What is the Mathematical Definition of Negentropy vs Entropy?

There is no single, universally accepted mathematical definition of negentropy, but it is always defined in relation to entropy. The two main interpretations come from thermodynamics (Boltzmann) and information theory (Shannon).

1. Thermodynamic Entropy (Boltzmann–Gibbs)

In thermodynamics, entropy (S) represents the measure of microscopic disorder in a system. It quantifies the number of possible microscopic arrangements (microstates, W) corresponding to a given macroscopic state.

Boltzmann formula:

S = k_B × ln(W)

• S — Entropy of the system
• k_B — Boltzmann constant (a universal physical constant)
• W — Number of microscopic states

A larger W means more possible arrangements, i.e., greater disorder and higher entropy. A smaller W means fewer arrangements, i.e., more order and lower entropy.

2. Negentropy Definition (Brillouin)

The concept of negentropy (N) was introduced by physicist Léon Brillouin as the opposite of entropy.

N = –S

A more practical definition compares a system’s actual entropy to its maximum possible entropy:

N = S_max – S_actual

• N — Negentropy (measure of order or information)
• S_max — Maximum possible entropy (total disorder)
• S_actual — Actual entropy of the system

Thus, negentropy indicates how far a system is from complete randomness. Greater negentropy means higher order; zero negentropy means total disorder.