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Entropy Explained on a Microscopic Level

Table of contents

Entropy, abbreviated with the letter $S$, is a fundamental concept in thermodynamics. To understand this concept, consider a very simple closed system. A closed system is an isolated part of the universe that does not interact with its surroundings.

The closed system has two subsystems A and B, in which four particles O can reside.

A system with four particles consists of two subsystems A and B.

Another concept that is necessary for understanding entropy is the microstate of a system. A microstate of this system is characterized by how the four particles are distributed between the two subsystems. A system usually has many possible microstates. Let's denote their number by $N$.

Examples of possible microstates

In subsystem A there are two particles O, O. The other two particles O, O are located in subsystem B. This is a possible microstate of the system. Another possible microstate is when all particles O, O, O, O can be found in A and none in B.

Microstate in which all particles are in subsystem A.

What is entropy?

The entropy $S$ is a measure of the number $N$ of possible micro states. The more states the system can occupy, the larger the entropy!

A system with minimum entropy

Let's construct a system in which no particles can be in subsystem B. All four particles must be in subsystem A:

The system has minimal entropy.

In order to be able to make a statement about the entropy of this system, the number of possible microstates must be considered. However, the system was constructed in such a way that all particles can ONLY be in A. There is therefore only one microstate of the system, namely the state: »All particles in A«.

This system has a single microstate: $N = 1 $.

A system with non-minimal entropy

Let's look at another system where there is more than a single microstate. Let us now assume that there can be a maximum of two particles in subsystem B. The first possible microstate is where all particles are in subsystem A:

$$ \begin{align} \text{A}(\class{red}{\text O}, \class{red}{\text O}, \class{red}{\text O}, \class{red}{\text O}) ~~~ \text{B}(~) \end{align} $$

The second allowed microstate is where one particle is in subsystem B:

$$ \begin{align} \text{A}(\class{red}{\text O}, \class{red}{\text O}, \class{red}{\text O}) ~~~ \text{B}(\class{red}{\text O}) \end{align} $$

And the last allowed microstate is when there are two particles in subsystem B:

$$ \begin{align} \text{A}(\class{red}{\text O}, \class{red}{\text O}) ~~~ \text{B}(\class{red}{\text O}, \class{red}{\text O}) \end{align} $$

Placing another particle in subsystem B is not permitted in this system. This system has a higher entropy than the previously considered system, where particles could only be in subsystem A.

A system with maximum entropy

Let us now consider a third system. Let us assume for this system that there are no restrictions on how particles can be distributed in subsystems A and B. Let's go through all possible microstates of the system.

In the first microstate, all particles are in subsystem A:

$$ \begin{align} \text{A}(\class{red}{\text O}, \class{red}{\text O}, \class{red}{\text O}, \class{red}{\text O}) ~~~ \text{B}(~) \end{align} $$

The second allowed microstate is where there is a single particle in subsystem B:

$$ \begin{align} \text{A}(\class{red}{\text O}, \class{red}{\text O}, \class{red}{\text O}) ~~~ \text{B}(\class{red}{\text O}) \end{align} $$

Der dritte erlaubte Mikrozustand ist, wo zwei Teilchen im Teilsystem B sind:

$$ \begin{align} \text{A}(\class{red}{\text O}, \class{red}{\text O}) ~~~ \text{B}(\class{red}{\text O}, \class{red}{\text O}) \end{align} $$

The next allowed microstate is where there are three particles in subsystem B:

$$ \begin{align} \text{A}(\class{red}{\text O}) ~~~ \text{B}(\class{red}{\text O}, \class{red}{\text O}, \class{red}{\text O}) \end{align} $$

And the last permitted microstate is where all four particles are in subsystem B:

$$ \begin{align} \text{A}(~) ~~~ \text{B}(\class{red}{\text O}, \class{red}{\text O}, \class{red}{\text O}, \class{red}{\text O}) \end{align} $$

This system has five possible microstates: $N = 5 $. With only four indistinguishable particles, a maximum of five microstates of this system can be realized.

Which of the three systems has the maximum and minimum entropy? The first system has a single microstate. This system has the lowest entropy $ S_1 $. The second system has three possible microstates. And the third system has five possible microstates. The third system therefore has the highest entropy $ S_3 $.

Entropy and the second law of thermodynamics

A fundamental law in physics is the second law of thermodynamics. It states the following:

The second law of thermodynamics

The entropy $S$ of an isolated system can only increase, but NEVER decrease.

In other words, the number of possible microstates of an isolated system can only increase, not decrease.

Example: Mixing water

Subsystem A contains hot water (fast particles). Subsystem B contains cold water (slow particles). The entropy of the overall system is minimal. Now the two subsystems are brought into thermal contact. Subsystem A becomes colder and subsystem B becomes warmer. The water temperatures in both subsystems equalize. According to the second law of thermodynamics, the entropy of the system has increased over time. The system that is in temperature equilibrium has the maximum entropy, that is it has the maximum number of possible microstates.

The second law of thermodynamics has a serious impact on the future of our universe. Because if the entire universe can be regarded as an isolated system, then entropy will increase over time until it reaches a maximum. At maximum entropy, the number of possible states of the universe will be maximum! And now philosophy comes into play...