Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Exact $q$-exponential Multi-Mode Solutions with Independent Centres and Power-Law Relaxation in the Plastino-Plastino Equation

Airton Deppman

TL;DR

This work derives the first exact, multimode solutions to the Plastino--Plastino nonlinear diffusion equation with arbitrary power-law drift by introducing independent time-dependent centres for each mode, yielding fully separable dynamics for centres and probabilities. A single attractor mode with fixed amplitude absorbs probability flux from transient modes, driving the system toward the stationary $q$-exponential state while transient modes decay with exact power-law relaxation and hold constant widths. The framework unifies Tsallis nonextensive thermodynamics, fractal diffusion, and multi-scale relaxation, with direct relevance to heavy-quark jets in quark-gluon plasma, Lévy flights in fractal media, and urban population redistribution. All previously known exact results emerge as special cases, and the approach opens avenues for higher-dimensional and multi-species generalizations along with data-driven validation in complex systems.

Abstract

We present the first exact, multi-mode solutions to the Plastino-Plastino nonlinear diffusion equation with arbitrary power-law drift. By allowing each $q$-exponential mode to have its own independent, time-dependent centre, all inter-mode couplings in the drift term vanish, yielding fully separable evolution equations for centre motion, probability content, and (for the attractor mode) width. Transient modes exhibit constant width and decay via exact q-exponential (power-law) relaxation, while a single attractor mode irreversibly absorbs the entire probability flux, with fixed amplitude and time-growing width, driving the system to the known stationary q-exponential state from arbitrary initial conditions. The hierarchy closes exactly without approximation. These analytic solutions unify Tsallis nonextensive thermodynamics, fractal-space diffusion, and multi-scale relaxation dynamics, with direct applications to heavy-quark jets in quark-gluon plasma, Lévy flights in fractal media, and urban population redistribution. All previous exact results are recovered as special cases.

Exact $q$-exponential Multi-Mode Solutions with Independent Centres and Power-Law Relaxation in the Plastino-Plastino Equation

TL;DR

This work derives the first exact, multimode solutions to the Plastino--Plastino nonlinear diffusion equation with arbitrary power-law drift by introducing independent time-dependent centres for each mode, yielding fully separable dynamics for centres and probabilities. A single attractor mode with fixed amplitude absorbs probability flux from transient modes, driving the system toward the stationary -exponential state while transient modes decay with exact power-law relaxation and hold constant widths. The framework unifies Tsallis nonextensive thermodynamics, fractal diffusion, and multi-scale relaxation, with direct relevance to heavy-quark jets in quark-gluon plasma, Lévy flights in fractal media, and urban population redistribution. All previously known exact results emerge as special cases, and the approach opens avenues for higher-dimensional and multi-species generalizations along with data-driven validation in complex systems.

Abstract

We present the first exact, multi-mode solutions to the Plastino-Plastino nonlinear diffusion equation with arbitrary power-law drift. By allowing each -exponential mode to have its own independent, time-dependent centre, all inter-mode couplings in the drift term vanish, yielding fully separable evolution equations for centre motion, probability content, and (for the attractor mode) width. Transient modes exhibit constant width and decay via exact q-exponential (power-law) relaxation, while a single attractor mode irreversibly absorbs the entire probability flux, with fixed amplitude and time-growing width, driving the system to the known stationary q-exponential state from arbitrary initial conditions. The hierarchy closes exactly without approximation. These analytic solutions unify Tsallis nonextensive thermodynamics, fractal-space diffusion, and multi-scale relaxation dynamics, with direct applications to heavy-quark jets in quark-gluon plasma, Lévy flights in fractal media, and urban population redistribution. All previous exact results are recovered as special cases.
Paper Structure (17 sections, 90 equations, 2 figures, 1 table)

This paper contains 17 sections, 90 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Theoretical Preliminaries
  3. Stationary solution
  4. Time-Dependent Single-Mode Solution
  5. $q$-Gaussian Dynamics
  6. Multi-mode solution
  7. Stretched $q-$-exponentials
  8. Centre-of-mass motion (per mode)
  9. Width evolution
  10. Amplitude and probability transfer
  11. Attractor mode
  12. Special Cases for Specific Drift Exponents
  13. Linear Drift ($\alpha = 1$, $\beta_0 = 2$): q-Gaussian Attractor
  14. Constant Drift ($\alpha = 0$, $\beta_0 = 1$): q-Exponential Attractor
  15. Fractional Drift ($\alpha = -0.5$, $\beta_0 = 0.5$): Sublinear Attractor
  16. ...and 2 more sections

Figures (2)

  • Figure 1: Example of application of the multi-modal solution. The hypothetical case of a socioeconomic output (wealth, skill, social influence, opinion, etc.) in an initially distorted probability density (blue line at $t=0$), which evolves in time reaching, after some time (in arbitrary units), the stationary regime in red ($t=32$). The example uses six modes: the attractor (mode 0) and five transient modes. The fit of the initial distribution (target) can be improved by including additional modes. See the text for details on the parameters used.
  • Figure 2: Time evolution of the modes and of the total probability density. The dynamical evolution time (arbitrary units) is: $t=0.0$ (top-left); $t=0.5$ (top-right); $t=2.0$ (bottom-left); and $t=10.0$ (bottom-right). The darker line represents the sum of the components, which are shown in the grey lines.