On the right side, kB is called the Boltzman constant, and omega (Ω) is the number of possible “microstates.” Let me explain with an example: Say I have four coins and four people. How many different ways can I share these coins? Well, two extreme cases would be that everyone gets one coin or that all four coins go to one person. Altogether, there are 35 possible distributions. These are the microstates (Ω). So you can think of them as possible arrangements of energy among particles while keeping the total energy the same.
If I drop a basketball, its gravitational potential energy declines as it falls and its kinetic energy increases—total energy is conserved. But it doesn’t bounce as high as it started. That’s because some energy leaks away as heat on impact. The ball warms up a little bit. When we take that thermal energy into account, we find that energy is still conserved. But the entropy is higher.
But what if the ball got colder and bounced higher? This would mean thermal energy is reduced and kinetic energy increases. Energy would still be conserved, but in this outcome, the entropy is reduced. This is actually possible, but you could run this experiment till the end of time and never get that outcome.
Here’s another fun example. Imagine putting ice in a glass of warm water. Is it possible that the water gets warmer and the ice gets colder? Again, the probability is non-zero, but it’s extreeemely unlikely. Boltzmann’s formula says that the more possible microstates there are, the greater the entropy.
Ulitimately, this leads to the second law of thermodynamics, which states that the total entropy of a closed system can only increase over time, or at least it can never decrease. So, like, your desk will only get messier and messier, unless you open that “system” and do some “work”—which, remember, means adding energy. Unfortunately the universe is a truly isolated system, so it can only end one way—in the total loss of all structure and life.
4. Ohm’s Law
Courtesy of Rhett Allen
This equation is used in many of our modern devices since it deals with electricity. Ohm’s law gives a relationship between the change in electric potential energy (ΔV) across some element in a circuit and the electric current (I) moving through that element. Because it’s tiring to say “the change in electric potential,” we often simply call it “the voltage,” as it’s measured in volts. The proportionality constant between the voltage and current is called the resistance (R); not surprisingly, it’s measured in units of ohms.