ἐν + τροπή

Entropy was coined in 1865 by the German physicist Rudolf Clausius from Greek ἐν (en, 'in') and τροπή (tropē, 'transformation, turning, change'), producing a word meaning roughly 'the content of transformation' or 'that which turns.' Clausius was deliberate in his choice of Greek roots: he wanted entropy to parallel 'energy' (from Greek ἐνέργεια, energeia, 'activity, operation'), a word that had been coined a century earlier from Greek en + ergon (work). Energy and entropy were to be companion concepts, both measured in thermodynamic systems, both expressed in Greek. Clausius wrote: 'I have intentionally formed the word entropy so as to be as similar as possible to the word energy; for the two magnitudes to be denoted by these words are so nearly allied in their physical meanings, that a certain similarity in designation appears to be desirable.' The naming was deliberate symmetry.

The concept of entropy emerged from Clausius's work on the second law of thermodynamics, a law he formulated with increasing precision through the 1850s and 1860s. The first law of thermodynamics states that energy is conserved — it cannot be created or destroyed, only transformed. The second law adds a crucial asymmetry: heat flows spontaneously from hot to cold, not from cold to hot. Perpetual motion machines of the second kind — devices that extract work from a single heat reservoir without degrading it — are impossible. Clausius sought a mathematical quantity that captured this asymmetry. Entropy was his answer: in any reversible thermodynamic process, entropy remains constant; in any irreversible process (which is every real process), entropy increases. The entropy of an isolated system never decreases. This is the second law stated as a formula: ΔS ≥ 0.

The statistical interpretation of entropy came from Ludwig Boltzmann in 1877, in one of the most profound single equations in the history of physics: S = k log W. In this formula, S is entropy, k is the constant now named after Boltzmann, and W (from German Wahrscheinlichkeit, 'probability') is the number of microscopic configurations of a system that correspond to its macroscopic state. Entropy, Boltzmann revealed, is a measure of disorder or probability: a high-entropy state is one that can be realized in many different microscopic arrangements; a low-entropy state is one that can be realized in only a few. The reason entropy increases is not a fundamental law of physics at the microscopic level (the underlying equations are time-symmetric) but a consequence of probability: high-entropy states are vastly more numerous than low-entropy states, so any random process is overwhelmingly likely to move toward higher entropy.

Claude Shannon's 1948 paper 'A Mathematical Theory of Communication' extended entropy into information theory. Shannon needed a measure of the average information content of a message — how much uncertainty a message resolves. He defined information entropy using the same logarithmic formula as Boltzmann's thermodynamic entropy: H = −Σ p log p. The story goes that Shannon showed the formula to John von Neumann, who said: 'You should call it entropy, for two reasons. In the first place your uncertainty function has been used in statistical mechanics under that name, so it already has a name. In the second place, and more important, no one knows what entropy really is, so in a debate you will always have the advantage.' Whether the story is true or apocryphal, the name stuck, and thermodynamic entropy and information entropy have been in productive conversation ever since.