∫dx/x
Let's start with the infinite series
(Eq'n 1)
in which we have used the usual technique of evaluating an infinite series to get the last term. Next integrate the series with respect to x and get
(Eq'n 2)
We now want to explore the properties of F(1+x).
We know right away that for x = 0, F(1) = 0 and that for x = -1, F(0) = -∞.
If we make the substitution 1+x = (1+y)(1+z), we have
(Eq'n 3)
which we divide by (1+y)(1+z) to get
(Eq'n 4)
so now we know that
(Eq'n 5)
That statement combined with the statement that F(1) = 0 tell us that
(Eq'n 6)
Thus, we infer that the function F has the properties of a logarithm.
To what base must we refer that logarithm? That is, for what number B does F(B) = 1? We know that we can always represent a number as a convergent infinite series, so let's define our base as
(Eq'n 7)
We know that we have an infinite range of possible values for the numbers an. Most of those values will equal very small fractions and some may be negative numbers. In order to determine what those values are we define
(Eq'n 8)
and multiply it by Equation 7 to get a partial product matrix
(Eq'n 9)
That equation has an infinite number of solutions and here I must leave a
logical gap
because I have failed in all of my efforts to parlay the knowledge that I have before me into criteria that will lead to the one solution that I want. Nonetheless, I know that solution. It comes from stating that every upper-right to lower-left diagonal, except the zeroth, of the matrix adds up to zero. Leaping another
logical gap,
I state that the set comprising those diagonals coincides with the set made up from the rows of the alternating Pascal's triangle. That statement necessitates that
(Eq'n 10)
and then that
(Eq'n 11)
which means that we have at last
(Eq'n 12)
We recognize that equation as expressing Euler's number,
e=2.7182818284590...,
the base of the natural (or Naperian) logarithms.
Now we can make the substitution t=1+x and dt=dx and write the logarithm as mathematicians do to make the definition
(Eq'n 13)
Example Application: Entropy
If we have a thermodynamic system at some absolute temperature T and we change the system's energy by the amount dE and the system's temperature changes by dT in consequence, then we can ascribe to the system an entropy defined as S = dE/dT. If we change the energy of the system without changing the temperature, we have dS = dE/T. Thus, we have taken E=ST, differentiated it (dE=SdT+TdS), and taken the cases dS=0 and then dT=0.
We know that the temperature of a thermodynamic system reflects the average energy of the particles comprising the system; that is, E=kT, in which k, Boltzmann's constant, through the indicated multiplication converts units of temperature into units of energy. For an N-particle system, then, we can describe the energy contained by the system as E=NkT. If we change the energy of the system, we thus have dE=NkdT. If we change the energy of the system by an amount that changes the system's temperature from T0 to T1, then we change the system's entropy by
(Eq'n 14)
In that system the number of ways in which a particle can participate in sharing the energy of the system, from having all of the energy to having none at all, however large that number may be, must be proportional to the average energy of the system, proportional to the system's temperature. If the system were to contain no energy, then each particle would have only one way to participate in the energy share-out. We know by Nernst's theorem that the temperature T0 at which that occurs cannot equal zero: that fact necessitates that there be a temperature below T0 separated from it by some quantity of energy that absolutely cannot be removed from the system (the zero-point energy). So the number of ways that the system can distribute its energy among N particles is W=(T1/T0)N and thus, by substituting that expression into Equation 14 we have Boltzmann's Equation
S=klnW.
(Eq'n 15)
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