Research: Science and Education
Boltzmann without Lagrange Carl W. David Department of Chemistry, University of Connecticut, Storrs, CT 06269-3060;
[email protected] In a wonderful article, F. T. Wall showed that one can avoid Lagrange multipliers in deriving the Boltzmann distribution (1). This article rephrases Wall’s arguments slightly, offering what appears to be a simpler approach to part of his derivation. The Derivation We start with the thermodynamic probability, Ω, of a state characterized by “occupation” numbers, {ni }, for the number of particles with energy εi in a given (not necessarily optimal) distribution
Ω ≡ N ! ∏ i
∑ ni
(1)
N! ∏ ni ! 1
In Figure 1 we have the non-degenerate ( g i = 1∀i) distribution that would have the thermodynamic probability of
i =1
i =2
Ω ≡ 5 !
2 1!
1
i =1
1 1!
12 2!
i =3
2
i =2
∑ ni εi
E =
=
∑ ni* εi
(2)
i
including the optimal distribution that maximizes the thermodynamic probability. The entropy of the system is
S = kB ln Ωmax
(3)
where, for Boltzmann statistics, eq 1 holds (this is different for Fermi–Dirac and Bose–Einstein particles). Then the Helmholtz energy, A, would be
which translates to B, the Helmholtz energy mirror function, by substituting eqs 1, 2, and 3 into this defining equation
11 1!
∑ ni εi i
3 2!
i =3
i =4
We assume that there is one (and only one) distribution {ni } that maximizes Ω. This special distribution is denoted as {ni* }, where each energy level’s occupancy number is optimal to make Ω the biggest possible value. Said another way,
Ω ({ni }) → Ωmax ({ni* }) This will mean that any change in occupancy numbers (from those of the thermodynamically realized state) must result in lowering Ω, which is the thrust of the argument that follows. Remember, one has to change at least two occupancy num-
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i
i
1
2 1!
− kBT ln N ! ∏
g i ni ni
(4)
i =4
In the lower half of Figure 1 we have a degenerate case, which would lead to 1
i
whether we are “at equilibrium” or not (N is the total number of particles or systems in the larger system being considered). The total energy of a system (for any distribution) such as this is given by
B = 11 1!
∑ ni *
A = E − TS
i
11 Ω ≡ 5 ! 1!
= N =
i
i
g i ni ni !
where g i is the degeneracy of the ith state and N is the sum over ni. For non-degenerate energy levels, the more common form of this probability expression is Ω =
bers, one going up, the other going down, in order to conserve the number of particles in the system, since
non-degenerate ni εi
4
8.1
1
3
7.4
2
2
4.2
1
1
3.2
1
i
degenerate εi ni
gi
4
8.1
1
2
3
7.4
2
3
2
4.2
1
1
1
3.2
1
2
Figure 1. Distributions of particles among energy levels.
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Research: Science and Education
which allows us to write
When {ni } → {ni* } then B → A, that is, eq 4 becomes A =
gi ni
− kBT ln N ! ∏
∑ ni* εi
Consider the arbitrarily chosen energy levels 7 and 14. Then i =6
∏ e
∑ ni* εi = −kBT ln
= −kBT ∑ ln i
i =1
fi i ni !
i =13
ni
×
= −kBT ∑ −
ni* εi kBT
×
i =8
∏
i = ∞
ni
∏
B = −kBT ln N !∏ e
−ni εi kB T
− kBT ln N ! ∏ i
i
g i ni ni !
A = −kBT ln N ! ∏ i
i
g i ni ni* !
f 1414
×
n14 !
f7 7 n7 !
= ln
kB T
n
f 14 14 n14 !
n7 + 1
f7 ( n7 + 1) !
f7
n14 −1
( n14 − 1) !
which can be rewritten in the form We note that (n7 + 1)! = (n7 + 1)n7! and
ni*
− εi
n !
( n14 − 1) ! = n14 14
gi e kBT A = − kB T ln N ! ∏ i
ni* !
so, we have
Since the Helmholtz free energy is only defined at thermodynamic equilibrium, we need a different function, which we have called B, that will approach A as {ni } → {ni* }. Alternatively, we argue that lim
{ni } → {ni *}
B = A
B before − Bafter kB T
n
n +1
f7 7 ( n7 + 1) !
To minimize B by choices of {ni } (i.e., different distributions) is our actual problem. Following Wall, we simplify the notation, defining f i ⬅ gi e
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1 = − ∆n = kB T − kB T f7 7 n7 !
= ln
− εi kB T
(7)
fi ni !
Our argument is, if n7 increases by one, and n14 compensates by decreasing by one, can we show that the Helmholtz free energy mirror function, B, goes up (i.e., away from the minimum) or, if we were already at a minimum, is zero?
B before − Bafter
*
− kBT ln N ! ∏
n7 ! n
n
−n i * εi ι e kB T
f7
×
fi ni !
i =15
(where the kBT cancels) so substituting this into the righthand side of eq 4) (and ultimately into eq 5 for A) gives us
n7
n
B = −kB T ln N ! ∏
i
i
−ni εi e kB T
−ni εi kB T
ni*
ni *!
i
Following Wall, we note that the first (energy) term can be rewritten (using eq 2) as
E =
fi
A = −kB T ln N !∏
(5)
ni*
i
i
*
f7
n14 −1
( n14 − 1) !
f 14 (n7 + 1) f 7 n14
≈ ln
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f14 14 n14 !
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If B, the Helmholtz free energy mirror function, were a function of n, and n was a continuous rather than a discrete variable, then (dB兾dn) → 0. Temporarily thinking of the derivative as a ∆, that is, (∆B兾∆n) → 0 and remembering that ∆n is numerically one, we could require that at the optimal values of {ni* }, (∆B兾1) = 0. Further, for large occupation numbers,
− ⇒
B before − Bafter dB ∆B ≈ − = → 0 dn 1 1
B before − Bafter
f 14 n14
≈ ln
− kB T
− ln
f7 n7
→ 0
(8)
but this must be zero, that is, (∆B兾∆n) → 0 at the minimum of B({ni}). If the natural logarithms have to be equal, then so do their arguments, and we have2
f 14 n14*
=
f7
pliers completely, although Nash (9) complained that such attempts were “clumsy”. Nash did the “standard derivation” with calculus methods, which are more appropriate for “students who command no more than the bare rudiments of the calculus”. Physicists have not been immune to the seductive call of a derivation without Lagrange (10). Yet one more attempt should be noted, by Friedman and Grubbs (11) based on Pascal’s triangle. In the derivation presented here, the attempt has been made to make a distinction between a function B({ni }) dependent on a set of occupation numbers, and A({ni* }), the Helmholtz free energy when the set of occupation numbers consists of those values that minimizes B. Also stressed is the arbitrary choice of two (separated) states to repopulate while investigating whether or not B is a minimum (i.e., becoming A); Equation 7 explicitly highlights this idea. Notes 1. The symbol ∀ stands for “for all”. 2. Alternatively,
n 7*
or, in general, defining a new constant “a” f1 n1*
=
f2 n 2*
=
f3 n 3*
1 = … ≡ a
ln
f 14 n14
− ln
f7 n7
= ln
f 14 n7 n14 f 7
so
Then using eq 6, we have
ni* = a fi = agi
− εi e kB T
f14 n7 = 1 or n14 f 7
which is the result sought!
= 0
f 14 n14
n7 f7
f 14 n14
= 1 or
f7 n7
= 1
Discussion
which forces n14 and n7 to become n14* and n7*, respectively.
It is important to reiterate that this derivation is not original with the author, and also to note that attempts other than Wall’s have succeeded to a greater or lesser degree in freeing the derivation from Lagrange multipliers (and Stirling’s approximation, incidentally). A similar derivation to Wall’s (and ours) was presented by Russell (2), whose final argument (paralleling eq 8 here) may be more comfortable for some readers. An argument based on integrating the equation dS = dU兾T led McDowell (3) to an alternative approach to the derivation. Gibbs (4) used a quasi chemical argument to effect the derivation. Hakala (5) attempted to obviate the use of Stirling’s approximation in the derivation while still using Lagrange multipliers, while he (6) amplified that discussion in a later publication, using a multiplicative probability argument to achieve the same end (7). Lorimer (8) used a reversal technique to bypass the Lagrange multi-
Literature Cited
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1. Wall, F. T. Proc. Nat. Acad. Sci. 1971, 68, 1720. 2. Russell, D. K. J. Chem. Educ. 1996, 73, 299. 3. McDowell, S. E. A. J. Chem. Educ. 1999, 76, 1393; but see Rioux, F. J. Chem. Educ. 2000, 77, 1559; with reference to Bent, H. A. The Second Law; Oxford University Press: New York, 1965; pp 163–165. 4. Gibbs, J. H. J. Chem. Educ. 1971, 48, 542. 5. Hakala, R. W. J. Chem. Educ. 1961, 38, 33. 6. Hakala, R. W. J. Chem. Educ. 1992, 39, 526. 7. Hakala, R. W. J. Chem. Educ. 1967, 44, 657. 8. Lorimer, J. W. J. Chem. Educ. 1996, 43, 39. 9. Nash, L. K. J. Chem. Educ. 1982, 59, 824. 10. Kleppner, D. Am. J. Phys. 1968, 36, 843. 11. Friedman, E.; Grubbs, W. T. Chem. Educator 2003, 8, 116.
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