Theoretical model to deduce a PDF with a power law tail using Extreme Physical Information

The theory of Extreme Physical Information (EPI) is used to deduce a probability density function (PDF) of a system that exhibits a power law tail. The computed PDF is useful to study and fit several observed distributions in complex systems. With this new approach it is possible to describe extreme and rare events in the tail, and also the frequent events in the distribution head. Using EPI, an information functional is constructed, and minimized using Euler-Lagrange equations. As a solution, a second order differential equation is derived. By solving this equation a family of functions is calculated. Using these functions it is possible to describe the system in terms of eigenstates. A dissipative term is introduced into the model, as a relevant term for the study of open systems. One of the main results is a mathematical relation between the scaling parameter of the power law observed in the tail and the shape of the head.

💡 Research Summary

The paper presents a novel theoretical framework for deriving a probability density function (PDF) that simultaneously captures the frequent events in the head of a distribution and the extreme, power‑law‑type events in its tail. The authors employ the Extreme Physical Information (EPI) methodology, which builds on Fisher information, to construct an information functional that is minimized to obtain the governing equation of the system.

First, Fisher information for a shift‑invariant variable y is written as I=∫(g′(y))²/g(y) dy. Two observable variables, y and z, are introduced with a linear relationship z = t(y)·y, where t(y) is a piecewise constant function. The EPI functional is defined as F = I(y) – κ I(z), with κ measuring the discrepancy between measuring y and z. By expressing I(z) in terms of y, the Lagrangian becomes L = g′(y)²/g(y) – κ t(y)² g(y) y². Substituting g(y)=q(y)² simplifies the Lagrangian to L = 4 q′(y)² – κ t(y)² q(y)² y², leading to the Euler‑Lagrange equation

  4 q″(y) + κ t(y)² q(y) y² = 0.

Because t(y) is piecewise constant, the solution is also piecewise:

  q_i(y) = c_{i,1} y^{½+k_i} + c_{i,2} y^{½−k_i},

with k_i = ½ √(1 – κ t_i²). Boundary conditions (q(1)=0, q→0 as y→∞) force the tail solution to behave as q₂(y)² ∝ y^{−(α_tail+1)}, identifying the tail exponent α_tail = 2 k₂ – 1. In the head region the solution takes an oscillatory form q₁(y) = c₁ y^{½} sin(k₁ ln y). Continuity of q and q′ at the junction y = y₀ yields the transcendental relation

  k₁ = –k₂ tan(k₁ ln y₀).

This equation directly links the tail scaling parameter α_tail to the shape parameter k₁ governing the head, implying that once the tail exponent is known, the head can be described by a discrete set of eigenstates (different possible k₁ solutions).

To account for open systems that exchange information with their environment, a dissipative term H = β q′(y) y is added, modifying the equation of motion to

  4 q″ + κ t² q y² = β q′ y.

The solution now involves an additional exponent λ = 4 – β/8, giving q_i(y) = c_{i,1} y^{λ+k_i} + c_{i,2} y^{λ−k_i}. The sign of (λ – ½) determines whether the system gains information (λ < ½), loses information (λ > ½), or is in equilibrium (λ = ½), providing a thermodynamic interpretation via Fisher information.

The authors validate the model on four empirical data sets: (1) number of customers affected by US power outages, (2) US city populations from the 2000 Census, (3) peak solar flare gamma‑ray intensities, and (4) simulated world‑wealth distribution from an agent‑based model. For each set, the tail exponent α_tail and the lower cutoff y₀ are estimated by maximum likelihood. Using the derived relation, possible k₁ values are computed, and the head is fitted as a linear combination of the first two eigenstates. Goodness‑of‑fit is assessed with the Kolmogorov‑Smirnov statistic, which remains below the critical value for all cases, demonstrating that the model captures both head and tail accurately.

In the discussion, the paper contrasts its approach with conventional practices that either discard extreme events or ignore the head, arguing that such truncations lose essential information about the underlying dynamics. By providing a unified, analytically derived PDF with interpretable parameters, the model offers insights into the microscopic mechanisms that generate power‑law tails while preserving the full distribution. The piecewise nature of t(y) can be extended to multiple segments, allowing the description of distributions with several scaling regimes in the tail.

Overall, the work showcases how EPI, a principle rooted in information theory, can be leveraged to derive a physically motivated, flexible probability model that bridges frequent and rare events, incorporates dissipation, and yields testable relations between tail exponents and head shapes—features that are valuable for the analysis of complex systems across physics, economics, and social sciences.