Drift Theory: Mathematical Model

2. Configuration Space of Experiential Frames

2.1 Experiential manifold

Let $\mathcal{X}$ denote a high-dimensional manifold representing the space of experiential frames. A point $x \in \mathcal{X}$ encodes a coarse-grained description of what it is like for a subject to inhabit a given moment, including:

  • perceptual interpretation of the environment,
  • autobiographical memory configuration (what is taken as past),
  • relational and affective context (who matters, in what way),
  • implicit expectations and predictions about near-future events.

Formally, one may write $$x = (q, m, r),$$ where $q$ represents coarse physical context (e.g. location, posture, visible surroundings), $m$ represents memory and self-narrative, and $r$ represents relational and emotional ties.

The manifold $\mathcal{X}$ is equipped with a metric $$d : \mathcal{X} \times \mathcal{X} \to \mathbb{R}_{\ge 0}$$ capturing how different two experiential frames are. Components of $d$ may weight physical similarity, memory differences, and relational differences differently.

2.2 Trajectory of subjective experience

A subject's lived reality over time is modeled as a continuous trajectory $$\gamma: \mathbb{R} \to \mathcal{X}, \quad t \mapsto X(t).$$ Here, $t$ denotes subjective time, and $X(t)$ is the experiential frame at time $t$. The continuity of $\gamma$ formalizes the intuition that experience usually changes smoothly: there are no instantaneous jumps between radically incompatible frames under ordinary conditions.

3. Drift Dynamics in Experience-Space

To describe how $X(t)$ evolves, Drift Theory introduces a stochastic dynamical equation combining structural constraints, emotional modulation, and random fluctuations.

3.1 Structural continuity potential

Let $\Phi: \mathcal{X} \to \mathbb{R}$ be a continuity potential. Intuitively, $\Phi$ penalizes frames that are hard to reach from the current one without violating basic physical and cognitive consistency (e.g. abrupt violation of physical laws, discontinuous memory changes with no narrative bridge). The gradient $\nabla \Phi(x)$ encodes directions in which experiential change is "costly" or "implausible", encouraging the trajectory to follow smoother evolution.

3.2 Affective and relational potential

Let $E(t) \in \mathbb{R}^k$ represent the subject's emotional state vector at time $t$, with components corresponding to affective dimensions such as fear, joy, shame, attachment, guilt, or love. Define the emotional intensity as $$e(t) = \lVert E(t) \rVert.$$

Let $C$ denote a set of relatively stable commitment parameters, encoding long-term attachments, values, and promises (e.g. commitment to a partner, moral principles). These shape how the subject evaluates frames in terms of coherence with those commitments.

Define an affective–relational potential $$\Psi: \mathcal{X} \times \mathcal{C} \to \mathbb{R}, \quad (x, C) \mapsto \Psi(x; C),$$ lower for frames that align with commitments and emotionally preferred outcomes, higher for frames that represent betrayals or ruptures of those commitments.

3.3 Drift equation

The dynamics of $X_t = X(t)$ are modeled as a Langevin-type stochastic differential equation (SDE):

$$ dX_t = -\nabla \Phi(X_t)\,dt - \alpha e(t) \nabla \Psi(X_t;C)\,dt + \sqrt{2D (1 + \kappa e(t))}\, dW_t. $$

Here:

  • $\alpha > 0$ controls the strength with which emotional intensity biases drift along $-\nabla \Psi$;
  • $D > 0$ is a baseline diffusion coefficient capturing everyday micro-variability in experience;
  • $\kappa \ge 0$ governs how emotion amplifies experiential variability;
  • $W_t$ is a Wiener process on $\mathcal{X}$ (Brownian noise).

6. Probability Distributions over Experiential Versions

In addition to a single realized trajectory, Drift Theory can consider a probability distribution $\rho(x,t)$ over $\mathcal{X}$, representing the density of possible experiential frames at time $t$ given the stochastic dynamics. The SDE for $X_t$ induces a corresponding Fokker–Planck equation:

$$ \frac{\partial \rho}{\partial t}(x,t) = \nabla \cdot \Big[ \rho(x,t) \, \nabla (\Phi(x) + \alpha e(t) \Psi(x;C)) \Big] + D (1 + \kappa e(t)) \, \Delta \rho(x,t). $$

This equation describes how probability mass flows across $\mathcal{X}$ over time as a function of baseline continuity constraints ($\Phi$), emotional and commitment influences ($\Psi$), and diffusion modulated by emotional intensity.