Each of these particles possess kinetic energy ($\frac{1}{2}m v^2$) and if you add up the total kinetic energy of each particle you get a new form of energy, thermal energy ($E_{th}$). Since accounting for 10²⁶ particles all with different speeds is impossible, we will instead employ the use of averages. In the equation below $\overline{K}$ is the average kinetic energy per particle.

$\overline{K} = \frac{1}{2}m^*v_{rms}^2$, where $m^*$ and $v_{rms}$ are the average mass and speed per particle

Technically $v_{rms}$, where the rms stands for root mean square, is a more complicated statistical average, but for the sake of this discussion you can simply think of it as an average speed. From here down I'll just write it as m and v, but realize they are the average of those quantities. If you have N number of particles then you have a total thermal energy of:

$E_{th}=N\overline{K}=N \frac{1}{2}mv^2$

All of this is well and good but we still haven't made any connection to the macroscopic world and real measurable - after all we can't shrink down to the size of a molecule and start using our radar gun to measure a bunch of particle speeds. That's where a bold postulate was formed by a physicist named Boltzman where he imagined each degree of freedom the particles have contribute $\frac{1}{2}k_B T$ to the total energy. Here $k_B$ is Boltzman's constant (the real heart of the micro/macro connection) and T is temperature measured in Kelvins. Degrees of freedom are related to the ways a particle can move. For a mono-atomic gas (bunch of individuals "balls") the particles can move in x, y, and z, but do not have any appreciable rotation or vibration. So for a mono-atomic gas the degrees of freedom (D) is equal to 3. Contrast this to a diatomic gas, where I picture two balls connected by a spring, so they collection can still move in x, y, and z, but they can also rotate about each other and vibrate towards then away from each other. This means they would have a degree of freedom higher than 3, most likely near 5. More complex molecules will have higher degrees of freedom which results in more ways to store energy without changing linear kinetic energy of the molecule. For this discussion though we will stick to mono-atomic gases. So, thermal energy can be written in the following way:

$E_{th}=N D \frac{1}{2} k_B T$

which for a mono-atomic gas with D = 3, is $E_{th}=N\frac{3}{2}k_B T$

Combine the microscopic view of particle speeds above with Boltzman's view of a macroscopic observable like temperature, and you get a fundamental connection between the two views.

$E_{th}=N\frac{1}{2}m v^2 = N \frac{3}{2}k_B T$, mono-atomic gas

Equilibrium: If there is a different average kinetic energy in one part of the container than in the another, the system will not be in equilibrium and will drive towards a constant average kinetic energy throughout the container. Once equilibrium has been obtained, average kinetic energy is constant, and thus temperature is also constant. It's important to think in terms of energy gradients cause systems to want to change. In this case the measurable that drives towards being constant is temperature.

Now, take a look at the pre-lecture reading and videos below.