Cyclones and Galactic Rotations: Two Scales of Prigogine Dissipative Structures ---- Galactic Rotation Curves and Cosmic Accelerated Expansion in the GEE

AbstractThis paper proposes a phenomenological framework (the GEE framework) based on gravitational dissipative structures and the holographic principle, in which the observational effects conventionally called “dark matter” and “dark energy” in the standard cosmological model are interpreted as effective projections of gravitational degrees of freedom during coarse-graining.The central physical picture of this paper is the isomorphic manifestation of Prigogine dissipative structures at two scales: at the thermodynamic scale, the most ubiquitous Prigogine dissipative structure is the cyclone; at the gravitational scale, the most ubiquitous manifestation is the galactic rotation, which is structurally isomorphic to the cyclone. Terrestrial typhoons, Jupiter’s Great Red Spot, Martian global dust storms, and Venusian polar vortices are all quasi-steady ordered structures formed by open systems maintained by a continuous energy flow, far from equilibrium, and self-organized through nonlinear positive feedback. Galaxies, galaxy clusters, and accretion disks are quasi-steady rotational structures formed by self-gravitating systems during virialization. Under the Prigogine framework, both share the same abstract structure: open system, far from equilibrium, nonlinear positive feedback, continuous entropy production, and self-organized ordered structure. The physical identity of the GEE framework is precisely a Prigogine dissipative-structure theory in a long-range-force system.However, structural isomorphism is not physical identity. From cyclone to galactic rotation there are three hard fractures: first, opposite signs of heat capacity—the cyclone is a positive-heat-capacity system, the galactic rotation is a negative-heat-capacity system; second, different energy currencies—the cyclone corresponds to the thermal energy of the canonical ensemble, the galactic rotation to the gravitational potential energy of the microcanonical ensemble; third, a time-scale gap of about 20 orders of magnitude—cyclone lifetimes are days to centuries, galactic-rotation lifetimes are billions of years. Of the three fractures, only the second can be softened by a temperature-equivalent reduction. This paper definesT_{g}^{local}=\frac{m_{b}{\sigma }^{2}}{3k_{B}},y=\frac{T_{local}^{local}}{T_{th}},{\theta }_{local}=\frac{y}{y+1}.Here T_{g}^{local} is the local gravitational temperature in the microcanonical-ensemble sense; when the self-gravitating system reaches local thermal equilibrium, it is approximately equal to the microcanonical statistical temperature T_{g}=(\partial S/\partial E)^{-1}; in the virial approximation, it is equivalent to the dynamical temperatureT_{dyn}=\frac{m_{b}{\sigma }^{2}}{3k_{B}}.y is the ratio of two temperature scales, used to compare the local gravitational potential depth with the thermal kinetic energy density; it does not represent a phase transition between two thermodynamic temperatures. T_{g}^{local} has negative heat capacity and does not have the stable thermodynamic limit of the canonical ensemble; T_{th} is the canonical-ensemble thermodynamic temperature.The first and third fractures remain hard fractures: negative heat capacity solves the mechanical stability problem of the galactic rotation, but not the energy-source problem: the galactic rotation still requires external energy flows Q_{th}, Q_{\exp} to maintain the far-from-equilibrium state, so that entropy production d_{i}S/dt\gt 0 persists. The cyclone and the galactic rotation are isomorphic in mathematical form, but opposite in the sign of heat capacity: after the cyclone’s energy flow is cut off, T_{th} drops, causing y to rise and the structure to collapse; after the galactic rotation’s energy flow is cut off, T_{g}^{local} rises, causing y to rise, but the structure is maintained by negative-heat-capacity self-regulation. They are two ends of the dissipative-structure spectrum, not the same mechanism; this paper calls this “structural isomorphism, opposite mechanisms.”As a Prigogine dissipative structure, the galactic rotation also requires a continuous external energy flow, and the input energy required to maintain the dissipative structure must be at least several times the dissipation rate; in the late epoch the main energy source is the cosmic expansion energy Q_{\exp}. Negative heat capacity solves only the mechanical stability problem of the galactic rotation, not the energy-source problem.Convention: Throughout this paper we adopt natural units\mathrm{\hbar }=c=k_{B}=1unless otherwise stated. GEE assumes zero particle mass for the dissipative-structure gravitational gradient:m_{diss}=0.(1.1)The framework introduces a dimensionless effective gravitational entropy order parameter S and a dissipative-structure order parameter Θ, and fuses gravitational relaxation with the holographic principle: gravitational relaxation provides the dynamical background for the formation of dissipative structures, while the holographic principle provides the energy-density formulas for the dissipative-structure gravitational gradient and for the effective energy density driving cosmic accelerated expansion. The two meet through the gravitational entropy S. E=mc^{2} serves as a specified proportionality relation, stipulating that the energy released by gravitational relaxation is proportional to the system mass, but it does not directly enter the fusion equation.Hubble parameter evolution: Planck’s 67.4 is a model-dependent inference of today’s H_{0}, not the actual expansion rate at recombination. Therefore this paper does not claim that GEE has eliminated the Hubble tension. In GEE, H_{0} is a shared fit parameter; H(z) is obtained by solving the full background equations together with the GEE components; the temperature-conversion order parameter {\Theta }_{cosmo}(z) serves only as a phenomenological function controlling the delayed onset of interaction, and is not directly identified with T_{CMB}/T_{cosmo}^{cosmo}.Core statement: GEE is currently a phenomenological dissolution framework. It provides a parameter table, a fitting pipeline, and schemes for MCMC and Bayes-factor calculations. Before the full background solution and MCMC are completed, it cannot claim to have explained flat rotation curves, to have solved the Hubble tension, or to have obtained support from three independent observational datasets.Prigogine virial equation: The standard virial theorem assumes a closed, dissipationless system, giving2K+U=0.(1.2)In the GEE framework, the competition between the local gravitational temperature T_{0}^{local} and the local thermodynamic temperature T_{th} introduces a Prigogine dissipative term D_{\Pr}, and the virial equation is generalized to2K+U=D_{\Pr},(1.3)whereD_{\Pr}=T_{diss}\cdot \Delta S_{diss}=2{\alpha }_{D}\frac{GM_{D}^{2}}{R}(y^{2}-1).(1.4)y=\frac{T_{0}^{local}}{T_{th}},{\Theta }_{local}=\frac{y}{y+1}.(1.5)The self-consistent equation gives y≥2, {\Theta }_{local}\geq 2/3. This is an algebraic domain lower bound, not an independent phase-transition criterion. Formation of a dissipative structure requires separately d_{i}S/dt\gt 0 and Q_{th}+Q_{\exp}\gt 0.y is the ratio of two temperature scales, used to judge the relative size of the local gravitational potential depth and the thermal kinetic energy density. T_{0}^{local} is the local gravitational temperature in the microcanonical sense and does not have canonical-ensemble heat capacity; T_{th} is the canonical-ensemble thermodynamic temperature.When y≥2, D_{\Pr}\geq 0; when y>2, D_{\Pr}\gt 0, and the dissipative term provides additional effective gravity. The Prigogine virial equation predicts that the standard virial theorem may overestimate mass; whether three observational datasets support it must be rechecked.Holographic dissolution of the dissipative-structure gravitational gradient: GEE does not use particle components. The observational effects conventionally attributed to “dark matter” in the standard cosmology are reinterpreted in GEE as the effective projection of the galactic dissipative-structure gravitational gradient under a holographic cutoff. We adopt the Granda-Oliveros (Ricci) cutoff holographic model. This cutoff is introduced as a phenomenological assumption; its first-principles derivation from the first law of thermodynamics on the horizon is left to Sec. 15.1. In a universe containing only baryons and radiation, the background interaction isQ=3\beta H_{0}\frac{{\rho }_{diss}{\rho }_{\exp}}{{\rho }_{diss}+{\rho }_{\exp}}{\Theta }_{cosmo}(z).(1.6)In the background continuity equations, the direction of Q is from the expansion energy density to the dissipative-structure gravitational gradient:{\dot{\rho }}_{diss}+3H{\rho }_{diss}=+Q,{\dot{\rho }}_{\exp}+3H[1+w_{\exp}]{\rho }_{\exp}=-Q.(1.7)The effective power-law index is defined as a derived quantity:{\alpha }_{RDM}(a)\equiv -\frac{d\ln⁡{\rho }_{diss}}{d\ln⁡a}.(1.8)During the dissipative-structure-dominated era, if {\alpha }_{RDM}(a) is approximately constant, its average is denoted {\alpha }_{RDM}. From the continuity equation one obtains(3-{\alpha }_{RDM})H{\rho }_{diss}=Q_{th}+Q_{exp},hence{\alpha }_{RDM}=3-\frac{Q_{th}+Q_{exp}}{H{\rho }_{diss}}.Since Q_{th}\gt 0, Q_{exp}\gt 0, algebraically one must have{\alpha }_{RDM}\lt 3.If early {\rho }_{diss} is growing, i.e. {\dot{\rho }}_{diss}\gt 0, then{\alpha }_{RDM}\lt 0.Therefore there is no epoch with {\alpha }_{RDM}\gt 3. \lambda ={\alpha }_{RDM}-3 is a derived quantity, andλ<0.The present density ratio\frac{{\rho }_{diss}}{{\rho }_{B}}\approx 5.3-5.4(1.9)only constrains normalization and cannot by itself determine {\alpha }_{RDM}. The dimension of the interaction term is[Q]=[\rho ][H]=[\text{energy}]^{5}.(1.10)Phenomenological construction of the k-essence Lagrangian: The k-essence Lagrangian is constructed phenomenologically from the holographic energy density and the k-essence definition (\rho (X)\propto X^{a_{k}} is a phenomenological assumption):P(X