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Phys4230 Tutorial 7 The Boltmann Factor
NAME_________________________
Consider the isolated system shown. The accessible microstates are labeled i = 1, 2, 3, 4,…
What is the probability that the system will be found in a particular microstate i:
P(i) =
U, V, N
= (U,V,N)
 P(i) =
all i
Now consider a NOT-isolated system s, in thermal contact with
a very large reservoir R at temperature T.
The “Universe” = Reservoir + system = (R + s) is isolated.
Total energy Utot = UR + Ei = fixed.
Are all accessible states of system s equally likely?
Claim: The probability that the system s is found in microstate i is
Reservoir R
temperature T
energy UR
R =R(UR)
P(i)  C e Ei /kT
system s
state i
energy Ei
, where C is a constant.
Here is the proof: All the states of the Universe = R+s are equally likely.
The probability that the Universe is found with the system s in state i is:
P(i)  # states of (R+s) with s in state i
 (multiplicity of R when s in state i)  (multiplicity of s when s in in state i)
Invent symbols for these factors: P(i) 
Now, entropy of reservoir SR  k ln R . [Don’t confuse entropy S with system s!] Solve for  R :
The entropy of the reservoir is a function of its energy: SR(UR) = SR(Utot – Ei). Write this as a Taylor series
expansion. Note that Ei << Utot.
Phys4230 Dubson, Tutorial 6
Recall that temperature is defined as
1 SR
. Write simplified expressions for SR(Utot – Ei) and P(i).

T U R
P(i)  C e Ei /kT . Use your knowledge of probabilities to solve for C.
The partition function is defined as
Z   e Ei /kT
. Write P(i) in terms of Z.
i
Sketch a graph of exp(E/kT) vs. E/kT.
eE/kT
E/kT
1
Is P(i) = Prob[s is in state i] the same as P(Ei) = Prob[s is in a state with energy Ei] ?
Define g(E) = degeneracy of states with energy E . What is the relationship between P(i) and P(Ei)?
Write an expression for P(E), the probability that the system S is found in a state with energy E.
Phys4230 Dubson, Tutorial 6
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