The Entropy of Light

If we want to treat light as a complete thermodynamic entity and not merely as a simple carrier of energy, then we need to describe its other thermodynamic properties. Let's start with pressure.

We know that a pulse of light
carries linear momentum. If the pulse carries energy E, then we can attribute to
it, by way of Einstein's mass-energy theorem, an equivalent mass of "m"=E/c^{2}.
That pseudo-mass moves at the speed of light, so it carries linear momentum

(Eq'n 1)

(In that equation I have used pi, instead of pee, to represent linear momentum because we use the letter pee to represent both linear momentum and pressure. In thermodynamics we refer to pressure more than we refer to momentum, so I have chosen to use pee to represent pressure and, to avoid confusion, I have used pi to represent linear momentum.) If a beam of light carrying energy at the rate dE/dt strikes a body and the body absorbs the beam, then the body absorbs linear momentum at the rate

(Eq'n 2)

That force cannot come into existence ex nihilo at the interface between vacuum and the absorbing body; Newton's third law of motion, the conservation law pertaining to linear momentum, necessitates that it must exist inherent in the light.

Imagine a flat plane crossing that beam such that every straight line in that plane that crosses the beam does so at a right angle. Through any given unit area of that plane the beam carries a pressure

(Eq'n 3)

But that equation gives us the instantaneous pressure, which fluctuates rapidly in an electromagnetic wave. We actually want the averaged out pressure, the steady push that we would infer from observing the acceleration of any object that absorbs the wave. We can get that by calculating the time average of Poynting's vector,

(Eq'n 4)

Thus we have

(Eq'n 5)

Making the appropriate substitutions from the third and fourth of Maxwell's Equations transforms that equation into

(Eq'n 6)

But cdt=dx, the minuscule element of distance that the wave propagates in the element of time dt. Thus we transform an integration of a time derivative over a time interval into the integration of a spatial derivative over a spatial interval. Instead of imagining integrating the wave as it passes through the plane defined above, then, we can imagine seeing the wave frozen in place and integrating it from the plane to some suitable point along the direction of propagation. Instead of integrating from an earlier time to a later time, we integrate from a larger coordinate to a smaller coordinate. Because we conventionally integrate from smaller to larger, we reverse the limits of our integration and thereby eliminate the minus sign from Equation 6.

I have also tacitly assumed,
and now make explicit, that the beam of radiation constitutes a plane wave
propagating in the positive x-direction with its electric-field vectors oriented
entirely in the y-direction. In that circumstance the fourth term of the
integral in Equation 6 becomes
. But E_{x}=0,
so the term drops out of the integral. By the same reasoning we also drop the
second term, so we now have Equation 6 as

(Eq'n 7)

in which represents the average energy density in the beam.

Imagine that we have radiation filling a cavity whose walls are perfectly black. The walls absorb and re-emit the radiation, thoroughly mixing it. In that circumstance we can describe the radiation as comprising plane waves propagating in the x-direction, plane waves propagating in the y-direction, and plane waves propagating in the z-direction. Because of the mixing by the walls of the cavity each of those components contributes the same amount to the overall energy density of the radiation field:

(Eq'n 8)

If we conceive the cavity as having the shape of a cube, then we see clearly that the pressure that the radiation field exerts upon any given wall comes from only one of the components of that sum. But a well-mixed radiation field must exert the same pressure in all directions, so we have for cavity radiation

(Eq'n 9)

To calculate the entropy of cavity radiation and, thus, of blackbody radiation we employ the master equation of thermodynamics,

(Eq'n 10)

Inside the cavity at uniform temperature we have . That fact and Equation 9 allow us to rewrite Equation 10 as

(Eq'n 11)

which gives us the increment of entropy as

(Eq'n 12)

We know that we cannot describe as a function of V; indeed, our description of the cavity and its contained radiation field necessitates an assumption that has the same value throughout the volume of the cavity. Thus, we obtain a description of the total entropy of the radiation in the cavity by simply adding up all of the increments: we get

(Eq'n 13)

in which E represents the total energy in the radiation field.

Finally, we apply the Stefan-Boltzmann representation of the energy density in the cavity,

(Eq'n 14)

and obtain

(Eq'n 15)

Now we can see how we might use light as a thermodynamic entity.

Imagine that we have a cavity with perfectly reflecting walls and that we have filled it with blackbody radiation at some temperature. We then expand that cavity to eight times its original volume, allowing the light's pressure to do work on the walls as they expand. This process does not change the entropy of the light (it's merely the expansion phase of a Carnot cycle), so Equation 15 tells us that the light's temperature decreases to half its original value. We might then remove some of the light's energy as discarded heat, recompress the light, and then reheat it to complete the Carnot cycle. We would then have used light as the working fluid in a heat engine.

Though certainly not practical, such an engine, as an abstract idea, leads us to begin thinking about light as though it were some kind of vapor. As we shall see, in the quantum theory physicists actually do conceive and refer to light as a photon gas that has entropy

(Eq'n 16)

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