What Is the Statistical Definition of Entropy (S = kB ln Ω)?

For IB Physics HL: The statistical definition of entropy is

$$S = k_B \ln \Omega$$

where $k_B$ is the Boltzmann constant and $\Omega$ is the number of possible microscopic arrangements consistent with the system's macroscopic state. A macrostate with more possible arrangements has greater entropy.

The Statistical Mechanism

A macrostate is described by measurable properties such as pressure, volume and temperature. A microstate specifies the detailed arrangement and energy of every particle. Many different microstates can produce the same macrostate.

Suppose a system changes from $\Omega_1=2$ to $\Omega_2=8$. Its entropy change is

$$\Delta S = k_B\ln\left(\frac{\Omega_2}{\Omega_1}\right) = (1.38\times10^{-23})\ln(4) =1.91\times10^{-23}\,\text{J K}^{-1}.$$

The entropy increases because the final macrostate can be produced by four times as many microstates. For an isolated system, spontaneous processes tend toward the macrostate with the greatest multiplicity because it is overwhelmingly the most probable. This statistical tendency explains the second law of thermodynamics: the entropy of a real isolated system tends to increase.

For example, consider four distinguishable particles shared between two equal compartments. The macrostate with two particles on each side has more arrangements than a macrostate with all four particles on one side. It is therefore more probable and has greater entropy, even though no physical law forbids the less probable arrangement.

Because $\Omega$ is dimensionless, the logarithm is valid and $k_B$ supplies entropy's unit. Each accessible microstate is treated as equally probable.

The logarithm also makes entropy additive. If two independent systems have multiplicities $\Omega_A$ and $\Omega_B$, their combined multiplicity is $\Omega_A\Omega_B$, so .

A common misconception is that entropy simply means “disorder.” Disorder is only an informal analogy; the precise statistical meaning is the logarithm of the number of accessible microstates.

Exam Technique

This is B.4 Thermodynamics (HL only). In an Explain question, link increasing $\Omega$ to increasing probability and therefore increasing entropy. In calculations, use natural logarithms and state the unit $\text{J K}^{-1}$.