The Helmholtz Theorem

Derived from considerations of Special Relativity, the principle of least action tells us that for any mechanical system the action played out by that system must manifest its minimum possible value; that is,

(Eq地 1)

If the system contains no potential energies ϕ and, therefore, only kinetic energies K, then we have

(Eq地 2)

If our system consists of a single particle, then we get the principle of least time, familiar from optics. In general we have, because of the product rule of differentiation,

(Eq地 3)

In the limit as t_{1} approaches t_{2} we have

(Eq地 4)

By the definition of the differential we know that d(Kt)<<Kt, so we also know that

(Eq地 5)

Subtracting Kt from both sides of that equation and multiplying the result by minus one gives us

(Eq地 6)

We divide that equation by Kt and get

(Eq地 7)

Because we have made the value of t_{1} approach the value of t_{2},
we have also made the value of t approach the value of dt (though obviously it
can稚 be less than dt), so we have

(Eq地 8)

That result seems rather obvious: if we add kinetic energy
to a system, we expect that the ratio of the change to the system痴 energy
content will be a positive number. The situation gets more interesting when we
have dK=dK_{1} -dK_{2}=0, which means that we have kinetic
energy moving between two subsystems of a larger system.

If our system consists of N components, then we know that K=N K , in which K represents the average energy of a component. If the average is determined as a time average, we can invoke the ergodic theorem to equate it to the ensemble average, the average calculated over the whole system at a given instant. So we can rewrite Equation 8 as

(Eq地 9)

Multiply that equation by N and the convenient constant 3/2 to get

(Eq地 10)

We define the temperature T of a system by stating that T=(2/3) K , so we can write Equation 10 as

(Eq地 11)

In that equation S represents the entropy of the system or an entropy-like quantity. If the energy that we put into the system includes potentials or potential-like energies (such as pressure exerted throughout the volume), we must take proper account of it. If our system has a pressure in it, then we have

(Eq地 12)

so we have Equation 11 as

(Eq地 13)

in which we can identify dE=pQ, the amount of heat put into the system. Thus we obtain the master equation of thermodynamics and we have the Helmholtz theorem. For any mechanical system Equation 13 tells us what happens when we put kinetic energy into the system.

Now go back to Equation 10, multiply and divide it by N, and integrate the result. We get

(Eq地 14)

That痴 an indefinite integral and we want the definite version,

(Eq地 15)

That result gives us a pure number, rather than the energy-divided-by-temperature that we see in Equations 11 and 13. Implicit in that fact is a conversion factor, a constant multiplier that we can imagine as invisible. Having noticed it, we can make it visible and at the same time absorb the factor of 3/2 into it, so we rewrite Equation 15 as

(Eq地 16)

Because N represents a variable of the system we take it into the logarithm and get

(Eq地 17)

in which

(Eq地 18)

If we make K_{0} represent the smallest amount of energy that we can
give a component of the system, then W represents the number of different ways
in which we can distribute the total energy K_{1} throughout the system
and Equation 17 stands before our eyes as Boltzmann痴 equation.

Again let痴 return to Equation 10 and see what it tells us
when we transfer energy between two separate parts of our mechanical system. In
this case conservation of energy tells us that dK_{1}+dK_{2}=0,
but we can稚 simply substitute that equality into Equation 10. If the average
kinetic energy differs between the subsystems, K_{1} < K_{2} (T_{1}<T_{2})
for example, then we must write

(Eq地 19)

For convenience I have taken the subscripts off the differential energies
because |dK_{1}|=|dK_{2}|.

That equation describes the change in entropy occurring in the system when we take a small amount of energy from the hotter subsystem and put it into the colder one. Because the first term on the right side of the equality sign is larger than the second term, the change in entropy comes out as a positive number, as the inequality requires. Under the right circumstances energy could move spontaneously from Subsystem-2 to Subsystem-1, usually manifested in the flow of heat.

On the other hand, if we have T_{1}>T_{2},
the change in entropy would be a negative number and would violate the
inequality. Certainly we believe in the possibility of taking energy from the
colder subsystem and putting it into the hotter one, so can we do that transfer
without violating Equation 19? We can if we take the energy dK out of the colder
subsystem and do work dU upon it as we put it into the hotter subsystem. We then
get Equation 19 as

(Eq地 20)

If we make the amount of work such that

(Eq地 21)

then we have satisfied the requirement of the inequality. But we can also turn that proposition around and note that if we have energy going from hotter to colder, we can convert part of it into work.

Thus we gain the statement that heat will not, of itself, go from a colder to a hotter body, which statement gives us the classical law of entropy as formulated by Rudolf Clausius.

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