# A Theory of Complex Stochastic Systems with Two Types of Counteracting Entities

## TL;DR

This paper develops a theoretical framework for analyzing complex stochastic systems with two counteracting types of entities, demonstrating conditions for equilibrium and stability based on their abundance and effects.

## Contribution

It introduces a novel theory combining kinetic and energetic equilibrium notions for systems with two intertwined counteracting constituents.

## Key findings

- Existence of a point satisfying kinetic equilibrium for less abundant entities.
- Regulation of more abundant entities to minimize energetic equilibrium.
- Unbounded growth of stabilizing/destabilizing entities affects system stability.

## Abstract

Many complex systems share two characteristics: 1) they are stochastic in nature, and 2) they are characterized by a large number of factors. At the same time, various natural complex systems appear to have two types of intertwined constituents that exhibit counteracting effects on their equilibrium. In this study, we employ these few characteristics to lay the groundwork for analyzing such complex systems. The equilibrium point of these systems is generally studied either through the kinetic notion of equilibrium or its energetic notion, but not both. We postulate that these systems attempt to regulate the state vector of their constituents such that both the kinetic and the energetic notions of equilibrium are met. Based on this postulate, we prove: 1) the existence of a point such that the kinetic notion of equilibrium is met for the less abundant constituents and, at the same time, the state vector of more abundant entities is regulated to minimize the energetic notion of equilibrium; 2) the effect of unboundedly increasing less (more) abundant constituents stabilizes (destabilizes) the system; and 3) the (unrestricted) equilibrium of the system is the point at which the number of stabilizing and destabilizing entities increase unboundedly with the same rate.