Theory of Order-Disorder Phase Transitions Induced by Fluctuations Based on Network Models

I sincerely thank you for reviewing my manuscript and identifying its shortcomings. To address these deficiencies, I have developed a systematic revision strategy structured as follows:

      Methodological Clarification: I incorporated concrete examples and schematic diagrams to elucidate the network model construction methodology, ensuring technical transparency.
      Fundamental Motivation: I explicitly articulated the core objective of investigating how microstructural mechanisms specifically manifest as macroscopic phenomena,

deliberately avoiding speculative assumptions throughout the analysis. Limitation Disclosure: I comprehensively supplemented the discussion to clarify methodological constraints and potential biases inherent in the proposed approach. Contextual Differentiation: I clarified that my previously published work constituted one specific validation case among three experimental benchmarks demonstrated in this study. This manuscript ascends to generalizable principles, particularly focusing on the application of high-order detailed balance theory to describe lattice model phase transitions between order and disorder. Notably, the analytical frameworks and result derivation approaches employed in these two studies demonstrate fundamental differences. Comparative Enhancement: To address the identified lack of comparative analysis, I systematically expanded the evidence base through extensive literature review and comparative experimentation. Benchmarking and Future Directions: The revised manuscript includes comprehensive comparisons with existing methodologies, followed by five distinct research avenues with significant potential for follow-up investigations.

-.The description of the novel method is very unclear and the arguments often did not make sense to me. If this method works then at least it is not explained well.

     This work presents a substantially different theoretical approach compared to conventional methods, dedicating significant attention to explicating the foundational theory.

The following sections detail the network model construction through concrete examples and schematic illustrations.

Network Model Construction

Example: 2D Ising Model with Nearest-Neighbor Interactions Consider a 2D Ising model where each lattice site exhibits spin-up/down states. Sites are classified based on their own spin state and the number of nearest-neighbor sites (4 neighbors) sharing the same spin: ​Spin-up classification: 5 categories (0-4 matching neighbors) ​Spin-down classification: 5 categories (0-4 matching neighbors) This results in 10 distinct classes (C₁₅-C₂₅). Mathematically, for the Hamiltonian H=1/2∑<i,j> Si Sj, I reorganize terms by these classes. The factor of 1/2 accounts for double-counting interactions. Importantly, this classification covers all possible configurations in the infinite 2D Ising model through 10 network nodes, where node weights represent configuration probabilities.

State Transitions and Network Dynamics Case 1: Spin Flip of Central Site

​Initial State: Central site has spin-up with 4 matching neighbors (Class C₁₅). ​Active Transformation: Flipping the central site changes its state to spin-down with 0 matches (Class C₂₁). ​Passive Transformation: This flip simultaneously alters neighboring sites' classes from C₁₅ → C₁₄ (each neighbor loses one match). Case 2: Neighbor Spin Flip

​Initial State: Central site remains spin-up with 4 matches (C₁₅). ​Passive Transformation: Flipping a neighboring site reduces its match count to 3 (Class C₁₄ for the neighbor), indirectly modifying the central site's class to C₁₄. These two cases illustrate all possible transformations under nearest-neighbor interactions. The complete network structure emerges from considering all such transitions.

Network Node Labeling Convention Nodes are labeled Cij : i∈{1,2}: Spin state (1=up, 2=down) j∈{1,5}: Number of matching neighbors (1=0 matches, 5=4 matches) Example: C₁₅​: Spin-up with 4 matches C₁₃​: Spin-down with 2 matches

Generalization to Higher Dimensions ​ 3D Ising Model: Classifies sites into 14 nodes (7 match counts × 2 spins) ​N-dimensional Models: Follows similar classification logic with 2(N+1) nodes This framework applies to various lattice models through analogous interaction-based classifications. Active vs. Passive Transformations

Transformation Type Definition Network Impact

​Active Direct spin flip of target site Horizontal transition between nodes ​Passive Indirect flip via neighbor changes Vertical transition within nodes

Key Insight: A single active transformation (e.g., flipping site A) corresponds to four simultaneous passive transformations (its four neighbors' state changes). Deriving High-Order Detailed Balance Unlike Monte Carlo simulations that use active transformations, this work focuses on passive transformations to establish:

​Microstate Transition Probabilities: Calculate joint probabilities for four-site passive transformations ​Balance Equations: Derive relationships between node weights during phase transitions

Phase Transition Analysis Using Waterfall Metaphor

   ​Initial State (T=0): All nodes in C₁₅​ with weight 1
  ​Slow Phase (C₁₅ → C₁₃): Gradual weight migration resembling water approaching a cliff edge
 ​Rapid Phase (C₁₃ → C₂₃): Abrupt weight redistribution analogous to water cascading

This demonstrates how passive transformations effectively capture critical transition dynamics missed by traditional active-only approaches.

-. The method is heuristic and it remained unclear what its limitations are.

     Research Objectives and Methodological Approach
       This work aims to investigate how microscopic fluctuations drive phase transitions, with the ultimate goal of uncovering the fundamental physical principles governing this process. 
    The validity of this theoretical framework requires further experimental verification and peer recognition within the scientific community.

Network Model Transformation I systematically convert lattice models into network representations to study phase transitions. This transformation process adheres to strict mathematical rigor:

   1)​Minimum Action Principle: Transition probabilities between network nodes follow the principle of least action
   2)​Node Classification: Network structures are categorized into three fundamental types:

​Single-node structures: Correspond to 0 K ground states ​Boundary structures: Intermediate configurations during phase transitions ​ Maximum entropy structures: Represent disordered states post-transition

Thermodynamic Interpretation

    The temperature evolution from 0 K to critical points can be understood as:
    Initial state: Dominance of single-node structures (C₁₅ weight = 1)
    Phase transition: Coupling between single-node and maximum entropy structures
   Critical regime: Boundary structures emerge where single-node weights undergo dramatic redistribution

(Note: Potential fractal connections remain unexplored in this work)

Special Case Analysis: 1D Ising Model The strictly converted network model for 1D Ising system: Lacks boundary structures Explains absence of phase transitions Maintains consistency with theoretical predictions

Methodological Advantages No algorithmic rules or additional assumptions introduced Focus on macroscopic phenomenon generation mechanisms Avoids conventional phase transition parameter calculations (e.g., critical exponents)

Limitations and Challenges 1)​Complex Interactions Handling: Multiple spin interactions (aligning/opposing) in Ising model increase node complexity Systems with multi-spin orientations lead to exponential growth of network nodes 2)​Mathematical Rigor Constraints: Angular interaction calculations (e.g., spin deflection angles) result in infinitely large network models No general solution exists for continuous interaction spectra

-. There appears to be significant overlap of the Ising model discussion with the results of the author's previous publication in AIP advances 14, 085308 (2024).

Theoretical Framework and Validation This work derives a general phase transition formula based on high-order detailed balance principles. Three specific experimental benchmarks validate this formulation: 1)​Ising model in different dimensions (overlapping with AIP Advances 14, 085308 (2024)) 2)​Frustrated triangular Ising model 3)​Edwards-Anderson Model Comparison with AIP Advances 14, 085308 (2024)

The referenced study focuses on: Constructing network models for 2D/3D/ND Ising systems Performing dimension-specific analyses Deriving phase transition formulas through case-by-case treatments ​No application of high-order detailed balance principles ​ No general lattice model framework

Methodological Differentiation Feature Present Work AIP Advances 14, 085308 (2024) ​Core Objective General lattice model phase equations Dimension-specific Ising system analysis ​Theoretical Foundation High-order detailed balance Empirical case studies ​Applicability Universal lattice models Restricted to Ising families ​Derivation Approach Abstract generalization Case-specific parameterization Innovation and Progression

1)Generalization from Specific Cases: Previous works analyzed particular instances (e.g., 2D Ising) This research establishes universal phase transition equations applicable to any lattice model

2)​Theoretical Unification:

Derives general formula → Specializes for Ising models through variable substitution Contrasts with the referenced study's dimension-specific derivations

3)​Methodological Advancement:

Utilizes high-order detailed balance for analytical solutions Avoids empirical fitting procedures used in AIP Advances approach Validation Strategy ​Dimensional Consistency Check: Validates formula predictions across 2D/3D/ND systems ​Network Model Cross-Validation: Compares analytical results with network-based simulations ​Critical Exponent Comparison: Demonstrates agreement with known thermodynamic limits

-. The results, also for the two other models in the appendix, are not compared against existing values in the literature. No side-by side comparison to an existing method is made.

Validation Results Across Dimensional Ising Models The phase transition formulas derived in this work have been validated through three-dimensional lattice systems: 1) 2D Ising Model: Achieves ​quantitative agreement with the analytical solutions by Yang-Zheng and Onsager (Physical Review 85, 808 (1952)).

2)3D Ising Model: No analytical solution exists; comparison with Monte Carlo simulations shows: Phase transition temperature difference: ​0.7%​(established theoretically in Physical Review B 62, 14837 (2000)) Critical exponent deviation: ​1/3 Note: Monte Carlo results remain empirical references due to lack of analytical benchmarks. 2)≥4D Ising Models: Confirms critical exponent ​α = 1/2 , matching our general formula predictions.

Appendix A: Frustrated triangular Ising model For systems with spin restricted to z-direction (±1), I prove: ​Existence of Minimum Energy Configuration: Total energy Etotal​≥N⋅Emin​, where N = total lattice sites. ​Achievability: Configurations with all spins aligned (either all +1 or all -1) realize this minimum. ​Formula Derivation: All results strictly follow from the phase transition equations proposed in this work.

Appendix B: Edwards-Anderson Model Validation W. F. Wreszinski's 2012 ground-state calculation for the 2D Edwards-Anderson model (with double-peaked distribution): ​Experimental Result: Egs​=−1.5 (Journal of Statistical Physics 146, 118 (2012)) ​Theoretical Prediction: This formula yields ​identical result through parameter substitution. ​

Methodological Distinction This work ​does not employ numerical simulation algorithms but instead: Proposes a ​universal phase transition formula eq 10 Derives specific results through dimensional parameterization Focuses on ​analytical derivations rather than empirical validation

-. Concerning the journal acceptance criteria: I did not see why the presented approach would constitute a breakthrough or detail a groundbreaking discovery. To my understanding, fluctuations are not taken into account in basic Landau theory but a large literature exists on extensions to fluctuations and other methods. So the motivation given in the manuscript, that the role of fluctuations in phase transitions would be unclear, appears not justified.

Fluctuation-Driven Phase Transitions: theoretical perspectives and unresolved questions Yes, while fundamental Landau theory neglects fluctuations, most phase transitions originate from thermodynamic or quantum fluctuations. The mechanism of topological phase transitions remains unclear to me within this framework. Current Research Limitations

The vast literature on fluctuation propagation and interaction methods: Fails to provide clear mechanisms linking fluctuations to phase transitions Primarily focuses on symmetry principles as explanatory frameworks In my view, symmetry considerations alone prove insufficient for complete understanding Fundamental Challenge If we were to fully understand how fluctuations drive phase transitions, we would presumably possess: The analytical solution for the 3D Ising model A generalized theory transcending traditional symmetry-based approaches

-. However, maybe the "opens a new pathway [...] with clear potential for multi-pronged follow-up work" condition may be fulfilled, if the method was more clearly presented and discussed in the light of related literature.

Innovative Framework and Future Perspectives This work presents the first systematic formulation of ​higher-order detailed balance equations. By combining these equations with network modeling, I demonstrate: Clear visualization of physical mechanisms in strongly correlated systems Vast potential for deriving ​analytical solutions for various lattice models

Key Validation Achievements Successful application to Ising models demonstrates theoretical consistency Network structure analysis reveals critical transition pathways Predictions align with experimental results in specific cases (See Appendices A-B)

Five Promising Research Directions

1) External Field Effects Unresolved Question: How do external fields modify node weight distributions in this network model? Potential Impact: Could enable control of phase transition thresholds through field manipulation

2) Frustration Phenomena Studies Application Basis: Appendix A's uniaxial spin framework Target Systems: Wannier's antiferromagnetism (Phys. Rev. 79, 357 (1950)) Anderson's localized spin systems (Mater. Res. Bull. 8, 153 (1973)) Modern frustrated systems (PRL 123, 207203 (2019); PRX 9, 031026 (2019))

3) Glassy Systems Analysis Methodological Transfer: Adapt network model to study: Aging effects Non-equilibrium dynamics Appendix B's Edwards-Anderson model extension (J. Stat. Phys. 146, 118 (2012))

4) Fractal Critical Phenomena New Insight: Boundary structures (e.g., C₁₄ in 2D Ising) exhibit: Nonlinear weight evolution near critical points Fractal dimension signatures (Complementary to: AIP Adv. 14, 085107 (2024); Phys. Rev. E 110, L062107 (2024))

5) Quantum Circuit Error Analysis Interdisciplinary Potential: Map quantum error processes to network models: Local bit flips

Correlated error propagation Reference frameworks: Quantum Error Mitigation (PRL 119, 180509 (2017),Rev. Mod. Phys. 95, 045005 (2023),Phys. Rev. X 7, 021050 (2017)) Theoretical and Practical Significance Analytical Power: Unifies microscopic mechanisms with macroscopic observables through network formalism Interdisciplinary applicability: Provides common mathematical framework for: Classical spin systems Quantum information devices Disordered materials Computational Efficiency: Enables analytical treatment of complex correlations previously requiring numerical simulations

Manuscript Revision Plan I will perform substantial revisions to the manuscript as follows:

1)Reference Integration

Systematically incorporate all cited works into: ​Introduction (contextual framing) ​Conclusion (theoretical implications and future directions) Ensure seamless integration with existing narrative flow

2)Methodological Appendix Transfer detailed model construction procedures and case study demonstrations to: ​ Appendix C: Comprehensive derivation of network model formalism Step-by-step validation with Ising model examples Comparative analysis with Monte Carlo simulations Include: Schematic diagrams illustrating transformation pathways Tabular summaries of critical exponent comparisons

3)Structural Optimization Streamline main text by: Removing redundant technical explanations Concentrating core innovations in theoretical framework Reserving experimental validations for dedicated sections Enhance reader navigation through: Updated table of contents Cross-referencing between main text and appendices Strategically placed summary paragraphs

I will submit the revised manuscript in the near future. 

Sincerely, Yonglong Ding

Attachment:

SciPost_Physics_Reply.pdf