# Entropy

**Entropy** is the amount of disorder or molecular chaos in a system.

The **microscopic** way to measure the disorder is by looking at the individual pieces of the system (i.e. the microstates) and counting the number of ways a system can be arranged in order to reach its current macrostate. The different possible arrangements are called the microstates.

The **macroscopic** definition describes the change in the entropy of the system in terms of the amount of heat required to raise the temperature of a system one degree.

## Microscopic Definition

The

microscopic definitionlooks at the individual components of a system in order to study the system as a whole:

- $S=k_B \ln\Omega$
- $k_B =$ Boltzmann constant
- $\Omega =$ Multiplicity of the macrostate of the system, that is, the number of microstates corresponding to the given macrostate

A box contains two particles. Does it have more entropy when it has 0, 1 or 2 units of energy?

Total Energy Possible Microstates Number of Microstates Entropy 0 units $E_1=0, E_2=0$ $\hspace{15mm}$ 1 $k_B \ln(1)=0$ 1 unit $E_1=1, E_2=0$ $E_1=0, E_2=1$ $\hspace{15mm}$ 2 $k_B \ln(2)$ 2 units $E_1=2, E_2=0$ $E_1=1, E_2=1$ $E_1=0, E_2=2$ $\hspace{15mm}$ 3 $k_B \ln(3)$ The system has the most entropy when it has 2 units of energy.

Rudolph the red-nosed, statistical mechanist is trying to maximize the entropy of his system. He can either fill his box with 2 particles and 3 unit of energy or 3 particles and 2 units of energy. Assuming that a unit of energy is quantized (i.e. a particle cannot have a part of a unit of energy) and that all energy is assigned to the particles, which arrangement provides for more entropy?

## Macroscopic Definition

The

macroscopic definitionlooks at the overall state of a system in order to study it:$\Delta S = \int \frac{dQ_{\text{rev}}}{T} .$

One way of understanding the second law is that the entropy (in an isolated system) tends to increase, that is, a system tends to be in a macrostate of maximum multiplicity.

Consider a partitioned box with one partition filled with a hot gas and another with a relatively cooler gas. The hotter portion has a relatively higher number of molecules with higher velocity compared to the colder one.

When the partition is removed, both come to thermal equilibrium. This is because now the faster and the slower molecules mix up and are evenly distributed.