Statistical theory of self-similarly distributed fields

TL;DR

This paper develops a field theory for self-similar statistical systems using Mellin transforms and Jackson derivatives, revealing the role of a fluctuating order parameter as a deformed logarithm of hydrodynamic modes.

Contribution

It introduces a novel theoretical framework combining Mellin transforms and Jackson derivatives to analyze self-similar statistical fields.

Findings

Derived deformed partition function and moments of the order parameter.

Formulated equations for the generating functional considering system constraints.

Established a connection between fluctuating order parameters and hydrodynamic modes.

Abstract

A field theory is built for self-similar statistical systems with both generating functional being the Mellin transform of the Tsallis exponential and generator of the scale transformation that is reduced to the Jackson derivative. With such a choice, the role of a fluctuating order parameter is shown to play deformed logarithm of the amplitude of a hydrodynamic mode. Within the harmonic approach, deformed partition function and moments of the order parameter of lower powers are found. A set of equations for the generating functional is obtained to take into account constraints and symmetry of the statistical system.