Physics
The principles of physical reversibility
Reversibility runs through physics at two levels: the time-reversal symmetry of the microscopic laws, and the idealised, entropy-conserving processes of thermodynamics. Where the two meet, we find entropy, the arrow of time, and the deep link between information and heat.
What reversibility means in thermodynamics
A reversible process is an idealisation carried out quasi-statically — infinitely slowly, through a continuous sequence of equilibrium states — so that it can be run backwards through exactly the same states, returning both the system and its surroundings to their initial conditions with no net change. In this limit, total entropy stays constant.
Real processes are never like this. They proceed at finite rates and involve friction, mixing and heat flowing across finite temperature differences, all of which generate entropy. For any real process the total entropy increases, ΔStotal > 0, with equality holding only in the reversible limit. Entropy production is therefore the quantitative measure of irreversibility, and it defines a thermodynamic arrow of time. See reversible processes and the second law of thermodynamics.
Loschmidt's paradox and its resolution
The microscopic equations of motion are time-reversal symmetric: reverse every velocity and the laws still hold. Yet macroscopic behaviour is manifestly not symmetric — cream mixes into coffee, never the reverse. This tension is Loschmidt's paradox.
The modern resolution is statistical. Given a low-entropy initial condition, coarse-grained evolution toward higher-entropy macrostates is overwhelmingly probable, simply because such macrostates correspond to vastly more microstates. This intuition is made quantitative by the fluctuation theorems of Jarzynski (1997) and Crooks (1999), which show precisely how rare entropy-decreasing fluctuations are.
Information is physical — Landauer's principle
In 1961 Rolf Landauer showed that erasing one bit of information must dissipate at least kT ln 2 of heat — about 2.9 × 10−21 J, or roughly 0.018 eV, at around 300 K. Erasure is logically irreversible: it maps two distinct states onto one, compressing phase space and reducing information entropy by k ln 2. To satisfy the second law, that entropy must be exported to the surroundings as heat.
Charles Bennett established the converse in 1973: logically reversible computation, which discards no information, carries no such lower bound. There is no fundamental thermodynamic cost to computing as long as nothing is erased. See Landauer's principle.
Exorcising Maxwell's demon
Maxwell's demon (1867) appears to violate the second law by sorting fast and slow molecules to create a temperature difference, seemingly lowering entropy for free. The resolution is that the demon must store the information it measures. To keep operating it must eventually erase that record to reset its memory — and, by Landauer's principle, erasing it dissipates at least kT ln 2 of heat per bit, exactly enough to rescue the second law. This is Bennett's resolution, linking the demon directly back to computation. See Maxwell's demon.
The same kT ln 2 bound is why our Programming programme takes reversible computing seriously: if no information is erased, computation can in principle approach zero energy.