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MULTIPLICATIVE PROPERTY
OF THE EXPONENTIAL
FUNCTION
An article in the September issue of
this journal1 presents a derivation of
the Boltzmann distribution function
(BDF) for introductory students.
However, a key proof in the derivation
is left to a graduate text in mathematics. A simpler proof is presented here.
It is based on the idea of separation of
variables for partial differential equations, but is here applied to a basic
example of a function of two variables.
The BDF is presented in our curriculum in the third course in the introductory sequence for physics majors (after
the first two courses in mechanics and
in electricity and magnetism), while
separation of variables is introduced in
the fourth and final course (to solve
Schr€
odinger’s equation for the hydrogen atom).2 It therefore does not seem
too much of a stretch to make use of
the technique in the derivation of the
BDF. In fact, it could serve as a gentle
preparation for follow-on courses
where separation of variables is applied
to more difficult problems.
The goal of this letter is to prove
that the only nonzero real and differentiable function that satisfies
f ðxÞf ðyÞ ¼ f ðx þ yÞ
991
(1)
Am. J. Phys. 83 (12), December 2015
for the two independent variables x
and y is the exponential function f ðuÞ
¼ expðcuÞ where c is an arbitrary constant. To do so, first take the partial derivative of both sides of Eq. (1) with
respect to x to get
@f ðx þ yÞ
@f ð xÞ
f ð yÞ ¼
:
@x
@x
the right-hand side is only a function
of the independent variable y. The only
way this can be true is if each side is a
constant, call it c, so that
df
¼ c dx;
f
(7)
(2)
which integrates to
Next define z x þ y so that
@f ð x þ yÞ df ðzÞ @z
;
¼
dz @x
@x
f ðxÞ ¼ Aecx :
and noting that @f ðxÞ=@x ¼ df ðxÞ=dx
and @z=@x ¼ 1, Eq. (2) can be rewritten as
df ðxÞ
df ðzÞ
f ð yÞ ¼
:
dx
dz
(4)
Repeating these steps for the derivative
of Eq. (1) with respect to y, one
obtains
f ð xÞ
df ð yÞ df ðzÞ
¼
:
dy
dz
Carl E. Mungan
U.S. Naval Academy, Annapolis,
Maryland 21042
1
(6)
However, the left-hand side of this
equation is only a function of x, while
http://aapt.org/ajp
(For the special case of c ¼ 1, this
result follows from the fact that the exponential is defined as that continuous
function equal everywhere to its derivative.3) Substituting Eq. (8) back into
Eq. (1) shows that A2 ¼ A whose only
nonzero solution is A ¼ 1, thereby
completing the proof.
(5)
Since the right-hand sides of Eqs. (4)
and (5) are equal, the left-hand sides
must also be equal, which leads to
1 df ð xÞ
1 df ðyÞ
¼
:
f ð xÞ dx
f ð yÞ dy
(8)
(3)
L. Engelhardt, M. L. del Puerto, and N.
Chonacky, “Simple and synergistic ways to
understand
the
Boltzmann
distribution
function,” Am. J. Phys. 83, 787–793 (2015).
2
We use the textbook P. A. Tipler and G. Mosca,
Physics for Scientists and Engineers, 6th ed.
(W. H. Freeman, NY, 2008) in our introductory
course sequence. The Boltzmann factor is introduced on page 583, and separation of variables
on page 1234.
3
C. E. Mungan, “Introducing the exponential
function,” Phys. Educ. 41, 373–374 (2006).
C 2015 American Association of Physics Teachers
V
991