Initially, only a few of the particles are moving.
The energy quickly spreads throughout all the particles as shown.
Each square represents the particle with the corresponding color.
The horizontal axis represents the magnitude of the velocity of each particle.
The vertical axis represents the number of different particles with that velocity magnitude.
In the graph, the velocity of each particle is rounded to the appropriate number of digits, so that all the squares neatly fit into columns.
This system quickly reaches what we call thermodynamic equilibrium, and the graph for the velocity distribution ends up looking as shown.
A system is in thermodynamic equilibrium when the entropy of the system has reached its maximum possible value, given the constraints of the system.
Entropy is a measure of how many different ways something can occur.
There are only a small number of ways in which all the energy can be given to only a few particles, and hence the situation at the beginning had a smaller entropy.
There is a much larger number of ways in which the energy can be dispersed as shown, and hence this situation has a higher entropy.
If all the particles hypothetically had the exact same energy, then this would have a lower entropy, because there is a smaller number of different ways in which all the particles can have the exact same velocity magnitude.
There is a much larger number of ways in which the velocity magnitudes can be dispersed on the graph as shown, and this velocity distribution is what we get when the entropy is at its maximum, for this given temperature.
In this simulation, all the particles are represented as spheres interacting through perfectly elastic collisions.
The total energy of the entire system is always constant.
The average kinetic energy per sphere is what call "temperature."
If we increase the temperature by adding energy to the system, the graph for the velocity distribution changes as shown.
This system now has a higher entropy than before, because with more energy, there are more different ways in which the energy can be distributed.
On average, the graph will look like this.
If we decrease the temperature, the graph changes as shown.
Here are the graphs for a variety of different temperatures.
Each of these graphs is for a system in thermodynamic equilibrium for that given temperature.
A system is in thermodynamic equilibrium when the entropy of the system has reached its maximum possible value, given the constraints of the system.
Entropy is a measure of how many different ways something can occur.
Each of the different ways something can occur is what we call a micro-state.
Here, a micro-state would consist of all the positions and velocities of all of the particles, at a given moment in time.
In Quantum physics, the way we count micro-states is different than in classical physics.
For example, consider the positions and velocities of several identical particles at a given moment in time.
Now, consider a situation where we have the exact same set of positions and velocities, but we swap the colors of the particles.
In a classical system, we can do this because the particles are distinguishable, and each of these cases is counted as a separate possible micro-state.
However, in a quantum system, identical particles are indistinguishable, and therefore all these cases are counted as only a single possible micro-state.
Hence, the thermodynamic equilibrium is different for a quantum system than for a classical system.
Also, in a quantum system, the thermodynamic equilibrium is different depending on if we are dealing with particles such as electrons, which obey the Pauli Exclusion principle, or if we are dealing with particles such as photons, which do not.
The Pauli Exclusion principle prevents multiple identical particles from occupying the exact same quantum state.
But this principle only applies to particles we call fermions, such as electrons.
It does not apply to particles we call bosons, such as photons.
Hence, the thermodynamic equilibrium will be different for a system of fermions than it will be for a system of bosons.
Though, at high temperatures and low particle concentrations, the energy distribution for a system of fermions or for a system of bosons will look very similar to that of a system of classical particles.
In this simulation, all the particles are represented as spheres interacting through perfectly elastic collisions.
Everything we have discussed so far assumes that the particles behave as independent spheres, with no rotational energy, and no potential energy in bonds between particles.
The situation is much more interesting when we instead have molecules composed of multiple atoms.
This is discussed in detail in the video: "Molecular Temperature & Degrees of Freedom"
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