Entropialla

Entropialla is a hypothetical scalar quantity used in theoretical discussions of information and complex systems. It is defined to combine Shannon entropy with a structural factor that accounts for correlations and spatial or relational patterns within a system. The term is not widely adopted in established physics, but it appears in thought experiments and exploratory work on how order and randomness interact.

Definition and formulation. For a system with a discrete set of states i with probabilities p_i, the Shannon entropy is H(p) = - sum_i p_i log p_i. Entropialla is defined as S_e = H(p) · F, where F is a dimensionless structural factor in [0,1] representing the degree of organization beyond the distribution of state probabilities. A common choice is F = 1 - C̄, where C̄ is the average pairwise mutual information between neighboring or related states, normalized by its maximum possible value for the given system.

Properties. Entropialla is nonnegative and bounded above by H(p). It vanishes when there is no uncertainty (H(p) = 0) or when structural correlations saturate (F = 0). It reaches the value H(p) when state probabilities are maximally uncertain and the system shows no extra structure beyond the distribution.

Examples. In a uniform distribution over N states with no detectable correlations, S_e ≈ log N (in the chosen logarithmic units) when F ≈ 1. If the same distribution exhibits strong, predictable correlations, F < 1 and S_e is reduced, reflecting the added order despite the same state probabilities.

Applications and notes. Potential uses include comparing complexity in ecological networks, communication systems, and social dynamics where both diversity and organization matter. As a speculative construct, practical estimation depends on choices about neighborhoods, normalization, and the logarithm base.