A principle called the second law of thermodynamics
derived from the works of people such as Carnot,

Clausius, Maxwell, Boltzman and Szilard, is
a fundamental concept in science and physics.

It describes how energy and everything related
to energy such as heat and work evolve over

time and interact with the environment, how
heat and energy is transferred from one side

to another and how it behaves in closed and
open environments.

To help us understand this principle and its
connections to the concept of information,

let's perform what is called a thought experiment,
that is, an experiment that one can run in

one's own mind to reach some sort of logical
conclusion.

In this case is going to rather be illustrated
by an actual computer simulation.

So let's imagine for an instant that we have
a room with gas.

Gas is made of particles so particles are
free in the room only constrained by the room

walls.

Particles are moving around bouncing against
each other.

Here is an illustration of 100 particles shown
in black inside a square of size 100 x 100

pixels.

For illustration purposes, the room can be
divided in 100 x 100 cells each of the same

size, making 10 thousand cells that can contain
one of the 100 particles.

To describe any particular state of the room
at any given time we would have to provide

the bit value of those 10 thousand particles.

Alternatively, we can simply give the 100
coordinates where particles are located.

If the room were empty, however, it would
just suffice to say that the room is empty

and nothing else, that would characterize
the state of the room in a very simple description,

but if the particles are distributed randomly
then one would need to provide the exact coordinate

for each particle and no simpler description
would be possible in order to reproduce the

same state of the room.

If you wanted to describe this room you would
need to provide the exact location of each

particle.

But there was an idea in quantum physics to
partially circumvent the need to study such

detailed descriptions that are impossible
to get in practice, and even in principle,

according to some aspects of quantum mechanics
when we are considering very small particles.

So, statistical mechanics makes qualitative
assessments instead of providing the exact

locations.

It would characterize the room statistically,
in this case, for example, as typically 'disordered'.

What the second law of thermodynamics would
then tell is that most configurations will

look disordered and those ordered will be
unlikely to appear by chance and, if they

did, they would be unstable and quickly evolve
into disordered configurations.

The room would then be said to be in a high
Entropy configuration.

Such qualitative description is statistical
in nature.

The idea is that if you plot the distribution
of the particles across all the possible places

in which they could be, the distribution would
look the same for most random configurations

and thus anything with such distribution would
be characterized as being in the same state

of disorder.

So this is the way in which the distribution
of the particles in our room looks like when

dividing the room into smaller square pieces
of equal size, so when looking at the room

in 100 places no particular place seems to
have any particular feature meaning that particles

look mostly distributed equally, the same
when dividing the room into 20, 10 and only

5 places to look at as if we were looking
at the corners and centre to see if there

is some arrangement of the particles to notice,
but basically everything looks the same in

any direction.

Imagine that you were able to place the same
number of particles into a smaller room:

What would happen is that the same number
of particles have now fewer places to be,

what is called less degrees of freedom or
less cells in the square to be or move into.

So because there is less room for the particles
in the new room it can be said that the room

has a lower Entropy compared to the larger
one for the same number of particles but different

volume or area.

If one reduced the number of particles it
would also lead to less Entropy because you

would need to describe less particles, so
Entropy is a function of at least 3 elements:

the number of particles, the size of the space
and the number of places in which the particles

can be.

Imagine you pushed all particles into a smaller
and smaller place in the corner of the room:

Eventually the particles would have no place
to move and particles would accumulate next

to each other in an increasingly simpler configuration.

Then the room would be said to be in a low
Entropy state because the configuration is

not typical, there are only a few cases among
all possible in which the statistical distribution

looks like the one distributing particles
in the corner of the room.

This is how the distribution plots similar
to the ones we made for the random configuration

would look like in this case:
Only moving one particle outside that corner

would make Entropy to increase.

So what does it tell us the second law of
thermodynamics?

It tell us that if we looked at a room with
some gas particles, the chances to find the

room in a disarranged or disordered state,
that is, of high Entropy, has much greater

chances because there are many of those configurations
that distribute particles in a random-looking

distribution compared to a particular arrangement
of low Entropy like the one with all the particles

in the corner.

In other words, there are many more possible
disordered arrangements than ordered ones,

and low Entropy configurations will tend to
produce higher Entropy configurations if left

unattended and no work is performed to place
them in a single non random-looking configuration.

So it is the case that to reach a special
configuration one has to invest work or apply

energy, to push all the particles to the corner.

Moreover, if the particles were moving at
constant speed, by pushing them into a corner

they would bounce with each other with higher
frequency producing heat that, when released,

would naturally cool down hence establishing
a strong connection with heat and temperature.

This is why this law or guiding principle
is so important, because it connects most

if not all the fundamental quantities in physics.

But it is also all about information as we
saw before, as it involves knowledge about

the position of the particles and ways to
steer them to one side or the other.

So we have connected all these measures with
information, heat, disorder, Entropy and temperature.

So the second law of thermodynamics tells
us that, in general, because particles will

tend to maximal Entropy, they will also dissipate
heat over time instead of gaining it, and

they will eventually reach thermodynamic equilibrium
by matching the environment's temperature,

just as it happens with hot drinks eventually
reaching room temperature.

This also means that if you disconnect your
fridge, it will reach room temperature soon

because of the second law of thermodynamics
even though there are chances that it can

remain cold or even colder, but those chances
are so incredibly small that it will almost

never happen.

It is right then to think of the second law
of thermodynamics as a law based on probability

and deeply connected to information about
the specific and overall configuration of

a system.

The traditional way in which the second law
of thermodynamics, for a closed system like

our room, is formally written like this.

The increase of Entropy represented by S over
time t will remain or increase but not decrease.