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Chapter 29
Statistical Distributions (Boltzmann, Fermi–Dirac)

Thermal and Entropic Flows
Author: Mark Anthony Higgins
© Mark Anthony Higgins 11 October 2025
Revised Date: 23 February 2026
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Flow Algebra 
flow "Statistical Distributions — Boltzmann, Fermi–Dirac" v1.0

# ---------- PURPOSE ----------
# Defines classical and quantum particle distributions as boundary population flows.
# Links occupancy probability to energy, temperature, and coherence index (IB).
# Demonstrates how Boltzmann, Fermi–Dirac, and Bose–Einstein forms emerge from the same
# exponential information-entropy structure.

# ---------- PARAMETERS ----------
params:
  units: "SI"
  k_B: 1.380649e-23       # Boltzmann constant [J/K]
  T: positive              # system temperature [K]
  mu: real                 # chemical potential [J]
  E_i[]: real              # energy levels [J]
  statistics: ["MB","FD","BE"]  # Maxwell–Boltzmann, Fermi–Dirac, Bose–Einstein
  IB: 0.0..1.0             # boundary coherence index (1 = equilibrium)
  include_coherence_term: true

# ---------- STATE ----------
state System:
  p_i[]: real              # occupancy probabilities per state
  Z: real                  # partition function
  n_i[]: real              # expected occupancy numbers
  S_phys: real             # entropy [J/K]
  D_info: real             # information divergence
  sigma: real              # entropy production rate

# ---------- INVARIANTS ----------
invariants:
  # --- 1. Partition function ---
  Z = Σ_i( exp(-(E_i - mu) / (params.k_B * params.T)) )

  # --- 2. Distributions (normalised probabilities) ---
  if params.statistics == "MB":
    p_i = (1/Z) * exp(-(E_i - mu) / (params.k_B * params.T))

  if params.statistics == "FD":
    p_i = 1 / ( exp((E_i - mu) / (params.k_B * params.T)) + 1 )

  if params.statistics == "BE":
    p_i = 1 / ( exp((E_i - mu) / (params.k_B * params.T)) - 1 )

  # --- 3. Coherence-corrected occupancy (optional) ---
  if params.include_coherence_term:
    p_i = p_i * params.IB + (1 - params.IB) / Z
    # blends coherent (equilibrium) and uniform (decohered) distributions

  # --- 4. Thermodynamic entropy (Boltzmann-Gibbs form) ---
  S_phys = -params.k_B * Σ_i( p_i * ln(p_i) )

  # --- 5. Information divergence relative to equilibrium (p_eq) ---
  p_eq_i = (1/Z) * exp(-(E_i - mu) / (params.k_B * params.T))
  D_info = Σ_i( p_i * ln(p_i / p_eq_i) )

  # --- 6. Entropy production rate ---
  sigma = params.k_B * params.T * ddt(D_info)
  sigma >= 0

# ---------- TRANSFORMS ----------
transforms:
  coherence_link:
    params.IB = exp(-D_info)
    # coherence index decreases as statistical distribution deviates from equilibrium

  thermal_equilibrium (when params.IB == 1):
    p_i = p_eq_i
    sigma = 0

  high_coherence_limit:
    params.statistics = "FD" or "BE"
    note: "Quantum distributions emerge as coherent occupation restrictions."

  classical_limit:
    if exp((E_i - mu)/(params.k_B * params.T)) >> 1:
      p_i ≈ exp(-(E_i - mu)/(params.k_B * params.T))
      params.statistics = "MB"
      note: "Classical (Boltzmann) behaviour recovered."

# ---------- MONITORS ----------
monitors:
  normalisation:
    assert abs(Σ_i(p_i) - 1) < 1e-9
  entropy_nonnegative:
    assert S_phys >= 0
  coherence_range:
    assert 0 <= params.IB <= 1

# ---------- SCENES ----------
scenes:
  boltzmann_example:
    given params:
      statistics: "MB"
      T: 300
      mu: 0
      E_i: [0, 1e-21, 2e-21, 3e-21]
      IB: 1.0
    expect:
      p_i decreasing with E_i
      sigma = 0
      comment: "Classical distribution — full coherence, equilibrium state."

  fermi_dirac_example:
    given params:
      statistics: "FD"
      T: 50
      mu: 2e-21
      E_i: [0, 1e-21, 2e-21, 3e-21]
      IB: 0.95
    expect:
      p_i ≈ 1 / (exp((E_i - mu)/(k_B*T)) + 1)
      sigma small but > 0
      comment: "Quantum occupancy — partial coherence with exclusion principle active."

  decohered_distribution:
    given params:
      statistics: "MB"
      T: 300
      mu: 0
      IB: 0.5
      E_i: [0, 1e-21, 2e-21, 3e-21]
    expect:
      p_i blended toward uniform
      D_info > 0
      sigma > 0
      comment: "Decoherence flattens population distribution — information divergence visible."

# ---------- NOTES ----------
notes:
  - The Boltzmann factor exp(-E/kT) represents the probability weight of a state under thermal equilibrium.
  - Fermi–Dirac and Bose–Einstein statistics add quantum coherence constraints: exclusion and accumulation.
  - The coherence index IB quantifies departure from ideal statistical equilibrium — a measure of microscopic memory.
  - Entropy increases as p_i diverges from its equilibrium reference distribution (information divergence).
  - In HST terms, each probability field corresponds to a *boundary microstate map*, and statistical mechanics becomes
    the geometry of information flow on that boundary.
Overview

All systems share one unifying rule: states distribute according to energy and coherence.
Whether describing a classical gas, an electron sea, or a photon field, the same exponential foundation appears — the balance between energy cost and information spread.
This flow unifies classical and quantum distributions under a single coherence framework, showing that Boltzmann, Fermi–Dirac, and Bose–Einstein statistics are not separate laws but different coherence geometries of the same exponential field.

⚖️ 1. Core Relations

At thermal equilibrium, each state ii has probability

$$ p_i = \frac{1}{Z}\, \exp\!\left[-\frac{E_i - \mu}{k_B T}\right]. $$

where

  • EiE_i is the state energy
  • μ\mu is the chemical potential,
  • is the temperature,
  • is Boltzmann’s constant
  • is the partition function ensuring normalisation:

$$ Z = \sum_i \exp\!\left[-\frac{E_i - \mu}{k_B T}\right]. $$

This is the Boltzmann distribution, describing independent, distinguishable particles in the classical limit.

🔸 2. Quantum Corrections — Fermi and Bose Statistics

When quantum coherence between particles becomes relevant, occupation no longer follows pure exponentials but adds phase constraints on how energy states fill.

\( f(E) = \begin{cases} \dfrac{1}{e^{(E - \mu)/k_B T} + 1}, & \text{Fermi–Dirac} \\ \dfrac{1}{e^{(E - \mu)/k_B T} - 1}, & \text{Bose–Einstein} \\ e^{-(E - \mu)/k_B T}, & \text{Boltzmann} \end{cases} \)

StatisticParticle TypeCoherence CharacterLimiting Condition
Boltzmann

Distinguishable
classical particles

Independent occupancy
\( e^{(E - \mu)/k_B T} \gg 1 \)
Fermi–Dirac
Fermions (e.g. electrons)
Exclusion symmetry
One particle per state
Bose–Einstein
Bosons (e.g. photons)
Collective coherence
Many particles per state

The form of f(E)f(E) reflects how coherence manifests at the quantum boundary:
fermions repel (anti-symmetric wavefunctions), bosons attract (symmetric wavefunctions), and classical particles remain non-interfering.

🧩 3. Coherence Adjustment (HST Extension)

In Flow Algebra, we treat the equilibrium distribution as the coherent baseline peqp_{\mathrm{eq}}, and introduce a boundary coherence index IB∈[0,1]I_B \in [0,1] that measures alignment between the actual and equilibrium population maps.

$$ p_i = I_B\, p_{\mathrm{eq},i} + (1 - I_B)\,\frac{1}{Z}. $$

  • IB=1I_B = 1: fully coherent equilibrium — pure Boltzmann/Fermi/Bose form.
  • IB<1I_B < 1: partial coherence — mixture of equilibrium and uniform distributions.
  • IB=0I_B = 0: total decoherence — every state equally likely, entropy maximised.

The entropy then grows with divergence from the equilibrium map:

$$ S_{\mathrm{phys}} = -k_B \sum_i p_i \ln p_i, \qquad \sigma = k_B\, T\, \frac{dD_{\mathrm{info}}}{dt}. $$

and coherence links directly to divergence:

$$ I_B = e^{-D_{\mathrm{info}}}. $$

Thus, the degree of equilibrium violation can be read as an information distance in phase space.

🔹 4. Thermodynamic and Informational Perspective
ConceptClassical MeaningCoherence Meaning
Temperature 𝑇
Mean kinetic energy
Rate of energy-information exchange
Chemical potential 𝜇
Energy to add a particle
Boundary bias toward occupancy
Partition function 𝑍
Normalisation of probabilities
Total coherence volume
Entropy 𝑆
Measure of disorder
Information divergence
from perfect alignment
IB
–
Coherence memory of the distribution

Entropy in this view measures how much the statistical field has forgotten its ideal equilibrium shape.
The closer pip_i remains to peq,ip_{\mathrm{eq},i} the smaller DinfoD_{\mathrm{info}}, the higher IBI_B, and the more reversible the microscopic flow.

🧮 5. Classical and Quantum Limits

(a) Classical (Boltzmann) Limit

$$ \text{When } e^{(E_i - \mu)/(k_B T)} \gg 1, $$

$$ p_i \approx \frac{1}{Z}\, e^{-\frac{E_i - \mu}{k_B T}}. $$

Particles act independently, coherence minimal but uniform — boundary alignment global rather than quantum.

(b) Quantum (Fermi/Bose) Regime

When occupation probabilities approach unity (low T or high density):

$$ p_i = \frac{1}{e^{(E_i - \mu)/(k_B T)} \,\pm\, 1}, $$

the system’s coherence enforces statistical symmetry:

  • +1+1 for fermions (exclusion coherence),
  • −1-1 for bosons (collective coherence).

Entropy reflects the structure of this symmetry —
it grows when coherence breaks and state occupation drifts toward uniformity.

🌐 6. Interpretation in HST Geometry

In the HST model, every energy level EiE_i corresponds to a boundary mode — a coherent standing wave on the system’s holographic surface.
Statistical distributions are not just probabilities, but field population densities on that boundary.

Thermal equilibrium (IB=1I_B = 1) means each boundary mode maintains correct phase population.
As coherence drops (IB<1I_B < 1), these modes flatten — the geometric curvature of energy-space smooths into noise, and information divergence manifests as entropy.

Summary

Statistical distributions are the fingerprints of coherence at the microscopic level.
Boltzmann shows energy balance in isolation; Fermi–Dirac and Bose–Einstein reveal how coherence geometry modifies occupancy.

When coherence is perfect, entropy vanishes — information is conserved within the boundary.
When coherence fades, populations blur, and energy becomes memoryless.
Through Flow Algebra, these curves are no longer empirical fits — they are the statistical shadow of boundary geometry in motion.

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