Thermodynamic Geometry Research Articles

Nonperturbative thermodynamic geometry of anyon gas

Following our earlier work on the Ruppeiner geometry of an anyon gas [B. Mirza and H. Mohammadzadeh, Phys. Rev. E 78, 021127 (2008)], we will derive nonperturbative thermodynamic curvature of a two-dimensional ideal anyon gas. At different values of the thermodynamic parameter space, some unique and interesting behaviors of the anyon gas are explored. A complete picture of attractive and repulsive phases of the anyon gas is given.

Measurement of supercritical CO 2 critical flow: Effects of L/D and surface roughness

The use of supercritical fluids (SCF) has been proposed for advanced power systems including advanced sodium reactors, since these fluids can provide higher thermal efficiency and reduced system component size. Data characterizing the behavior of SCF during a blowdown or rapid depressurization are essential to validate certain aspects of safety analyses. This paper describes the results of an experiment to measure the critical mass flux for numerous stagnation thermodynamic conditions, geometry and outlet tube roughness. It was found that a 1D homogeneous equilibrium model (HEM) was capable of relatively good (less than 10% error) prediction of the test data.

The geometry of black hole thermodynamics in Gauss–Bonnet theory

In this work, the five-dimensional black hole solutions are presented in Einstein–Maxwell–Gauss–Bonnet theory. The geometry of the black hole thermodynamics has also been studied briefly.

Thermodynamic geometry and extremal black holes in string theory

We study a generalisation of thermodynamic geometry to degenerate quantum ground states at zero temperatures exemplified by charged extremal black holes in type II string theories. Several examples of extremal charged black holes with non degenerate thermodynamic geometries and finite but non zero state space scalar curvatures are established. These include black holes described by D1-D5-P and D2-D6-NS5-P brane systems and also two charged small black holes in Type II string theories. We also explore the modifications to the state space geometry and the scalar curvature due to the higher derivative contributions and string loop corrections as well as an exact entropy expression from quantum information theory. Our construction describes state space geometries arising out of a possible limiting thermodynamic characterisation of degenerate quantum ground states at zero temperatures.

On Geometro-thermodynamics of Dilaton Black Holes

In this talk we present the latest results from our ongoing project on geometro-thermodynamics (also known as information geometry of thermodynamics or Ruppeiner geometry) of dilaton BHs in 4D in both Einstein and string frames and a dyonic dilaton BH and at the end we report very briefly results from this approach to the 2D dilaton BHs.

On the thermodynamic geometry of BTZ black holes

We investigate the Ruppeiner geometry of the thermodynamic state space of a general class of BTZ black holes. It is shown that the thermodynamic geometry is flat for both the rotating BTZ and the BTZ Chern Simons black holes in the canonical ensemble. We further investigate the inclusion of thermal fluctuations to the canonical entropy of the BTZ Chern Simons black holes and show that the leading logartithmic correction due to Carlip is reproduced. We establish that the inclusion of thermal fluctuations induces a non zero scalar curvature to the thermodynamic geometry.

Experimental and theoretical studies of monoalkylammonium hexafluorosilicate [CH 3(CH 2) nNH 3] 2SiF 6 ( n = 2, 3) and ethylammonium hexafluorosilicate [(C 2H 5) 2NH 2] 2SiF 6

The infrared spectra of [CH 3(CH 2) 2NH 3] 2SiF 6, [CH 3(CH 2) 3NH 3] 2SiF 6 and [(C 2H 5) 2NH 2] 2SiF 6 compounds, synthesized by us, were recorded at room temperature and discussed. For the butylammonium hexafluorosilicate which crystallized in orthorhombic system with P321 ( Z = 1), and ethylammonium hexafluorosilicate crystallizes at room temperature in the monoclinic system with P2 1 as space group the discussions are made in relation to previously mentioned crystal structures. The vibrational spectra of the cations indicate that they are disordered and hydrogen bonded to the anions. Hydrogen bonding in these compounds is indicated by overtone and combination modes possessing appreciable intensities. The results are compared to density functional theory B3LYP and semi-empirical PM3 calculations. Full thermodynamic properties and geometries are also located and characterized.

Geometry of higher-dimensional black hole thermodynamics

We investigate thermodynamic curvatures of the Kerr and Reissner-Nordstr\"om (RN) black holes in spacetime dimensions higher than four. These black holes possess thermodynamic geometries similar to those in four dimensional spacetime. The thermodynamic geometries are the Ruppeiner geometry and the conformally related Weinhold geometry. The Ruppeiner geometry for $d=5$ Kerr black hole is curved and divergent in the extremal limit. For $d \geq 6$ Kerr black hole there is no extremality but the Ruppeiner curvature diverges where one suspects that the black hole becomes unstable. The Weinhold geometry of the Kerr black hole in arbitrary dimension is a flat geometry. For RN black hole the Ruppeiner geometry is flat in all spacetime dimensions, whereas its Weinhold geometry is curved. In $d \geq 5$ the Kerr black hole can possess more than one angular momentum. Finally we discuss the Ruppeiner geometry for the Kerr black hole in $d=5$ with double angular momenta.

Riemannian geometry of thermodynamics and systems with repulsive power-law interactions

A Riemannian geometric theory of thermodynamics based on the postulate that the curvature scalar R is proportional to the inverse free energy density is used to investigate three-dimensional fluid systems of identical classical point particles interacting with each other via a power-law potential energy gamma r(-alpha) . Such systems are useful in modeling melting transitions. The limit alpha-->infinity corresponds to the hard sphere gas. A thermodynamic limit exists only for short-range (alpha>3) and repulsive (gamma>0) interactions. The geometric theory solutions for given alpha>3 , gamma>0 , and any constant temperature T have the following properties: (1) the thermodynamics follows from a single function b (rho T(-3/alpha) ) , where rho is the density; (2) all solutions are equivalent up to a single scaling constant for rho T(-3/alpha) , related to gamma via the virial theorem; (3) at low density, solutions correspond to the ideal gas; (4) at high density there are solutions with pressure and energy depending on density as expected from solid state physics, though not with a Dulong-Petit heat capacity limit; (5) for 3<alpha<3.7913 , the solution goes from the low to the expected high density limit smoothly; (6) for alpha>3.7913 a phase transition is required to go between these regimes; (7) for any alpha>3 we may include a first-order phase transition, which is expected from computer simulations; and (8) if alpha-->infinity, the density approaches a finite value as the pressure increases to infinity, with the pressure diverging logarithmically in the density difference.

Geometry of mesoscopic dynamics and thermodynamics

Contact geometry provides a natural setting for classical thermo-dynamics. In this paper we use it to derive the structure of mesoscopic dynamics (GENERIC) expressing its compatibility with thermodynamics. In the second part we derive kinematics (Poisson brackets) of a large family of mesoscopic state variables.

Theoretical and experimental vibrational analysis of [C 6H 5NH 3] 2SiF 6

Phenylammonium hexafluorosilicate ([C 6H 5NH 3] 2SiF 6) single-crystal compound was synthesized by us; and it crystallized in orthorhombic system with Pmmm ( Z=2) as space group. The vibrational spectra of this compound were recorded and discussed. Strong hydrogen bonding in this compound is indicated by overtone and combination modes possessing appreciable intensities. The results are compared to semi-empirical PM3 and B3LYP density functional theory (DFT) calculations. Full thermodynamic properties and geometries are also located and characterized.

Free Energy as a Dynamical Invariant (or Can You Hear the Shape of a Potential?)

Classical lattice spin systems provide an important and illuminating family of models in statistical physics. An interaction Φ on a lattice L⊂ℤd determines a lattice spin system with potential AΦ. The pressure P(AΦ) and free energy FAΦ(β)=−(1/β)P(βAΦ) are fundamental characteristics of the system. However, even for the simplest lattice spin systems, the information about the potential that the free energy captures is subtle and poorly understood. We study whether, or to what extent, (microscopic) potentials are determined by their (macroscopic) free energy. In particular, we show that for a one-dimensional lattice spin system, the free energy of finite range interactions typically determines the potential, up to natural equivalence, and there is always at most a finite ambiguity; we exhibit exceptional potentials where uniqueness fails; and we establish deformation rigidity for the free energy. The proofs use a combination of thermodynamic formalism, algebraic geometry, and matrix algebra. In the language of dynamical systems, we study whether a Holder continuous potential for a subshift of finite type is naturally determined by its periodic orbit invariants: orbit spectra (Birkhoff sums over periodic orbits with various types of labeling), beta function (essentially the free energy), or zeta function. These rigidity problems have striking analogies to fascinating questions in spectral geometry that Kac adroitly summarized with the question ``Can you hear the shape of a drum?''.

The effect of the ring size of fused chelates on the thermodynamic and spectroscopic properties of peptide complexes of copper(II)

Copper(II) complexes of the tripeptides GlyGly-β-Ala, Gly-β-AlaGly, β-AlaGlyGly, Gly-β-Ala-β-Ala, β-AlaGly-β-Ala, β-Ala-β-Ala-β-Ala and the tetrapeptides GlyGlyGly-β-Ala, GlyGly-β-AlaGly, Gly-β-AlaGlyGly and β-AlaGlyGlyGly were studied by potentiometric, EPR and UV–Vis spectroscopic methods. The stoichiometry of the complexes of peptides containing β-alanine residues are very similar to those of oligoglycines; [CuL] +, [CuL 2], [CuH −1L], [CuH −2L] −, [CuH −1L 2] − and [CuH −3L] 2− were detected as the major species in all cases. The presence of β-alanine residues, however, influenced both thermodynamic stability and coordination geometry of various complexes. In most cases the formation of six-membered chelate rings resulted in a decrease of thermodynamic stability and distortion of coordination geometry of peptide complexes, especially if β-alanine residues were present in N-terminal or adjacent positions. On the contrary, the formation of the mixed (5,6,5) and (5,5,6) linked chelate systems of tripeptides is favoured over the pure five-membered rings.

Large-Scale Circulation and Production of Stratification: Effects of Wind, Geometry, and Diffusion

The combined effects of wind, geometry, and diffusion on the stratification and circulation of the ocean are explored by numerical and analytical methods. In particular, the production of deep stratification in a simply configured numerical model with small diffusivity is explored. In the ventilated thermocline of the subtropical gyre, the meridional temperature gradient is mapped continously to a corresponding vertical profile, essentially independently of (sufficiently small) diffusivity. Below this, as the vertical diffusivity tends to zero, the mapping becomes discontinuous and is concentrated in thin diffusive layers or internal thermoclines. It is shown that the way in which the thickness of the main internal thermocline (i.e., the diffusive lower part of the main thermocline), and the meridional overturning circulation, scales with diffusivity differs according to the presence or absence of a wind stress. For realistic parameter values, the ocean is in a scaling regime in which wind effects are important factors in the scaling of the thermohaline circulation, even for the single hemisphere, flat-bottomed case. It is shown that deep stratification may readily be produced by the combined effects of surface thermodynamic forcing and geometry. The form of the stratification, but not its existence, depends on the diffusivity. Such deep stratification is efficiently produced, even in single-basin, single-hemisphere simulations, in the presence of a partially topographically blocked channel at high latitudes, provided there is also a surface meridional temperature gradient across the channel. For sufficiently simple geometry and topography, the abyssal stratification is a maximum at the height of the topography. In the limit of small diffusivity, the stratification becomes concentrated in a thin diffusive layer, or front, whose thickness appears to scale as the one-third power of the diffusivity. Above and below this diffusive abyssal thermocline are thick, largely adiabatic and homogeneous water masses. In two hemisphere integrations, the water above the abyssal thermocline may be either “intermediate” water from the same hemisphere as the channel, or “deep” water from the opposing hemisphere, depending on whether the densest water from the opposing hemisphere is denser than the surface water at the equatorward edge of the channel. The zonal velocity in the channel is in thermal wind balance, thus determined more by the meridional temperature gradient across the channel than by the wind forcing. If the periodic channel extends equatorward past the latitude of zero wind-stress curl, the poleward extent of the ventilated thermocline, and the surface source of the mode water, both then lie at the equatorial boundary of the periodic circumpolar channel, rather than where the wind stress curl changes sign.

Thermodynamic curvature of the BTZ black hole

In this paper we apply the concept of thermodynamic geometry to the Ba\~nados-Teitelboim-Zanelli (BTZ) black hole. We find the thermodynamic curvature diverges at the extremal limit of the black hole, which means the extremal black hole is the critical point with the temperature zero. We also study the effective dimensionality of the underlying statistical model. Near the critical point, the picture is clear; the spatial dimension of the underlying statistical model is just one, which agrees with other results. However, far from the critical point, the dimension becomes less than one and even negative. In order to interpret this result, we resort to a qualitative analogy with the Takahashi gas model.

Memorial: Nikolai Nikolaevitch Chentsov

Memorial: Nikolai Nikolaevitch Chentsov

Riemannian geometry of equilibrium thermodynamics in a liquid mixture

The metric tensor ɡ ij and the scalar curvature R in the Riemannian geometry of the thermodynamic parameter space are investigated in a two-component liquid mixture. The rigorous expressions for ɡ ij and R are obtained in terms of volume change, compressibility and concentration fluctuation. On the basis of these expressions, the behaviours of ɡ ij and R are discussed, in particular, in the vicinity of the critical point of phase separation. The result shows that the absolute value of R goes to infinity near the point and its sign depends on the manner the point is approached.

A void fraction correlation for generalized applications

A void fraction correlation has been developed to cover the full range of pressures, flows, void fractions, and fluid types (steamwater, air-water, hydrocarbons, and oxygen). The correlation, referred to here as the Chexal-Lellouche correlation, has been qualified against several sets of steady-state two-phase/two-component flow test data that cover a wide range of thermodynamic conditions and geometries typical of PWR and BWR fuel assemblies as well as for pipes up to 450 mm in diameter. The correlation is based on a drift flux model and determines the drift flux parameters, C o and V gj both cocurrent and countercurrent two-phase flows for the full range of pressures, flows and void fractions. The correlation is available as a source code module for inclusion into any thermal-hydraulic computer program, and as an interactive personal computer program. The correlation is continuous and does not depend on flow regime maps or spline fitting.

Global and local metric geometry of ligand binding thermodynamics.

An abstract metric representation of ligand binding thermodynamics reveals the basic connections between the classical thermodynamic theory of binding and linkage [Wyman, J. (1964) Adv. Protein Chem. 19, 223-286] and its site-specific formulation [Di Cera, E. (1990) Biophys. Chem. 37, 147-164]. These two approaches are limit cases of a general metric formulation of binding and linkage that also includes a number of intermediate "mixed" representations.

Influence of the Thermodynamic Parameters and Geometry on the Degree of Separation and Energy Consumption in a Gas Thermal Diffusion Column under Static Conditions

Abstract The influence of the thermodynamic parameters of a binary gas mixture and the geometry of the column on the degree of separation and energy consumption in a gas thermal diffusion column (TDC) was investigated under static conditions (no gas production). The theoretical results are illustrated with experimental data for a nitrogen-helium mixture in a cylindrical TDC, in the temperature interval 300 to 800 K, and pressures ranging from 24 kPa to 1.1 MPa.