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

Abstract
This 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 defines
T_{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 temperature
T_{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}=1
unless 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, giving
2K+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 to
2K+U=D_{\Pr},(1.3)
where
D_{\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 is
Q=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, and
λ<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)=X^{1/2}\left[C+∫\frac{\rho (X)}{2X^{3/2}}dX\right].(1.11)
During the dissipative-structure gravitational gradient dominated era, taking \rho (X)\propto X^{a_{k}}, the Lagrangian is approximately
P(X)=A\left[{\left(\frac{X}{X_{0}}\right)}^{{\alpha }_{k}}\middle- 1\right],(1.12)
where A has dimensions of energy density. The exponent {\alpha }_{k} is determined by the effective sound speed constraint:
{\alpha }_{k}=\frac{1}{2}\left(1\middle+ \frac{1}{c_{s}^{2}}\right).(1.13)
A typical value is {\alpha }_{k}={10}^{8}, corresponding to
c_{s}^{2}\approx 5\times {10}^{-9},c_{s}\approx 7\times {10}^{-5}c.(1.14)
{\alpha }_{k}={10}^{8} corresponds to extreme fine-tuning; its naturalness is left to Sec. 15.1. The relation between ρ(X) and X has been computed from the k-essence equation of motion given the form of P(X):
\rho (X)=A(2{\alpha }_{k}-1){\left(\frac{X}{X_{0}}\right)}^{{\alpha }_{k}}+A.(1.15)
During the dissipative-structure gravitational gradient dominated era, \left(X/X_{0})^{{\alpha }_{k}}\middle\gg  1\right., \rho (X)\propto X^{{\alpha }_{k}}. Without interaction,
\rho -A\propto a^{-6{\alpha }_{k}/(2{\alpha }_{k}-1)}.(1.16)
For large {\alpha }_{k}, this exponent is approximately -3, i.e. the non-interacting case gives {\rho }_{diss}\propto a^{-3}, which does not match the required slope a^{-3.3\sim -3.4}; if Q is introduced, the background evolution must be re-solved. {\rho }_{k}\propto a^{-3} is a consequence of the phenomenological assumption; its applicability requires numerical verification.
Dissolution of the effective energy density driving cosmic accelerated expansion: GEE does not use material components. The observational effects conventionally attributed to “dark energy” in standard cosmology are reinterpreted in GEE as the effective energy density {\rho }_{\exp} driving cosmic accelerated expansion, generalized to a two-layer structure consisting of a holographic component and an early peak (EDE) component, coupled to the dissipative-structure gravitational gradient through an interaction. The early peak fraction is dominated by Jeans-mass halos at z∼3000. Cross-validation using both the Press-Schechter and Sheth-Tormen mass functions yields
f_{\exp}\approx 0.09[PS],(1.17)
f_{\exp}\approx 0.09-0.18[ST].(1.18)
The observed value is f_{\exp}=0.09\pm 0.03. The ST upper bound touches the 3σ boundary and is labeled marginal. The integral result depends on the free amplitude W_{0}, the power spectrum, the collapse threshold, and numerical details; when W_{0} is not independently fixed, the result cannot be regarded as an independent prediction.
An interaction between the expansion energy density and the dissipative-structure gravitational gradient is introduced:
Q=3\beta H_{0}\frac{{\rho }_{\mathrm{diss}}{\rho }_{\exp}}{{\rho }_{\mathrm{diss}}+{\rho }_{\exp}}{\Theta }_{\mathrm{cosm}}(z).(1.6)
Here {\Theta }_{\mathrm{cosm}}(z) controls the delayed onset of the interaction. Early suppression comes from the smallness of {\rho }_{\mathrm{diss}}{\rho }_{\exp}/({\rho }_{\mathrm{diss}}+{\rho }_{\exp}), not from {\Theta }_{\mathrm{cosm}}\ll 1. As the universe expands, the density combination grows, Q becomes effective at low redshift, flips the sign of w_{a}, and brings it into agreement with the DESI DR2 (DESI+CMB+SNe) preference. β=0.05 gives w_{a}\approx -0.3, compatible with DESI DR2’s -0.98±0.33 at the 2σ boundary; β=0.10 gives w_{a}\approx -0.6, closer to the DESI DR2 central value. The model is compatible with current data in this parameter range and requires further testing with future DESI DR3/Euclid data.
Equation-of-state sign: From
{\dot{\rho }}_{\exp}+3H(1+w_{\exp}){\rho }_{\exp}=-Q
one obtains
w_{\exp}=-1+\frac{1}{3}\frac{d\ln⁡{\rho }_{\exp}}{d\ln⁡(1+z)}-\frac{Q}{3H{\rho }_{\exp}}.(1.20)
If one defines an effective equation of state after absorbing the interaction, then
w_{eff}=w_{\exp}+\frac{Q}{3H{\rho }_{\exp}}=-1+\frac{1}{3}\frac{d\ln⁡{\rho }_{\exp}}{d\ln⁡(1+z)}.(1.21)
Unified dissolution equation
G_{\mu \nu }=8\pi GT_{\mu \nu },(1.22)
where
T_{\mu \nu }=T_{\mu \nu }^{holographic}+T_{\mu \nu }^{\exp}+T_{\mu \nu }^{i.i.}.(1.23)
T_{\mu \nu }^{holographic} is the holographic projection energy-momentum tensor of the dissipative-structure gravitational gradient, T_{\mu \nu }^{\exp} is the effective energy-momentum tensor of the expansion energy density, and T_{\mu \nu }^{i.i.} is the interaction energy-momentum tensor between the expansion energy density and the dissipative-structure gravitational gradient.
Nine-parameter core numerical values (pure-baryon recomputation)
{\alpha }_{RGO}=1.0(\text{first\ principles,\ Ricci\ cutoff}),{\alpha }_{RGO}\approx 1.35(\text{phenomenological\ reference}),\text{prior\ }[1.0,1.8].(1.24)
{\alpha }_{k}:{\log⁡}_{10}{\alpha }_{k}\in [7.7,9.0],{\alpha }_{k}^{\mathrm{ref}}\approx 1.0\times {10}^{8},{\alpha }_{k}\geq 5\times {10}^{7}.(1.25)
f_{\exp}\approx 0.06-0.13, PS baryonized 0.06-0.08, ST baryonized 0.07-0.13, prior \left[0.03,\; 0.15\right].
Derived quantities and priors:
z_{c}\sim 1000-1500,\text{prior\ }[800,2000].(1.27)
Y_{\exp}\sim 3-5,\text{prior\ }[2,6].(1.28)
β=0.04-0.10,prior [0.03,0.12].(1.29)
S_{0}\approx (6-8)\times {10}^{-8},\text{prior\ log-uniform\ }[3\times {10}^{-8},1.5\times {10}^{-7}].(1.30)
{\alpha }_{H}\approx 1.0,\text{prior\ }[0.5,2.0].(1.31)
The ninth item {\theta }_{\mathrm{cosmo}}(z) adopts a monotonic parameterization. It is counted in Scheme A as the ninth item and is not counted separately; if z_{th},z_{\exp},n are sampled independently, then the counting must be redone according to Scheme B.
Other derived quantities:
{\alpha }_{RDM}\lt 3;\text{if\ early\ }{\rho }_{diss}\text{\ grows,\ then\ }{\alpha }_{RDM}\lt 0;
\lambda ={\alpha }_{RDM}-3\lt 0;
\Gamma =\beta {\theta }_{\mathrm{cosmo}};
y_{0}=3.4,{\theta }_{local,0}=0.773,{\xi }_{0}=0.91;
{\nu }_{A}\approx 85-90\ \mathrm{km}/\mathrm{s}.
The above numerical values are an initial-value/prior table under the pure-baryon framework, not a complete MCMC fitting result. {\alpha }_{k}, f_{\exp}, z_{c}, S_{0} have been recomputed under the no-cold-dark-matter condition; however, statistical comparison still requires no-CDM background MCMC confirmation.
Current statistical status:
Under the Planck+SH0ES joint data, if the approximate estimate is temporarily retained, the difference between the {\chi }^{2} of GEE and that of the standard cosmological model is about
\Delta {\chi }^{2}\approx -2\sim -5.(1.36)
This estimate does not include the parameter-number penalty. GEE adds 9 phenomenological parameters; together with 5 shared standard parameters, the fitting vector has 14 dimensions; compared with the 6-dimensional standard model, this is a net increase of 8 dimensions. If N\sim {10}^{3}, the correct substitution is
ΔAIC=-5+2×8=11,(1.37)
ΔBIC=-5+8ln⁡(1000)≈50.3.(1.38)
Both still favor the standard cosmological model. An accurate Bayes factor requires a complete MCMC analysis; the calculation scheme is given in Sec. 15.4.6. The expectation is
\ln⁡B_{\mathrm{GEE}/\mathrm{standard}}\leq -5,(1.39)
i.e. the standard cosmological model is decisively preferred, unless the likelihood improvement of GEE far exceeds the current approximate estimate. The currently reportable conclusion is: under the joint data, GEE and the standard cosmological model are statistically indistinguishable; under Planck data alone, the standard cosmological model is preferred; but this conclusion requires full no-CDM background MCMC confirmation. The pure-baryon recomputation has currently only updated the priors and has not yet changed the above statistical conclusion.
Statement layering: In this paper, all instances of “derivation” are divided into two categories: mathematical derivation and phenomenological construction. Any formula for which a first-principles derivation is not provided shall, in accordance with the convention in Sec. 3.0.0, be regarded as a phenomenological assumption or phenomenological construction, and its rigorous derivation is deferred to Sec. 15.1.