Second Law of Thermodynamics

Idea

Unlike the first law of thermodynamics (which follows directly from energy conservation and entropy maximization under constraints), the Second Law is a statistical statement about the direction of physical processes:

\frac{d S}{d t} \geq 0

for a closed system.

It says: the entropy of an isolated system never decreases.

Classical Statistical Mechanics Perspective

Microscopic dynamics (classical Hamiltonian or quantum unitary evolution) are time-reversal symmetric. They do not themselves force entropy to increase.
Nevertheless, in statistical mechanics we distinguish between:
- microstates: detailed states of the system.
- macrostates: coarse-grained descriptions (energy, volume, particle number, etc.).
The key fact: there are far more microstates compatible with high entropy than with low entropy. Indeed, entropy is a kind of measure of the compatible microstates.
If the system starts in a special low-entropy macrostate, then under time evolution it is overwhelmingly likely to move into macrostates of higher entropy.
Thus, entropy increase is not an absolute law but an almost certain statistical tendency, given appropriate initial conditions.

Arrow of Time Problem

Because microscopic laws are reversible (Liouville dynamics), there always exists a time-reversed trajectory where entropy decreases. The Second Law is not a mathematical consequence of the laws alone, but of the special initial state of our universe (low entropy at the Big Bang).

This is the source of the arrow of time problem: why does the future differ from the past?

Consequence: Heat Flow

Consider two systems in equilibrium, with entropies $S_{1}, S_{2}$ , expected energies $⟨ E_{1} ⟩, ⟨ E_{2} ⟩$ , and temperatures $T_{1}, T_{2}$ . We have total entropy $S = S_{1} + S_{2}$ and total energy $E = E_{1} + E_{2}$ .

The Second Law implies that when they exchange energy while the total energy is fixed,

\frac{d S}{d t} = \frac{1}{T_{1}} \frac{d ⟨ E_{1} ⟩}{d t} + \frac{1}{T_{2}} \frac{d ⟨ E_{2} ⟩}{d t} \geq 0.

This leads to the condition:

(\frac{1}{T_{1}} - \frac{1}{T_{2}}) \frac{d ⟨ E_{1} ⟩}{d t} \geq 0,

since $\frac{d ⟨ E ⟩}{d t} = 0$ . The condition means:
energy flows spontaneously from the hotter system to the colder one.

Related: first law of thermodynamics.