Thermodynamics of Information Processing

20.1 Landauer Bound in Practice

The minimum energy dissipation per irreversible bit operation depends on the physical implementation:

• CMOS transistor (2000s-era): $\sim 10^4\,k_BT$ per switch (vastly above the Landauer limit)
• Modern CMOS (7 nm node): $\sim 10^2$—$10^3\,k_BT$ per switch
• Adiabatic / reversible logic proposals: $\sim 1$—$10\,k_BT$ per operation (approaching the limit)

The gap between theory ($k_BT\ln 2 \approx 0.018$ eV at 300 K) and practice ($\sim 1$—$10$ fJ per switch) spans 5—6 orders of magnitude. Closing this gap requires fundamentally different computing paradigms.

20.2 Bennett”s Clock and Reversible Computing

Bennett (1982) showed that a computer can be made logically reversible at every step if it never erases information. Such a computer dissipates energy only during the initialisation of bits and during optional output, not during computation.

A logically reversible computation can be embedded in a thermodynamically reversible process by driving the system slowly enough that it remains near equilibrium at all times. The energy cost is then:

$E = \int_0^\tau \frac{\partial F}{\partial \lambda(t)}\,\dot{\lambda}(t)\,dt$

For a quasi-static process: $E \to \Delta F$ (minimum possible).

Fredkin and Toffoli gates are examples of logically reversible logic gates. Any computation can be made reversible by saving all intermediate results and running the computation in reverse to restore the input tape.

Worked Examples

Example 1: Carnot efficiency

Problem. A Carnot engine operates between $500 \mathrm{ K}$ and $300 \mathrm{ K}$. Find the maximum efficiency.

Solution. $\eta = 1 - T_C/T_H = 1 - 300/500 = 1 - 0.6 = 0.4 = 40\%$.

$\blacksquare$

Example 2: Entropy change

Problem. Find the entropy change when $2 \mathrm{ mol}$ of ice at $0°\mathrm{C}$ melts ($\Delta H_{\text{fus}} = 6.01 \mathrm{ kJ/mol}$).

Solution. $\Delta S = \frac{Q}{T} = \frac{n \Delta H_{\text{fus}}}{T} = \frac{2 \times 6010}{273} = 44.0 \mathrm{ J/K}$.

$\blacksquare$

Summary

• First law: $\Delta U = Q - W$; conservation of energy.
• Second law: $\Delta S_{\text{universe}} \geq 0$; entropy of an isolated system never decreases.
• Carnot efficiency: $\eta = 1 - T_C/T_H$ (maximum possible for given temperatures).
• Statistical mechanics: $S = k_B \ln \Omega$; Boltzmann distribution: $p_i \propto e^{-E_i/(k_BT)}$.

Cross-References