Critical Conditions for Ordered Structures in Open Nonequilibrium Systems and a Set-Theoretic-Thermodynamic Unified Framework

1. Introduction The spontaneous formation and maintenance of ordered structures in open nonequilibrium systems—from Bénard convection and chemical oscillations to biological metabolic networks—is a central problem in statistical physics and complex systems science. Prigogine's dissipative structure theory [1] and Haken's synergetics [2] laid the conceptual groundwork, positing that systems reduce internal entropy through energy dissipation. However, a fundamental question remains unanswered in a general sense: How strong must the external driving be to maintain an ordered state? Does a system-independent, principled criterion exist? Existing models are often system-specific, describing bifurcation behaviors through phenomenological kinetic equations [3,4]. While insightful, these models typically presuppose specific forms of driving and dissipation, lacking necessity derived from more fundamental physical and mathematical principles. Notably, a unified, dimensionally correct measure of driving intensity and its quantitative relationship with intrinsic system properties (structural impedance) has not been explicitly proposed and proven. This paper aims to fill this gap. We begin by proving two principle-level theorems on critical conditions from a few self-evident physical premises. Subsequently, to concretize these abstract principles, we construct a novel set-theoretic-thermodynamic model, where the system's dynamic processes are mapped to set operations within a measure space, and thermodynamic constraints are naturally embedded into the evolution dynamics. This dual "theorem + model" framework not only provides a universal "existence proof" for order but also offers a computable, simulable "constructive scheme." The paper is structured as follows: Section 2 states the physical premises and proves the two core theorems. Section 3 elaborates on the construction of the set-theoretic-thermodynamic model. Section 4 presents the coupled thermodynamic-constrained dynamics and its phase transition analysis. Section 5 demonstrates numerical simulation results. Section 6 discusses connections with classical theories. Finally, we conclude and outline future directions. 2. Principle-Level Theorems: Critical Power-Load Pressure 2.1. Physical Premises and Definitions Our theory is built upon four basic physical premises: · P1 (Openness): The system continuously receives power input \Pi (dimension: [ML^2T^{-3}]) from its environment. · P2 (Structural Impedance): The system possesses an intrinsic property S \geq 0, characterizing the "difficulty" or "cost" of being organized and maintaining an ordered state. · P3 (Nonequilibrium Essence): A stable ordered state cannot exist in an isolated equilibrium state; it must rely on continuous power dissipation. · P4 (Scalar Criterion): The classification of the system's macroscopic state as ordered or disordered must be determined by a comparable scalar function Q. Based on P1-P2, we define the core variable: · Power-Load Pressure: P := \Pi / S. It represents the driving intensity per unit structural impedance. 2.2. Theorem 1: The Unique Form of the Driving Intensity Measure Theorem 1 (Uniqueness Theorem): Under premises P1-P4, any continuous and differentiable scalar function Q = f(\Pi, S) that can serve as an order-chaos criterion for the system, and satisfies the scale invariance of physical systems, must be a monotonic function of P = \Pi/S. Proof (Outline): Consider a linear scaling of the system size by k > 0. Physical equivalence requires the macroscopic state classification to remain unchanged, i.e., Q(k\Pi, kS) = Q(\Pi, S). Therefore, Q is a homogeneous function of degree zero. By Euler's theorem for homogeneous functions, its solution must be Q = F(\Pi/S). According to P1-P3, increasing \Pi promotes order, while increasing S suppresses it. Hence, F is a monotonically increasing function of its argument. ∎ This theorem shows that P is the natural and unique dimensionless form for describing driving intensity. 2.3. Theorem 2: Existence of a Critical Threshold Theorem 2 (Criticality Theorem): For any open system that can maintain a local low-entropy (ordered) state in a nonequilibrium steady state, there exists a positive critical power-load pressure P_c > 0 such that the ordered state can be sustained if and only if P > P_c. Proof (Outline by Contradiction): Assume no such P_c > 0 exists. Then, there exists a sequence of systems \{\mathcal{S}_n\} with driving intensity P_n = \Pi_n / S_n \to 0 that can still maintain order. Consider the system's entropy balance: dS_{\text{int}}/dt = \Phi - J_{\text{in}}, where \Phi is the internal entropy production rate and J_{\text{in}} is the negative entropy inflow rate. For a steady ordered state, dS_{\text{int}}/dt \leq 0, implying J_{\text{in}} \geq \Phi. 1. By the second law of thermodynamics and the principle of finite efficiency, entropy production has a positive lower bound: \Phi \geq \phi_0 S, where \phi_0 > 0. 2. Negative entropy inflow is limited by the input power: J_{\text{in}} \leq \Pi / T_0, where T_0 is a characteristic environmental temperature. Combining J_{\text{in}} \geq \Phi with the inequalities above yields \Pi / T_0 \geq \phi_0 S, i.e., P = \Pi/S \geq \phi_0 T_0 > 0. This contradicts the assumption P_n \to 0. Therefore, such a P_c \geq \phi_0 T_0 > 0 must exist. ∎ This theorem establishes, from fundamental thermodynamic principles, the necessity of a minimum driving threshold. 3. The Set-Theoretic-Thermodynamic Model: A Formal Realization Theorems 1 and 2 provide the principled framework. This section constructs a concrete mathematical model to furnish it with mathematical entities and dynamical details. 3.1. Set-Theoretic Foundation: Mapping Processes to Sets Consider the set of all microscopic or mesoscopic processes of the system, forming a measure space (\Omega, \mathcal{F}, \mu), where \mu is a finite measure (interpretable as "process intensity" or "resource occupancy"). Define two measurable subsets: · H \subset \Omega: The set of processes executing mode H. · G \subset \Omega: The set of processes executing mode G. Their measures are denoted h = \mu(H), g = \mu(G), j = \mu(H \cap G). Key derived measures are (see Fig. 1 for illustration): · Synergy Basis: j (intersection), representing processes supporting both modes, the root of synergy. · Specific Processes: \Delta_h = h - j (unique to H), \Delta_g = g - j (unique to G). · Structural Impedance Origin: The symmetric difference s = \mu(H \Delta G) = \Delta_h + \Delta_g = h + g - 2j, characterizing mismatch and conflict between modes. · Total Subsystem Scale: The union u = \mu(H \cup G) = h + g - j. 3.2. Definition of Core State Variables Based on set measures, macroscopic state variables are defined: 1. Synergy Degree: C = \beta j, where \beta \in [0,1] is the synergy efficiency coefficient. 2. Structural Impedance: S = \gamma s + S_0 = \gamma(h + g - 2j) + S_0, with \gamma > 0, S_0 > 0 as the base impedance. 3. Internal Kinetic Energy: E_k = \frac{1}{2} M C^2 = \frac{1}{2} M (\beta j)^2, where M \geq 0 is the subsystem's "inertia" or complexity. 4. Total Driving Power: \Pi = E_k + X, where X is the external direct power input. 3.3. Thermodynamic Quantities and Their Constraints The system's entropy exchange with the environment is described by: · Negative Entropy Inflow Rate: J_{\text{in}} = \xi (X + \epsilon E_k). Here, \xi>0 is a conversion coefficient, \epsilon \in [0,1] characterizes the contribution of internal kinetic energy to attracting negative entropy. · Internal Entropy Production Rate (Dissipation): \Phi = \eta S^\gamma, with \eta>0, \gamma \geq 1, using a power-law form to encompass various dissipation mechanisms. Accordingly, the power-load pressure is specified as: P = \frac{\Pi}{S} = \frac{E_k + X}{S}. Theorem 3 (Condition for Negative Entropy Existence): A subsystem can maintain a negative entropy state (dS_{\text{int}}/dt < 0) if and only if \boxed{J_{\text{in}} > \Phi \quad \Longleftrightarrow \quad \xi(X + \epsilon E_k) > \eta S^\gamma} \tag{TC} Condition (TC) is the core thermodynamic constraint of the model, defining the "feasible region" for system evolution. 4. Dynamical Evolution and Phase Transitions under Thermodynamic Constraints 4.1. Thermodynamic Feasibility Factor To smoothly embed the hard constraint (TC) into continuous dynamics, we define the thermodynamic feasibility factor: \Theta = \sigma\left( \frac{J_{\text{in}} - \Phi}{J_{\text{in}} + \Phi + \epsilon_0} \right) where \sigma(\cdot) is a Sigmoid-type function, and \epsilon_0 > 0 is a small constant. When J_{\text{in}} \gg \Phi, \Theta \approx 1 (dynamics fully activated); when J_{\text{in}} \ll \Phi, \Theta \approx 0 (dynamics strongly suppressed). 4.2. Evolution Equations The system's dynamics are described by the following coupled equations, where each term has a clear physical/set-theoretic meaning: Variable Evolution Equation Key Term Explanation h (Mode H) \frac{dh}{dt} = \Theta\left[\alpha_h P(1 - \frac{h}{H_{\max}}) + \varepsilon_h \Delta_h(1 - \frac{h}{H_{\max}})\right] - \delta_h h - \kappa_h h g Drive-induced growth + Specific process feedback - Decay - Competition g (Mode G) \frac{dg}{dt} = \Theta\left[\alpha_g P(1 - \frac{g}{G_{\max}}) + \varepsilon_g \Delta_g(1 - \frac{g}{G_{\max}})\right] - \delta_g g - \kappa_g h g Drive-induced growth + Specific process feedback - Decay - Competition j (Synergy Basis) \frac{dj}{dt} = \Theta\left[\beta_J P \cdot \min(\Delta_h, \Delta_g) + \zeta(\min(h,g) - j)j(1-j)\right] - \delta_J j Drive-induced synergy + Saturated logistic growth - Decay M (Complexity) \frac{dM}{dt} = \Theta \eta_M P - \delta_M M - \kappa_M S \frac{M}{M + M_0} Drive promotes complexity - Decay - Impedance dissipation 4.3. Fixed Points and Phase Transition Analysis · Disordered State (Extinction): The trivial fixed point (h^*, g^*, j^*, M^*) = (0,0,0,0).