constraint is a constant so dN/d(anything) = 0. Poor choice of symbols.

21. To eliminate A we use,

22. Undefended for now, I will assert ?=1/kT

23. conclusion The most probable distribution of molecules among available energy levels can be calculated by maximizing W or ln W, subject to the constraints of constant energy and constant number of molecules. This most probable distribution is referred to as the Boltzmann Distribution. Assignment:5.23-5.27