Emergence of order in selection-mutation dynamics.

Area of Science:

• Quantum Information Theory
• Statistical Mechanics
• Dynamical Systems

Background:

• Characterizing the time evolution of probability distributions is crucial in various scientific fields.
• Entropy serves as a key metric to classify these evolutions as either mixing or purifying.
• Understanding the influence of system dynamics on entropy is essential for predicting system behavior.

Purpose of the Study:

• To investigate the general features of time evolution for d-dimensional probability distributions governed by a simple nonlinear equation.
• To analyze how the distribution of final entropy values is affected by random matrix elements.
• To explore the impact of restricting dynamical matrices to specific subspaces (e.g., diagonal, triangular) on entropy evolution.

Main Methods:

• Modeling the time evolution of a d-dimensional probability distribution using a nonlinear equation with a d x d matrix as input.
• Assigning uniformly distributed random numbers to matrix elements within a specified range [0, upper bound].
• Conducting computer simulations to observe the distribution of final entropy values across a field of random matrices.

Main Results:

• For random matrix elements with an upper bound of unity, the final entropy distribution concentrates at the maximum possible value, indicating mixing dynamics.
• When dynamical matrices are restricted to specific regions, such as diagonal or triangular matrices, the entropy distribution peaks near zero.
• Restricted matrix dynamics predominantly result in purifying evolutions.

Conclusions:

• The nature of the dynamical matrix significantly influences the final entropy of a probability distribution.
• Random matrix elements tend to drive systems towards maximum entropy (mixing).
• Specific matrix constraints can lead to purifying dynamics, characterized by low final entropy.