One of the greatest misconceptions in all of science
is the idea that entropy is disorder.
Who is to say what counts as disorder?
After all, if this specific arrangement of shattered glass
is precisely the exact arrangement of glass objects that we wanted,
then this arrangement is the most ordered combination possible.
Also, the probability of randomly finding all the pieces of glass in this
exact specific arrangement is just as unlikely as randomly finding all the pieces
of glass in the original configuration where all the glass is in one piece.
The original configuration where all the glass is in one piece is
no more or less probable than any other particular specific configuration.
Every possible combination of positions for the glass particles is equally special.
The original combination, before the glass is shattered, is no different in this regard.
The key to understanding entropy is to realize that it does
not actually deal with the specific arrangements of the particles.
In a gas, for example, we do not have the ability to control
the specific position and velocity of each of the gas atoms.
But, we do have the ability change properties such as the volume and the pressure
of the gas by performing an action such as lifting this barrier.
Similarly, we do not have the ability to measure the position and velocity
of all of the gas atoms, but we do have the ability to measure properties
such as the volume, pressure, and temperature of the entire gas as a whole.
Suppose we don’t know the positions of these three objects,
but we know the size of the area they are located inside.
There are a number of different combinations of positions
for the objects that could be inside this area.
Now, suppose we have another situation.
Here, the size of the area in which the objects are located is slightly larger.
But, the number of different combinations of positions for the objects is far,
far greater than before.
Only a tiny fraction of the possibilities are shown here.
When there is a larger number of different ways in which something can happen,
we give this a name, and we say that this has a higher entropy.
However, the probability of the objects being in each of these configurations…
…is still exactly equal to the probability of the objects
being in this original configuration.
Every possible combination of positions is equally likely.
The concept of entropy does not apply to a
specific combination of positions of the three objects.
Rather, entropy applies to a measurement
such as the size of the area that the three objects are confined to.
If we know that the three objects are confined to the smaller area,
then we say that the system has less entropy
than if we know that the three objects are confined to the larger area.
This is because there are more combinations of positions that are possible
inside the larger area than inside the smaller area.
However, each individual combination of positions in the larger area
is still equally likely to each individual combination of positions in the smaller area.
This is because the objects could be anywhere in the original 10x10 grid.
Each individual combination of positions in the larger area
is still equally likely to each individual combination of positions in the smaller area.
Similarly, each possible combination of positions for the broken glass
has an equal probability, including the combination of positions
where all the glass is together in one piece.
But, if we measure the volume across which all the pieces of glass are spread out,
then the greater this volume,
the greater the number of combinations of positions that are possible.
Therefore, after the glass has been shattered, it is far more likely
that the glass will always continue to be spread out over a large area,
than that the glass particles will ever return to the one combination
where all the glass is together in one piece.
This is not because any of the combinations of positions with the glass spread out
are more likely, but simply because there are more of these combinations available.
In the case of the gas, we have the ability to measure not just the volume,
but also properties such as the temperature and pressure of the gas.
Let us call each possible combination of measurements for
volume, temperature, and pressure a “macro-state” of the system.
Let us call each possible combination of positions and velocities
for all of the atoms a “micro-state” of the system.
For each combination of volume, temperature, and pressure measurements,
there are many different possible combinations
for the positions and velocities of all the molecules.
In other words, for each possible macro-state,
there are many different possible micro-states.
The specific number of possible micro-states is different for each macro-state.
For example, the number of possible micro-states
is much larger for a macro-state with a larger volume.
If the number of possible micro-states is larger,
then we say that this particular macro-state has a higher entropy.
The concept of entropy only applies to macro-states, not to micro-states.
All micro-states are equally likely, but not all macro-states are equally likely.
A macro-state of a system need not necessarily be defined by
the measurements of volume, pressure, and temperature.
We can instead pick any set of measurements on a system as a whole,
and use these to define the “macro-state.”
Regardless of which set of measurements we pick,
if our system happens to be the entire Universe as a whole,
the Second Law of Thermodynamics states that
the entropy of the Universe can only increase,
and that the entropy of the Universe can never decrease.
One of the unfortunate consequences of this law is that
it means that all life in the universe will eventually end.
Details of this are discussed in the video titled
“Thermodynamics and the End of the Universe.”
Also, a discussion of the philosophical aspects of entropy
are available in the video titled
“Entropy: Why the 2nd Law of Thermodynamics is a fundamental law of physics.”
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