A Relativistic Entropic Hamiltonian–Lagrangian Approach to the Entropy Production of Spiral Galaxies in Hyperbolic Spacetime

Abstract

Double-spiral galaxies are common in the Universe. It is known that the logarithmic double spiral is a Maximum Entropy geometry in hyperbolic (flat) spacetime that well represents an idealised spiral galaxy, with its central supermassive black hole (SMBH) entropy accounting for key galactic structural features including the stability and the double-armed geometry. Over time the central black hole must accrete mass, with the overall galactic entropy increasing: the galaxy is not at equilibrium. From the associated entropic Euler–Lagrange Equation (enabling the application of Noether’s theorem) we develop analytic expressions for the galactic entropy production of an idealised spiral galaxy showing that it is a conserved quantity, and we also derive an appropriate expression for its relativistic entropic Hamiltonian. We generalise Onsager’s celebrated expression for entropy production and demonstrate that galactic entropy production (entropy production corresponds to the intrinsic dissipation characteristics) is composed of two parts, one many orders of magnitude larger than the other: the smaller is comparable to the Hawking radiation of the central SMBH, while the other is comparable to the high entropy processes occurring within the accretion disks of real SMBHs. We conclude that galaxies cannot be isolated, since even idealised spiral galaxies intrinsically have a non-zero entropy production.

The Geometry of Thermodynamics III

“…( 6) & (9). Note that entropy production as such was similarly handled analytically by Parker & Jeynes [3], and an isomorphism between the Schrödinger equation and an entropic Partition Function was also specified (by Parker & Jeynes [6], their Eq.15) from the geometrical (QGT) perspective.…”

“…( 4)). But even though black holes appear to be non-conservative systems, Parker & Jeynes [3] have proved that entropy production Π is a Noether-conserved quantity. In all these cases, and in many others, the system geometry entails the conservation laws, whether or not dissipation is present.…”

“…Such geometries are necessarily Maximum Entropy, and real entities with such geometries, such as the DNA backbone or (heavily idealised) spiral galaxies [1], or the alpha particle [2], or black holes [3], or Buckminsterfullerene [4]), must therefore be stable. The QGT formalism was applied quantitatively to these examples, in particular underlining the isomorphism in the formalism between energy-based representations (with the quantum given by Planck's constant) and entropy-based representations (with the quantum given by Boltzmann's constant).…”

“…We start ( §2) by exploring the isomorphism between SM and thermodynamics just as QGT shows [1] that there exists an isomorphism between QM and thermodynamics. "Isomorphism" is a strong term that we use in its full mathematical sense but applied to physical entities, and the isomorphism we see here is that between energetic and entropic descriptions of the world, and in particular between energy and entropy production (it has long been known that the former is Noetherconserved: it has recently been shown by Parker & Jeynes [3] that the latter is too). In §3 we show that it is correct to think that irreversibility is basic to reality essentially because entropy production is Noether-conserved.…”

“…Up to now it is reversible systems that have been treated as archetypes of reality, but QGT has shown that the fundamental eigenfunctions of the entropic Hamiltonian have either double-helix or double logarithmic spiral geometries, where the double-helix is mathematically a special case of the double logarithmic spiral [1]. Both are Maximum Entropy geometries but the double helix has zero entropy production (indicating reversibility) whereas the double logarithmic spiral has non-zero entropy production (implying irreversible behaviour: see the calculations of Parker & Jeynes [3]). That is, reversibility should be regarded as a special case of irreversibility.…”