Entropy and the Second Law

Clausius’s entropy

For a reversible process in which heat $\delta Q_\text{rev}$ flows into the system at temperature $T$:

$$dS = \frac{\delta Q_\text{rev}}{T}.$$

This is a state function — for any cycle, $\oint dS = 0$.

The second law: equivalent statements

1. Clausius: heat does not spontaneously flow from cold to hot.
2. Kelvin–Planck: no heat engine, drawing on a single reservoir, can convert all heat into work.
3. Entropy version: the entropy of an isolated system never decreases:

$$\Delta S_\text{tot} \ge 0,$$

with equality only for reversible processes.

Boltzmann’s statistical definition

In terms of the number of microstates $W$ consistent with the macrostate:

$$\boxed{\; S = k_B \ln W \;}$$

(the formula carved on Boltzmann’s tombstone in Vienna). Consistent with Clausius’s thermodynamic definition.

Example: free expansion of an ideal gas

Volume $V_1 \to V_2 = 2V_1$ at constant temperature. The number of microstates scales as $W \propto V^N$, so

$$\Delta S = N k_B \ln\frac{V_2}{V_1} = N k_B \ln 2 > 0.$$

Irreversible, and entropy increases.

Carnot cycle and maximum efficiency

A reversible heat engine operating between reservoirs at $T_H$ and $T_C$ achieves

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

No engine can do better without violating the second law.

Information entropy (Shannon)

For a probability distribution $\{p_i\}$:

$$S = -k_B \sum_i p_i \ln p_i.$$

Equiprobable microstates reduce this to $S = k_B \ln W$. Entropy is information missing about the microstate.

Quantum: von Neumann entropy

For a density matrix $\hat\rho$,

$$S(\hat\rho) = -k_B \,\mathrm{tr}(\hat\rho \ln \hat\rho).$$

Under unitary evolution from the Schrödinger equation, the entropy of a pure state is conserved — but a subsystem’s entropy generally grows (entanglement entropy, information leak into the environment).

The arrow of time

The fundamental laws of physics are time-reversible, yet entropy growth picks out a direction. The reconciliation: the universe was born in an extraordinarily low-entropy state. Cosmology, not microphysics, supplies the arrow.