Stochastic Emergence Lab

1. Underlying Stochastic Dynamics

The system is governed by multiplicative stochastic evolution of amplitudes:

with: $\bar{\lambda} = \sum_j |A_j|^2 \lambda_j$

2. Probability Mapping

$\boxed{ P_i = |A_i|^2 }$

This defines the measurable quantities of the system.

3. Induced Probability Dynamics

Applying Itô calculus:

where: $\mu(P_i) \propto P_i, \qquad D(P_i) \propto P_i^2$

4. Fokker–Planck Equation

The probability density evolves as:

5. Stationary Solution

At equilibrium: $\boxed{ \rho(P) \propto P }$.

This corresponds to the Born distribution.

6. Dynamical Convergence

The system evolves toward equilibrium: $\boxed{ \rho(P,t) \;\longrightarrow\; \rho_{\text{stationary}}(P) }$ independently of initial conditions.

7. Scaling Constraint

Generalized mapping: $P = |A|^\alpha$. Consistency requires: $\boxed{ \alpha = 2 }$

8. Symmetry Invariance

Under phase transformation $A \rightarrow A e^{i\theta}$: $\boxed{ P = |A|^2 \text{ remains invariant} }$

9. Uniqueness Condition

The stationary distribution satisfies

• drift–diffusion consistency,
• no-flux boundary conditions,
• and normalization.

10. Physical Interpretation

• Drift → structural selection
• Diffusion → stochastic exploration
• Equilibrium → balance of both

Axiomatic Uniqueness Theorem

Axiomatic Setup

Consider a system of complex amplitudes $A_i(t) \in \mathbb{C}$ evolving under stochastic dynamics of the form:

• The evolution is multiplicative, i.e. both deterministic and stochastic terms scale proportionally with $A_i$.
• The stochastic component is modeled by Wiener processes, ensuring well-defined continuous stochastic evolution.

Define a generalized probability mapping:

Axioms

The system is required to satisfy the following axioms:

Axiom 1 — Normalization
The total probability is conserved: $\sum_i P_i = 1 \quad \text{for all } t$

Axiom 2 — Markovian Stochastic Dynamics
The induced probability dynamics are Markovian and admit a Fokker–Planck description: $$\frac{\partial \rho}{\partial t} = -\frac{\partial}{\partial P}[\mu(P)\rho] + \frac{1}{2}\frac{\partial^2}{\partial P^2}[D(P)\rho]$$

Axiom 3 — Multiplicative Scaling
The induced drift and diffusion satisfy: $\mu(P) \propto P, \qquad D(P) \propto P^2$

Axiom 4 — Continuity and Positivity
The probability density satisfies: $\rho(P) \ge 0$ and $\rho(P)$ continuous on $[0,1]$

Axiom 5 — Boundary Condition (No-Flux)
The probability current vanishes at the boundaries: $J(0) = J(1) = 0$

Axiom 6 — Ergodicity
The stochastic process admits a unique stationary distribution: $\rho(P,t) \to \rho_{\text{stationary}}(P)$ independent of initial conditions.

Axiom 7 — Phase Invariance
Observable quantities depend only on the modulus: $P_i = f(|A_i|)$ and are invariant under: $A_i \to A_i e^{i\theta}$

Theorem (Uniqueness)

Under Axioms 1–7, the only admissible probability mapping of the form $P_i = |A_i|^\alpha$ that is consistent with the induced stochastic dynamics, normalization, boundary conditions, ergodicity, and symmetry is:

Proof

Step 1 — Induced Dynamics
From multiplicative amplitude dynamics, the induced probability dynamics satisfy $dP = \mu(P)\,dt + \sqrt{D(P)}\,dW$ with scaling: $\mu(P) \propto P, \quad D(P) \propto \alpha^2 P^2$

Step 2 — Stationary Fokker–Planck Equation
At equilibrium: $0 = -\frac{d}{dP}[\mu(P)\rho] + \frac{1}{2}\frac{d^2}{dP^2}[D(P)\rho]$

Step 3 — Zero-Flux Condition
$J(P) = \mu(P)\rho - \frac{1}{2}\frac{d}{dP}[D(P)\rho] = 0$

Step 4 — General Solution Form
The equation admits power-law solutions $\rho(P) = C P^k$ for some exponent $k$ determined by $\alpha$.

Step 5 — Boundary and Normalization Constraints
The conditions $\int_0^1 \rho(P)\,dP = 1$ and $J(0)=J(1)=0$ restrict admissible values of $k$, ensuring finiteness and stability.

Step 6 — Ergodicity Constraint
Ergodicity ensures that only one stationary solution is physically admissible.

Step 7 — Compatibility Condition
Consistency between multiplicative scaling ($D \propto P^2$), normalization, phase invariance, and stationary distribution requires: $k = 1$

Step 8 — Determination of $\alpha$
The relation between $k$ and $\alpha$ from the Fokker–Planck structure uniquely yields: $\alpha = 2$

Conclusion

The only probability mapping consistent with all axioms is: $\boxed{P_i = |A_i|^2}$