# How to Win Friends and Influence Functionals: Deducing Stochasticity From Deterministic Dynamics

## TL;DR

This paper reviews how the influence functional formalism can derive stochastic behavior from deterministic Hamiltonian dynamics, bridging quantum and classical regimes and providing a rigorous foundation for stochastic equations of motion.

## Contribution

It introduces a formalism that derives stochastic equations directly from microscopic Hamiltonians, unifying quantum and classical descriptions of open systems.

## Key findings

- Stochastic terms arise exactly from the influence functional formalism.
- The classical limit of quantum stochastic dynamics leads to a generalized Langevin equation.
- The approach unifies open quantum systems with classical stochastic models.

## Abstract

The longstanding question of how stochastic behaviour arises from deterministic Hamiltonian dynamics is of great importance, and any truly holistic theory must be capable of describing this transition. In this review, we introduce the influence functional formalism in both the quantum and classical regimes. Using this technique, we demonstrate how irreversible behaviour arises generically from the reduced microscopic dynamics of a system-environment amalgam. The influence functional is then used to rigorously derive stochastic equations of motion from a microscopic Hamiltonian. In this method stochastic terms are not identified heuristically, but instead arise from an exact mapping only available in the path-integral formalism. The interpretability of the individual stochastic trajectories arising from the mapping is also discussed. As a consequence of these results, we are also able to show that the proper classical limit of stochastic quantum dynamics corresponds non-trivially to a generalised Langevin equation derived with the classical influence functional. This provides a further unifying link between open quantum systems and their classical equivalent, highlighting the utility of influence functionals and their potential as a tool in both fundamental and applied research.