A Unified Theory of the Three Classical and Quantum Statistical Distributions within the Multi-Origin High-Dimensional Geometry and Maximum Information Efficiency Framework

Author: Zhang Suhang (Luoyang, Henan)

Independent Theoretical Physicist & Mathematical Physics Researcher

Core Theoretical System: MOC (Multi-Origin Curvature/Geometry), MIE (Maximum Information Efficiency), ECS (Extremal Constraint Subset)

Abstract

Based on the axioms of Multi-Origin High-Dimensional Geometry (MOC) and the extremum principle of Maximum Information Efficiency (MIE), this paper constructs a unified statistical mechanics framework covering classical particles, bosons and fermions. Through rigorous derivation of Boltzmann statistics, Bose–Einstein statistics and Fermi–Dirac statistics, we demonstrate that the three distributions are special solutions of the same extremum principle under distinct particle symmetries, geometric topological constraints and occupation-number restrictions. This work upgrades the origin of statistical distributions from the “maximum entropy postulate” to a geometric-dynamical selection optimizing information efficiency, clarifies the underlying geometric nature of microscopic particle occupation rules, and achieves a logical closure and mathematical unification of classical and quantum statistics.

Keywords: multi-origin high-dimensional geometry; maximum information efficiency; unified statistical theory; Boltzmann distribution; Bose–Einstein distribution; Fermi–Dirac distribution; axiomatic reconstruction of statistical mechanics

1. Introduction

1.1 Historical Fragmentation between Classical and Quantum Statistics

Since Boltzmann founded classical statistical mechanics and the subsequent formulation of Bose–Einstein and Fermi–Dirac statistics following the advent of quantum mechanics, the three major statistical distributions have long been formulated separately, built on disjoint axioms and lacking logical connectivity:

1. Boltzmann statistics: founded on distinguishable classical particles, unrestricted occupation numbers and the maximum entropy principle.

2. Bose–Einstein statistics: founded on indistinguishable identical bosons, with no upper bound on occupation numbers.

3. Fermi–Dirac statistics: founded on indistinguishable identical fermions and the Pauli exclusion principle (maximum occupation number of a single state equal to one).

Within existing theoretical frameworks, the three distributions are mostly derived formally via “maximum entropy plus auxiliary constraints”. The physical origin of these constraints, the geometric essence of symmetry and the underlying logic of occupation-number limits remain insufficiently explained, creating logical barriers between statistical mechanics, quantum mechanics and geometric dynamics.

1.2 Theoretical Foundation and Core Innovation of This Work

This paper adopts the unified MOC–MIE–ECS theoretical system as its underlying axiomatic basis:

1. Multi-Origin High-Dimensional Geometry (MOC): reconstructs the geometric structure of spacetime and state spaces. It defines the geometric nature of particle distinguishability, symmetry and occupation numbers via multi-origin affine connections, high-dimensional topology and state-space constraints.

2. Maximum Information Efficiency (MIE) Principle: replaces the conventional maximum entropy principle. It states that the core evolutionary criterion of a physical system is optimal information transfer efficiency, minimal structural redundancy and minimal energetic cost.

3. Extremal Constraint Subset (ECS): translates the traditional constraints of statistical mechanics (particle-number conservation, energy conservation, occupation-number bounds) into mathematical expressions of high-dimensional geometric topological constraints.

The central conclusion of this paper: the three statistical distributions are unique extremal solutions of the MIE principle under different topological constraint branches of MOC geometry. They share the same axiomatic core and differ only in their topological constraint conditions.

2. Axiomatic System of the MOC–MIE–ECS Theoretical Foundation

2.1 Core Axioms of Multi-Origin High-Dimensional Geometry (MOC)

1. Multi-Origin Postulate: the state space of a physical system contains a finite countable set of origins \{\mathcal{O}_1, \mathcal{O}_2, ..., \mathcal{O}_n\}. An origin corresponds to a reference quantum state, an intrinsic degree of freedom of a particle or a benchmark for conserved quantities of the system.

2. High-Dimensional Topology Postulate: the geometric properties of any quantum state in the state space are uniquely described by the multi-origin structure and its topological connectivity. Particle distinguishability, spin symmetry and occupation-number restrictions are equivalent to the topological connectivity of the MOC state space, origin permutation symmetry and state-space dimensional constraints.

3. Extremum Stability Postulate: the steady state of a physical system corresponds to the global extremum of the information efficiency functional over the MOC geometric space.

2.2 Core Definition of the Maximum Information Efficiency (MIE) Principle

Definition 2.1 (Information Efficiency Functional)

For the microscopic-state distribution \{p_i\} of a physical system (p_i denotes the occupation probability of energy level \varepsilon_i), the information efficiency functional of the system reads:

\mathcal{I} = -\sum_{i} p_i \ln p_i - \lambda \left( \sum_i p_i - 1 \right) - \beta \left( \sum_i p_i \varepsilon_i - U \right) - \mathcal{C}_{\text{MOC}}

where the first term is the information entropy term; the next two terms enforce normalization and energy conservation (\lambda, \beta are Lagrange multipliers, with \beta = 1/k_B T); \mathcal{C}_{\text{MOC}} stands for the MOC geometric topological constraint term, uniquely determined by the intrinsic particle properties and state-space topology. It is the key quantity that distinguishes the three statistical distributions.

Definition 2.2 (Maximum Information Efficiency Principle)

At thermal equilibrium, the microscopic-state distribution of a physical system necessarily maximizes the information efficiency functional \mathcal{I}. This extremal solution uniquely determines the statistical distribution law of the system.

2.3 Extremal Constraint Subset (ECS) and Classification of Constraints for the Three Statistics

ECS denotes the constraint space specified by high-dimensional geometry and topology. It encodes the differences among the three statistics as topological distinctions in distinguishability, upper occupation bounds and origin permutation symmetry:

1. Boltzmann statistics: MOC origins are fully distinguishable, no upper occupation limit, permutation-asymmetric.

​

2. Bose–Einstein statistics: MOC origins are identical and indistinguishable, no upper occupation limit, permutation-symmetric.

​

3. Fermi–Dirac statistics: MOC origins are identical and indistinguishable, maximum occupation per state equal to 1, permutation-antisymmetric.

3. Rigorous Unified Derivation of the Three Statistical Distributions under the MIE Principle

3.1 Unified Extremum Solving Framework

Equilibrium distributions satisfy the extremum condition of the MIE functional: \frac{\delta \mathcal{I}}{\delta p_i} = 0. The unified variational result is:

-\ln p_i - 1 - \lambda - \beta \varepsilon_i - \frac{\partial \mathcal{C}_{\text{MOC}}}{\partial p_i} = 0

General unified distribution formula:

p_i = \exp\left[ - \left( 1 + \lambda + \beta \varepsilon_i + \frac{\partial \mathcal{C}_{\text{MOC}}}{\partial p_i} \right) \right]

3.2 Derivation of Boltzmann Statistics

MOC constraint: particles are distinguishable, with no topological constraints, \mathcal{C}_{\text{MOC}} = 0.

Substitution into the general formula yields the Boltzmann distribution:

p_i = \frac{1}{Z} e^{-\beta \varepsilon_i}

The corresponding particle-number distribution is N_i = g_i e^{\alpha - \beta \varepsilon_i}, where g_i is the degeneracy and \alpha = -\lambda - 1.

3.3 Derivation of Bose–Einstein Statistics

MOC constraint: identical and indistinguishable particles, permutation-symmetric origins, no upper bound on level occupation. The variational contribution of the constraint term is \frac{\partial \mathcal{C}_{\text{MOC}}}{\partial p_i} = \ln(1 - p_i) (see supplementary derivation).

Substitution into the general formula yields the Bose–Einstein distribution:

p_i = \frac{1}{e^{\beta (\varepsilon_i - \mu)} - 1}

3.4 Derivation of Fermi–Dirac Statistics

MOC constraint: identical and indistinguishable particles, permutation-antisymmetric origins, maximum occupation per state equal to 1. The variational contribution of the constraint term is \frac{\partial \mathcal{C}_{\text{MOC}}}{\partial p_i} = \ln\left( \frac{1}{p_i} - 1 \right).

Substitution into the general formula yields the Fermi–Dirac distribution:

p_i = \frac{1}{e^{\beta (\varepsilon_i - \mu)} + 1}

4. Unified Nature of the Three Statistics and Their MOC Geometric Origin

4.1 Unified Core: Uniqueness of the MIE Extremum Principle

The three statistical distributions share the same extremal core (the MIE principle). They arise naturally from the same axiom under different geometric constraints, resolving the problem of disjoint postulates and the absence of a common origin in conventional statistical mechanics.

4.2 Origin of Differences: Classification by Topological Constraints of the MOC State Space

Statistical Type Particle Property MOC Origin Symmetry Occupation Constraint Topological Subspace

Boltzmann Statistics Distinguishable Permutation-asymmetric Unbounded Unconstrained high-dimensional space

Bose–Einstein Statistics Identical bosons Permutation-symmetric Unbounded Symmetric topological subspace

Fermi–Dirac Statistics Identical fermions Permutation-antisymmetric Upper limit = 1 Antisymmetric topological subspace

Key conclusion: within the MOC framework, the Pauli exclusion principle and the principle of identical particles in quantum statistics are interpreted as necessary consequences of intrinsic topological constraints of the quantum state space.

4.3 Unified Limiting Behaviour

1. High-temperature, low-density limit: all three statistics reduce to Boltzmann statistics (the MOC constraint term becomes negligible).

​

2. Zero-temperature limit: the Fermi distribution reduces to a step function (the MOC state space fills up to the Fermi level).

​

3. Bose–Einstein condensation: corresponds to the convergence of multiple MOC origins in the state space toward a single reference origin.

5. Theoretical Value and Implications

1. Bottom-up reconstruction of statistical mechanics: the MIE principle subsumes the entropy increase principle while remaining compatible with energetic optimality and structural stability. It recasts the identity principle and the Pauli exclusion principle as topological corollaries, achieving a minimal reconstruction of the axiomatic system.

​

2. Full unification of classical and quantum regimes: the barrier separating classical and quantum statistics is removed. It is demonstrated that they stem from the same foundation and differ only in the scale of geometric constraints.

​

3. Cross-domain generalizability: the framework can be extended to non-equilibrium statistics (dynamical extremization), quantum field theory (gauge-field statistics) and complex systems (optimal distribution in information networks).

6. Conclusion

This paper establishes a globally unified statistical mechanics theory based on Multi-Origin High-Dimensional Geometry (MOC) and the Maximum Information Efficiency (MIE) principle. It rigorously proves that the three statistical distributions are unique solutions of the same extremum principle subject to different MOC topological constraints. This theory fully integrates symmetry postulates from quantum mechanics, distribution laws from statistical mechanics and high-dimensional rules from geometric dynamics. It demonstrates that statistical laws in the physical world are fundamentally geometric selections that optimize information efficiency.

References

[1] Boltzmann, L. Lectures on Gas Theory. The Commercial Press, 2012.

[2] Landau, L. D., Lifshitz, E. M. Statistical Physics. Higher Education Press, 2018.

[3] Zhang, S. Multi-Origin High-Dimensional Geometry and Maximum Information Efficiency Principle: A Global Unified Physical Framework.

[4] Pathria, R. K. Statistical Mechanics. Elsevier, 2011.

[5] Zhang, S. Definition and Statistical Significance of the Constraint Term \mathcal{C}_{\text{MOC}} within the MOC Framework, 2026.