arXiv subjects

George Ruppeiner

Publications and source records attributed to George Ruppeiner.

At least 19 recordsLinked to original sources

arXiv2026

Could black hole thermodynamics play a role in black hole mergers?

Gravitational waves detected from binary black hole mergers by the LIGO/Virgo/KAGRA collaboration yield values for both the black hole remnant masses $M$ and spins $a$, with the $169$ spin values collected so far crowding significantly around their average $\bar{a}=0.6869\pm 0.0135$. Could this crowding relate directly to the Davies Point (DP) from black hole thermodynamics? The DP results from the Kerr-Newman black hole model, and has heat capacity diverging at $a=0.68125$, a value very close to the measured $\bar{a}$. In this paper I construct a consistent thermodynamic fluctuation theory for black holes and use it to write the thermodynamic curvature $R$. $R$ immediately yields the correlation length $\xi$. $\xi$ is found to diverge at the DP, and I propose that this divergence brings on critical slowing down that retards the emission of gravitational waves. The spin drifts of the remnants slow in proportion, leading to a piling up of spin value at the DP, as observed. If correct, my work would combine general relativity with black hole thermodynamics in an observational setting.

gr-qc↗

arXiv2023

Black hole microstructures in the extremal limit

The microstructure of black holes is a mystery. There is yet no resolution of basic questions such as what the constituent particles are. We work here with black hole thermodynamics (BHT), and the metric geometry of thermodynamics, which connects to interparticle interactions via the invariant thermodynamic Ricci scalar curvature $R$. $R$ may be calculated with BHT. In ordinary thermodynamics (OT), $R$ is positive/negative for interparticle interactions repulsive/attractive. Its magnitude is the correlation length. The basic universality of thermodynamics leads us to expect similar relations for BHT. Our contribution here is motivated by a physical simplification that frequently occurs at low temperatures $T$ in OT: complicated interactions tend to freeze out, leaving only basic quantum statistical interactions, those of ideal Fermi or Bose gasses. Our hope is that similar simplification happens in black holes in the extremal limit, where the BHT temperature $T \to 0$. We evaluate the extremal $R$ for twelve BHT literature models, working with the independent variables mass, angular momentum, charge, and the cosmological constant, $\{M,J,Q,Λ\}$, respectively. We allowed only two of these variables to fluctuate at a time, with the other two fixed. $M$ always fluctuated, either $J$ or $Q$ fluctuated, and $Λ$ was always fixed. At constant average $M$, $R$ has limiting divergence $R=c\,T^{-1}$, with the nonsingular constant $c$ depending only on $M$ and the two fixed parameters. $c$ is positive for $11/12$ of the models we examined, and negative only for the tidal charged model. The positive sign for $R$ indicates a BHT microstructure composed of particles with repulsive (fermionic) interactions. The limiting BHT expression for $R$ resembles that for 2D and 3D ideal Fermi gasses at constant volume, which also have limiting divergence $R=c\,T^{-1}$, with positive $c$.

gr-qc↗

arXiv2021

Thermodynamic geometry of the Gaussian core model fluid

The three-dimensional Gaussian core model (GCM) for soft-matter systems has repulsive interparticle interaction potential $ϕ(r) = \varepsilon\, {\rm exp}\left[ -(r/σ)^{2} \right]$, with $r$ the distance between a pair of atoms, and the positive constants $\varepsilon$ and $σ$ setting the energy and length scales, respectively. $ϕ(r)$ is mostly soft in character, without the typical hard core present in fluid models. We work out the thermodynamic Ricci curvature scalar $R$ for the GCM, with particular attention to the sign of $R$, which, based on previous results, is expected to be positive/negative for microscopic interactions repulsive/attractive. Over most of the thermodynamic phase space, $R$ is found to be positive, with values of the order of $σ^3$. However, for low densities and temperatures, the GCM potential takes on the character of a hard-sphere repulsive system, and $R$ is found to have an anomalous negative sign. Such a sign was also found earlier in inverse power law potentials in the hard-sphere limit, and seems to be a persistent feature of hard-sphere models.

cond-mat.soft↗

arXiv2020

Thermodynamic curvature of the binary van der Waals fluid

The thermodynamic Ricci curvature scalar $R$ has been applied in a number of contexts, mostly for systems characterized by 2D thermodynamic geometries. Calculations of $R$ in thermodynamic geometries of dimension three or greater have been very few, especially in the fluid regime. In this paper, we calculate $R$ for two examples involving binary fluid mixtures: a binary mixture of a van der Waals (vdW) fluid with only repulsive interactions, and a binary vdW mixture with attractive interactions added. In both these examples, we evaluate $R$ for full 3D thermodynamic geometries. Our finding is that basic physical patterns found for $R$ in the pure fluid are reproduced to a large extent for the binary fluid.

cond-mat.stat-mech↗

arXiv2019

Thermodynamic metric geometry of the two-state ST2 model for supercooled water

Liquid water has anomalous liquid properties, such as its density maximum at 4\degree C. An attempt at theoretical explanation proposes a liquid-liquid phase transition line in the supercooled liquid state, with coexisting low-density (LDL) and high-density (HDL) liquid states. This line terminates at a critical point. It is assumed that the LDL state possesses mesoscopic tetrahedral structures that give it solid-like properties, while the HDL is a regular random liquid. But the short-lived nature of these solid-like structures make them difficult to detect directly. We take a thermodynamic approach instead, and calculate the thermodynamic Ricci curvature scalar $R$ in the metastable liquid regime. It is believed that solid-like structures signal their presence thermodynamically by a positive sign for $R$, with a negative sign typically present in less organized fluid states. Using thermodynamic data from ST2 computer simulations fit to a mean field (MF) two state equation of state, we find significant regimes of positive $R$ in the LDL state, supporting the proposal of solid-like structures in liquid water. In addition, we review the theory, compute critical exponents, demonstrate the large reach of the MF critical regime, and calculate the Widom line using $R$.

cond-mat.soft↗

arXiv2018

Thermodynamic calculation of spin scaling functions

Critical phenomena theory centers on the scaled thermodynamic potential per spin $ϕ(β, h)=|t|^{p}Y(h|t|^{-q})$, with inverse temperature $β=1/T$, $h=-βH$, ordering field $H$, reduced temperature $t=t(β)$, critical exponents $p$ and $q$, and function $Y(z)$ of $z=h|t|^{-q}$. I discuss calculating $Y(z)$ with the information geometry of thermodynamics. Scaled solutions obtain with three admissible functions $t(β)$: 1) $t=e^{-Jβ}$, 2) $t=β^{-1}$, and 3) $t=β_C-β$, where $J$ and $β_C$ are constants. For $p=q$, information geometry yields $Y(z)=\sqrt{1+z^2}$, consistent with the one-dimensional (1D) ferromagnetic Ising model.

cond-mat.stat-mech↗

arXiv2018

Thermodynamic Black Holes

Black holes pose great difficulties for theory since gravity and quantum theory must be combined in some as yet unknown way. An additional difficulty is that detailed black hole observational data to guide theorists is lacking. In this paper, I sidestep the difficulties of combining gravity and quantum theory by employing black hole thermodynamics augmented by ideas from the information geometry of thermodynamics. I propose a purely thermodynamic agenda for choosing correct candidate black hole thermodynamic scaled equations of state, parameterized by two exponents. These two adjustable exponents may be set to accommodate additional black hole information, either from astrophysical observations or from some microscopic theory, such as string theory. My approach assumes implicitly that the as yet unknown microscopic black hole constituents have strong effective interactions between them, of a type found in critical phenomena. In this picture, the details of the microscopic interaction forces are not important, and the essential macroscopic picture emerges from general assumptions about the number of independent thermodynamic variables, types of critical points, boundary conditions, and analyticity. I use the simple Kerr and Reissner-Nordstrom black holes for guidance, and find candidate equations of state that embody a number of the features of these purely gravitational models. My approach may offer a productive new way to select black hole thermodynamic equations of state representing both gravitational and quantum properties.

gr-qc↗

arXiv2017

Solid-like features in dense vapors near the fluid critical point

The phase diagram (pressure versus temperature) of the pure fluid is typically envisioned as being featureless apart from the presence of the liquid-vapor coexistence curve terminating at the critical point. However, a number of recent authors have proposed that this simple picture misses important features, such as the Widom line, the Fisher-Widom line, and the Frenkel line. In our paper we discuss another way of augmenting the pure fluid phase diagram, lines of zero thermodynamic curvature $R=0$ separating regimes of fluid solid-like behavior ($R>0$) from gas-like or liquid-like behavior ($R<0$). We systematically evaluate $R$ for the $121$ pure fluids in the NIST/REFPROP (version 9.1) fluid database near the saturated vapor line from the triple point to the critical point. Our specific goal was to identify regions of positive $R$ abutting the saturated vapor line ("feature D"). We found: a) $97/121$ of the NIST/REFPROP fluids have feature D. b) The presence and character of feature D correlates with molecular complexity, taken to be the number of atoms $Q$ per molecule. c) The solid-like properties of feature D might be attributable to a mesoscopic model based on correlations among coordinated spinning molecules, a model that might be testable with computer simulations. d) There are a number of correlations between thermodynamic quantities, including the acentric factor $ω$, but we found little explicit correlation between $ω$ and the shape of a molecule. e) Feature D seriously constrains the size of the asymptotic fluid critical point regime, possibly resolving a long-standing mystery about why these are so small. f) Feature D correlates roughly with regimes of anomalous sound propagation.

cond-mat.stat-mech↗

arXiv2016

Some Early Ideas on the Metric Geometry of Thermodynamics

It is a pleasure to write for this 90'th anniversary volume of Journal of Low Temperature Physics dedicated to Horst Meyer at Duke University. I was a PhD student with Horst in the period 1975-1980, working in experimental low temperature physics. While in Horst's group, I also did a theoretical physics project on the side. This project in the metric geometry of thermodynamics was motivated by my work in Horst's lab, and helped me to understand the theory of critical phenomena, very much in play in Horst's lab. In this paper, I explain the essence of my theory project and give a few accounts of its future development, focussing on topics where I interacted with Horst. I pay particular attention to the pure fluid critical point.

physics.hist-ph↗

arXiv2015

Unitary thermodynamics from thermodynamic geometry II: Fit to a local density approximation

Strongly interacting Fermi gasses at low density possess universal thermodynamic properties which have recently seen very precise $PVT$ measurements by a group at MIT. This group determined local thermodynamic properties of a system of ultra cold $^6\mbox{Li}$ atoms tuned to Feshbach resonance. In this paper, I analyze the MIT data with a thermodynamic theory of unitary thermodynamics based on ideas from critical phenomena. This theory was introduced in the first paper of this sequence, and characterizes the scaled thermodynamics by the entropy per particle $z= S/N k_B$, and energy per particle $Y(z)$, in units of the Fermi energy. $Y(z)$ is in two segments, separated by a second-order phase transition at $z=z_c$: a "normal" segment for $z>z_c$, and a "superfluid" segment for $z<z_c$. For small $z$, the theory obeys a series $Y(z)=y_0+y_1 z^{α}+y_2 z^{2 α}+\cdots,$ where $α$ is a constant exponent, and $y_i$ ($i\ge 0$) are constant series coefficients. For large $z$, the theory obeys a perturbation of the ideal gas $Y(z)= \tilde{y}_0\,\mbox{exp}[2γz/3]+ \tilde{y}_1\,\mbox{exp}[(2γ/3-1)z]+ \tilde{y}_2\,\mbox{exp}[(2γ/3-2)z]+\cdots$ where $γ$ is a constant exponent, and $\tilde{y}_i$ ($i\ge 0$) are constant series coefficients. This limiting form for large $z$ differs from the series used in the first paper, and was necessary to fit the MIT data. I fit the MIT data by adjusting four free independent theory parameters: $(α,γ,\tilde{y}_0,\tilde{y}_1)$. This fit process was augmented by trap integration and comparison with earlier thermal data taken at Duke University. The overall match to both the data sets was good, and had $α=1.21(3)$, $γ=1.21(3)$, $z_c=0.69(2)$, scaled critical temperature $T_c/T_F=0.161(3)$, where $T_F$ is the Fermi temperature, and Bertsch parameter $ξ_B=0.368(5)$.

cond-mat.quant-gas↗

arXiv2014

Thermodynamic $R$-diagrams reveal solid-like fluid states

We evaluate the thermodynamic curvature $R$ for fluid argon, hydrogen, carbon dioxide, and water. For these fluids, $R$ is mostly negative, but we also find significant regimes of positive $R$, which we interpret as indicating solid-like fluid properties. Regimes of positive $R$ are present in all four fluids at very high pressure. Water has, in addition, a narrow slab of positive $R$ in the stable liquid phase near its triple point. Also, water is the only fluid we found having $R$ decrease on cooling into the metastable liquid phase, consistent with a possible second critical point.

cond-mat.stat-mech↗

arXiv2014

Thermodynamic curvature for a two-parameter spin model with frustration

Microscopic models of realistic thermodynamic systems usually involve a number of parameters, not all of equal macroscopic relevance. We examine a decorated $(1+3)$ Ising spin chain containing two microscopic parameters: a "stiff" $K$ mediating the long-range interactions, and a "sloppy" $J$ operating within local spin groups. $K$ dominates the macroscopic behavior, and varying $J$ has weak effect except in regions where $J$ brings about transitions between phases through its conditioning of the local spin groups with which $K$ interacts. We calculate the heat capacity $C_H$, the magnetic susceptibility $χ_T$, and the thermodynamic curvature $R$. For large $|J/K|$, we identify four magnetic phases: ferromagnetic, antiferromagnetic, and two ferrimagnetic ones, according to the signs of $K$ and $J$. We argue that for characterizing these phases, the strongest picture is offered by the thermodynamic geometric invariant $R$, proportional to the correlation length $ξ$. This picture has correspondences to other cases, such as fluids.

cond-mat.stat-mech↗

arXiv2014

Thermodynamic curvature and black holes

I give a relatively broad survey of thermodynamic curvature $R$, one spanning results in fluids and solids, spin systems, and black hole thermodynamics. $R$ results from the thermodynamic information metric giving thermodynamic fluctuations. $R$ has a unique status in thermodynamics as being a geometric invariant, the same for any given thermodynamic state. In fluid and solid systems, the sign of $R$ indicates the character of microscopic interactions, repulsive or attractive. $|R|$ gives the average size of organized mesoscopic fluctuating structures. The broad generality of thermodynamic principles might lead one to believe the same for black hole thermodynamics. This paper explores this issue with a systematic tabulation of results in a number of cases.

gr-qc↗

arXiv2013

Unitary thermodynamics from thermodynamic geometry

Degenerate Fermi gases of atoms near a Feshbach resonance show universal thermodynamic properties, which are here calculated with the geometry of thermodynamics, and the thermodynamic curvature $R$. Unitary thermodynamics is expressed as the solution to a pair of ordinary differential equations, a "superfluid" one valid for small entropy per atom $z\equiv S/N k_B$, and a "normal" one valid for high $z$. These two solutions are joined at a second-order phase transition at $z=z_c$. Define the internal energy per atom in units of the Fermi energy as $Y=Y(z)$. For small $z$, $Y(z)=y_0+y_1 z^α+y_2 z^{2 α}+\cdots,$ where $α$ is a constant exponent, $y_0$ and $y_1$ are scaling factors, and the series coefficients $y_i$ ($i\ge 2$) are determined uniquely in terms of $(α, y_0, y_1)$. For large $z$ the solution follows if we also specify $z_c$, with $Y(z)$ diverging as $z^{5/3}$ for high $z$. The four undetermined parameters $(α,y_0,y_1,z_c)$ were determined by fitting the theory to experimental data taken by a Duke University group on $^6$Li in an optical trap with a Gaussian potential. The very best fit of this theory to the data had $α=2.1$, $z_c=4.7$, $y_0=0.277$, and $y_1=0.0735$, with $χ^2=0.95$. The corresponding Bertsch parameter is $ξ_B=0.462(40)$.

cond-mat.stat-mech↗

arXiv2012

Thermodynamic curvature: pure fluids to black holes

Thermodynamics unavoidably contains fluctuation theory, expressible in terms of a unique thermodynamic information metric. This metric produces an invariant thermodynamic Riemannian curvature scalar $R$ which, in fluid and spin systems, measures interatomic interactions. Specifically, $|R|$ measures the size of organized fluctuating microscopic structures, and the sign of $R$ indicates whether the interactions are effectively attractive or repulsive. $R$ has also been calculated for black hole thermodynamics for which there is no consensus about any underlying microscopic structures. It is hoped that the physical interpretation of $R$ in fluid and spin systems might offer insight into black hole microstructures. I give a brief review of results for $R$ in black holes, including stability, the sign of $R$, R=0, diverging |R|, and various claims of "inconsistencies" in thermodynamic metric geometry.

gr-qc↗

arXiv2012

Thermodynamic curvature from the critical point to the triple point

I evaluate the thermodynamic curvature $R$ for fourteen pure fluids along their liquid-vapor coexistence curves, from the critical point to the triple point, using thermodynamic input from the NIST Chemistry WebBook. In this broad overview, $R$ is evaluated in both the coexisting liquid and vapor phases. $R$ is an invariant whose magnitude $|R|$ is a measure of the size of mesoscopic organized structures in a fluid, and whose sign specifies whether intermolecular interactions are effectively attractive ($R<0$) or repulsive ($R>0$). I discuss five principles for $R$ in pure fluids: 1) near the critical point, the attractive part of the interactions forms loose structures of size $|R|$ proportional to the correlation volume $ξ^3$, and sign of $R$ negative, 2) in the vapor phase, there are instances of compact clusters of size $|R|$ formed by the attractive part of the interactions and prevented from collapse by the repulsive part of the interactions, and sign of $R$ positive, 3) in the asymptotic critical point regime, the $R$'s in the coexisting liquid and vapor phases are equal to each other, i.e., commensurate, 4) outside the asymptotic critical point regime incommensurate $R$'s may be associated with metastability, and 5) the compact liquid phase has $|R|$ on the order of the volume of a molecule, with sign of $R$ negative for a liquidlike state held together by attractive interactions and sign of $R$ positive for a solidlike state held up by repulsive interactions. These considerations amplify and extend the application of thermodynamic curvature in pure fluids.

cond-mat.stat-mech↗

arXiv2011

Thermodynamic Geometry, Phase Transitions, and the Widom Line

We construct a novel approach, based on thermodynamic geometry, to characterize first-order phase transitions from a microscopic perspective, through the scalar curvature in the equilibrium thermodynamic state space. Our method resolves key theoretical issues in macroscopic thermodynamic constructs, and furthermore characterizes the Widom line through the maxima of the correlation length, which is captured by the thermodynamic scalar curvature. As an illustration of our method, we use it in conjunction with the mean field Van der Waals equation of state to predict the coexistence curve and the Widom line. Where closely applicable, it provides excellent agreement with experimental data. The universality of our method is indicated by direct calculations from the NIST database.

cond-mat.stat-mech↗

arXiv2010

Thermodynamic curvature measures interactions

Thermodynamic fluctuation theory originated with Einstein who inverted the relation $S=k_B\lnΩ$ to express the number of states in terms of entropy: $Ω= \exp(S/k_B)$. The theory's Gaussian approximation is discussed in most statistical mechanics texts. I review work showing how to go beyond the Gaussian approximation by adding covariance, conservation, and consistency. This generalization leads to a fundamentally new object: the thermodynamic Riemannian curvature scalar $R$, a thermodynamic invariant. I argue that $|R|$ is related to the correlation length and suggest that the sign of $R$ corresponds to whether the interparticle interactions are effectively attractive or repulsive.

cond-mat.stat-mech↗