Second Law of Thermodynamics and Entropy

The Arrow of Time

Break an egg, stir cream into coffee, let perfume fill a room. You never see the reverse, though mechanics allows it. The second law governs this, driven by sheer probability.

Disordered arrangements so vastly outnumber ordered ones that with $10^{23}$ particles, the unlikely becomes impossible.

Core Statements

Entropy is time's compass. Entropy measures disorder, defined for reversible paths as $\Delta S = \int \frac{dQ_{\text{rev}}}{T}$. For isolated systems, $\Delta S \geq 0$.

Engines and Microstates

The Carnot engine sets the absolute efficiency limit for engines operating between $T_H$ and $T_C$:

$$\eta_{\text{Carnot}} = 1 - \frac{T_C}{T_H}$$

No real engine can exceed this value. Boltzmann linked entropy to statistics via Boltzmann's entropy formula:

$$S = k_B\ln\Omega$$

Here, $\Omega$ is the number of microstates. Systems naturally evolve toward the macrostate with the largest $\Omega$.

Common Pitfall

Entropy can decrease locally, like inside your freezer, provided more entropy is exported to the surroundings. The second law binds only the total system plus surroundings, never a subsystem alone.