Why Entropy Happens

In the beginning, there was order, a very hot singularity, but as time progressed the universe expanded and cooled--and became more disorderly. Scientists predict a "big freeze." It's all due to entropy. As you read this blog, entropy continues. Why? That's what we will explore below. First, let's define the variables we will use:

We begin with the partition function, which has the Boltzmann factor, an exponent with a thermodynamic beta power over the base e:

If we want to determine the probabilities of the energies in a system, we make sure the probabilities add up to 1, so we normalize the partition function by dividing it by itself (Z):

However, if we want to model the universe's evolution, we need to make a slight change to the partition function. Instead of using the thermodynamic beta, we use its reciprocal. We also change the i index to a time (t) index:

Also, we want the universe's total energy to be conserved. We know dark energy is increasing and radiation energy is decreasing, so we put together an energy-conservation equation:

At equation 5, notice how an increase in the universe's volume (V) reduces the radiation energy (Er) but increases the dark energy (pV). Multiply the two energies, add a little dark and baryonic matter, and take the square root and we get a constant energy (E).

We define temperature as follows:

As volume (V) increases, the universe's temperature decreases. Equation 7 below gives us the probability of the temperature at a given time t:

A high temperature has a low probability. A low temperature has a high probability. So there is a high probability the universe's temperature will continue to decrease, and a low probability the temperature will increase. Thus, an expanding universe has a higher probability.

Now, let's take a look at entropy. We define it as follows:

We see that entropy increases as temperature decreases--so it has the same probability as temperature:

So why does entropy happen? Greater entropy has a higher probability than lower entropy. We can also say that reverse entropy is possible but less probable. A good example is the one Tyson discussed in the above video. There are pockets of order caused by star energy, so life is possible.

How Did the Ancients Build Those Mysterious Stone Structures?

Several years ago I watched a documentary (I don't recall the title) about ancient pagan sun-worshipers who built these mysterious stone houses and shrines in Britain. I don't recall who they were exactly. They may have been Celts or Druids--or a different group. I do remember the problem the archeologists were having. They were trying to figure out how these ancient people managed to build these stone structures.

They weren't just ordinary stone structures. The stones were heat-fused together. An incredible feat by these ancients, considering their stone-age/bronze-age technology. The popular theory, of course, is that ancient people were lunkheads who couldn't do anything without the help of visiting extraterrestrials.

Being scientists, the makers of the documentary set out to prove the popular theory wrong. They tried to construct one of these stone structures, using only the materials available in ancient times. They first created a mound of dirt, then piled stones on top to get a kind of igloo shape:

Next, they surrounded the structure with timber:

They then set it ablaze:

What happened next? Well not much. They hoped the stones would fuse together after being exposed to a pyre that would make any Viking funeral a hit. Unfortunately, the stones failed to fuse together. But these archeologists were stubborn and did not give up. They tried the same procedure again and again, hoping for a different result: success. But the stones "stubbornly" refused to fuse together.

At the time I watched this documentary, I wasn't even a physics student yet, so I, like the archeologists, had no clue what went wrong. But that was then. Now it's patently obvious what went wrong. There wasn't enough heat because the second law of thermodynamics reared its ugly head!

When the fire burned, most of the heat was lost to the open air. The convective-heat-transfer equation below spells this out:

The fire's temperature is approximately 1571 degrees Fahrenheit or 855 degrees Celsius. The air temperature was far colder, so nearly 100% of the heat was lost. What these archeologists needed to do was somehow raise the air temperature to 855C, then little or no heat would have been lost.

What I suspect the ancients did was something the archeologists failed to do: The ancients used some form of insulation. If insulation is used, the following equation shows the benefits:

Thick insulation with low thermal conductivity will not only save energy, but can also cause the temperature to increase. This is good news, considering the minimum temperature needed to melt stones is higher than the fire's temperature. Here are the numbers:

So now the question is what materials were available that could be used to insulate? Here is a short list that includes each material's thermal conductivity (kt):

Using stones for insulation seems like an obvious choice, but if there is no mortar to work with, then we would need to heat-fuse these stones together so we can insulate the fire so the stones we started with will be heat-fused.

Wood has a slightly higher thermal conductivity: .5. It is easy to build a wooden insulating frame around the stone structure. The main drawback is it will burn up. That leaves soot. Soot has a very low thermal conductivity: .07. Insulation made from soot will raise the temperature approximately seven times higher than stone or wood--up to 10,500 degrees Fahrenheit or 6,000 degrees Celsius.

I don't know what those ancient people did exactly, but here's how I would engineer a heat-fused stone structure. I would pile stones on a mound of dirt, leaving openings where I want doors and windows. I'd put the timber on top, and over all that I'd build a wooden insulating structure. (I use cutaway views in the diagrams below.)

Of course the wooden planks I'd use would be coated with a paste containing soot, or be charred wood. I would also use a bellows to feed the fire more oxygen. I'd also bury the whole shebang under a pile of mud that would be allowed to harden. Finally, I'd light the fire and insert the bellows and pump away.

To save some time, it might not be necessary to coat the wood with soot or use burnt planks. Fresh wood could be used. It would no doubt burn and may become soot that sticks to the mud structure that remains. This would be ideal.

When enough time has lapsed, the fire would be doused, the mud structure removed. The final step involves removing the dirt from inside the stone structure. If all goes well, it should stand firm because the stones would be heat-fused together.

Entropy Succeeds Where the Ideal Gas Law Fails

The universe is getting bigger and colder. We are told that one day it could suffer a heat death. By contrast, the early universe was smaller and hotter. All this implies an inverse correlation between the universe's average temperature and its size.

We might be tempted to model it with the Ideal Gas equation. We could think of the universe as a big balloon filled with gas. As the volume of the balloon increases, the pressure decreases along with the temperature. Below is the equation (P=pressure; T=temperature; V=volume; R=universal gas constant; n=kilomoles of gas):

Right away the equation bursts our balloon. When volume (V) increases, pressure (P) decreases but the temperature can remain constant. So why isn't our universe the same temperature as it was in its early years? Its volume is increasing; its pressure is decreasing; yet, the temperature isn't remaining constant.

So what's the problem with the Ideal Gas Law? We know that pressure is equivalent to energy density, so let's plug energy density (E/V) into the equation in lieu of pressure (P):

The second equation above reveals the problem: The volume (V) in the denominator cancels the volume in the numerator. Change in volume in this case has no effect on the temperature.

We know that entropy increases as the universe expands. Perhaps lower temperature is a function of increased entropy? Let's take a look at the entropy equations (E=heat or energy; s=change in entropy; T=temperature; k=Boltzmann's constant; omega=number of different arrangements of particles):

If we equate the two equations above, we can derive a temperature equation:

Temperature appears to be a function of energy (E) over the number of ways particles can arrange themselves (omega). Once again, temperature is not a function of volume, but it seems like it should be. We know from experience that a candle can heat up and maintain the temperature of pickle jar, but a candle in an aircraft hanger can be a much colder space.

Let's explore the omega variable and see if we can make it a function of volume. Imagine a single particle with one state in a single space (see first diagram below). In that case, omega equals one. Now double the space. The particle will have two spaces it can occupy (see second diagram below). Omega now equals two.

Entropy has increased due to increased space. We can define omega in the following way:

Below are some examples that include the above diagrams, a coin toss, and dice:

Consider the coin-toss example above. Normally we think of a coin as having two states: heads or tails, so omega equals two, right? Well that's only true if the coin lands on the same spot on your floor because there is no other spot it can land. But suppose you have a big floor and you divide it into a grid of 20 spaces where the coin can land. Omega would then be 2^2 * 20 = 40 states.

Now suppose you have the same floor space, but you now have two coins. Omega becomes 1520 states! We are now ready to put together a temperature equation:

Notice the temperature is now a function of volume because entropy is now a function of volume. If the volume increases, the temperature decreases and so does the pressure (Boyle's Law) if the other omega variables and E are held constant. Remember pressure is E/V. The difference here is that E/V is no longer canceled by V*E/V.

Taking the temperature of the Universe or any other system can be done by this system of equations. The first equation is for systems that have more space than particles. The second is for singularities, high pressure systems, where the energy density is high. The third equation allows you to make a quick-and-dirty calculation. It is similar to the Ideal Gas equation albeit it includes entropy.