Thermodynamic Geometry Research Articles

Information geometry for the strongly degenerate ideal Bose–Einstein fluid

The thermodynamic geometry of the Bose–Einstein fluid in the framework of information geometry is revisited, and particularly the strongly degenerate case is considered for a finite volume. Therefore, in the construction of the metric, the term related to the number of particles that accumulate in the ground state is taken into account, and we allow to explore its contribution to the curvature in highly quantum conditions, namely for temperature values where the ratio λ3∕V (thermal de Broglie wavelength cubed over volume) is greater than unity. It is found that in this regime, the ground state contribution is relevant in the limit of condensation and it strongly affects the behavior of the scalar curvature R. We show, numerically and analytically, that R is finite and smoothly approaches to zero in the limit η→1, namely as the fugacity tends to the numerical value where the condensation occurs the quantum effects are stronger. Consequently, when both phases are taken into account, there exist a region for extremely low temperatures where the curvature is regular, similarly to other quantum phase transitions reported in the literature.

Consequences of thermal geometries in Brane-World black holes

Abstract The implications of thermal geometries on Brane-World black hole (consisting of cosmological and dark matter effects) will be investigated by extracting the stability conditions (small and large roots) and divergences. To achieve this, we develop the temperature and heat capacity with respect to horizon radius, cosmological and dark matter parameters, respectively, which help in analyzing the small and large roots (stability conditions) and divergences. Furthermore, we study various methods of thermodynamical geometry such as Weinhold, Ruppeiner, Hendi–Panahiyah–Eslam–Momennia (HPEM), and Geometrothermodynamics (GTD) formulations and derive corresponding scalar curvatures for Brane-World black hole. It is found that Ruppeiner and HPEM formulations provide more information as compared to Weinhold and GTD methods for Brane-World black holes.

Thermodynamic bounds on coherent transport in periodically driven conductors

Periodically driven coherent conductors provide a universal platform for the development of quantum transport devices. Here, we lay down a comprehensive theory to describe the thermodynamics of these systems. We first focus on moderate thermoelectrical biases and low driving frequencies. For this linear response regime, we establish generalized Onsager-Casimir relations and an extended fluctuation-dissipation theorem. Furthermore, we derive a family of thermodynamic bounds proving that any local matter or heat current puts a nontrivial lower limit on the overall dissipation rate of a coherent transport process. These bounds do not depend on system-specific parameters, are robust against dephasing, and involve only experimentally accessible quantities. They thus provide powerful tools to optimize the performance of mesoscopic devices and for thermodynamic inference, as we demonstrate by working out three specific applications. We then show that physically transparent extensions of our bounds hold also for strong biases and high frequencies. These generalized bounds imply a thermodynamic uncertainty relation that fully accounts for quantum effects and periodic driving. Moreover, they lead to a universal and operationally accessible bound on entropy production that can be readily used for thermodynamic inference and device engineering far from equilibrium. Connecting a broad variety of topics that range from thermodynamic geometry over thermodynamic uncertainty relations to quantum engineering, our work provides a unifying thermodynamic theory of coherent transport that can be tested and utilized with current technologies.Received 23 September 2020Revised 26 January 2021Accepted 25 February 2021DOI:https://doi.org/10.1103/PhysRevX.11.021013Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.Published by the American Physical SocietyPhysics Subject Headings (PhySH)Research AreasNonequilibrium & irreversible thermodynamicsQuantum thermodynamicsTransport phenomenaPhysical SystemsFloquet systemsQuantum heat engines & refrigeratorsTechniquesLinear response theoryStatistical Physics

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)=εexp[−(r/σ)2], with r the distance between a pair of atoms, and the positive constants ε 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 σ3. 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.

Thermodynamic Curvature of AdS Black Holes with Dark Energy

In this paper, we study the effect of dark energy on the extended thermodynamic structure and interacting microstructures of black holes in AdS, through an analysis of thermodynamic geometry. Considering various limiting cases of the novel equation of state obtained in charged rotating black holes with quintessence, and taking enthalpy H as the key potential in the extended phase space, we scrutinize the behavior of the Ruppeiner curvature scalar R in the entropy-pressure (S,P)-plane (or equivalently in the temperature-volume (T,V)-plane). Analysis of R empirically reveals that dark energy parameterized by α, significantly alters the dominant interactions of neutral, charged and slowly rotating black hole microstructures. In the Schwarzschild-AdS case: black holes smaller than a certain size continue to have attractive interactions whereas larger black holes are completely dominated by repulsive interactions which arise to due dark energy. For charged or rotating AdS black holes with quintessence, R can change sign at multiple points depending upon the relation between α and charge q or angular momentum J. In particular, above a threshold value of α, R is never negative at all, suggesting heuristically that the repulsive interactions due to quintessence are long ranged as opposed to the previously known short ranged repulsion in charged AdS black holes. A mean field interaction potential is proposed whose extrema effectively capture the points where the curvature R changes sign.

Thermodynamic identities with sunray diagrams

One of the hurdles in teaching undergraduate thermodynamics is a plethora of complicated partial derivative identities. Students suffer from difficulties in deriving, justifying, or interpreting the identities, misconceptions about partial derivatives, and a lack of in-depth understanding of the meanings of identities. We propose a novel diagrammatic method for the calculus of differentials and partial derivatives called the ‘sunray diagram’ that resolves the difficulties above. The sunray diagram technique relates a partial derivative with ‘arrow sliding’, which enables an aesthetic and intuitive manipulation of partial derivative expressions in the form of successive arrow slidings. Furthermore, the sunray diagram is more than an ad hoc or abstract machinery but is based on the symplectic structure of thermodynamics; the sunray diagram admits a direct physical interpretation on the P–V (or T–S) plane. We elaborate on such physical semantics of the sunray diagram by taking Maxwell’s approach to the geometry of thermodynamic structures—reinterpreted in terms of differential geometry—as a reference point. We anticipate that our discussion introduces the geometry of thermodynamics to learners and enriches the graphical pedagogy in physics education.

Thermodynamics of ( 2+1 )-dimensional Coulomb-like black holes from nonlinear electrodynamics with a traceless energy momentum tensor

In this work we study thermodynamics of 2+1-dimensional static black holes with a nonlinear electric field. Besides employing the standard thermodynamic approach, we investigate the black hole thermodynamics by studying its thermodynamic geometry. We compute the Weinhold and Ruppeiner metrics and compare the thermodynamic geometry with the standard description on the black hole thermodynamics. We further consider the cosmological constant as an additional extensive thermodynamic variable. In the thermodynamic equilibrium three dimensional space, we compute the efficiency of the heat engine and show that it is possible to be built with this black hole.

Dilaton black holes with logarithmic source in gravity’s rainbow

Starting from the suitable action of four-dimensional Einstein gravity, which is coupled to a scalar dilaton field and logarithmic nonlinear electrodynamics, the related field equations have been solved in a spherically symmetric and energy-dependent spacetime. It has been found that the scalar potential, as the exact solution of the scalar field equations, must be written as the linear combination of two Liouville-type potentials. Three families of novel exact black holes have been introduced as the exact solutions to the gravitational equations, which are asymptotically non-flat and non-AdS. It has been illustrated that our solutions, recover their corresponding quantities in the Einstein–Maxwell-dilaton gravity theory when the nonlinearity parameter of electrodynamics is chosen very large. The thermodynamic and conserved quantities of the black holes have been calculated under the influence of the rainbow function. Then, through calculation of the Smarr mass formula, we have shown that the first law of black hole thermodynamics remain valid for all of the new dilatonic black holes. Thermal stability or phase transition of the black holes have been investigated by use of the canonical ensemble and thermodynamic geometry approaches, separately. By comparison of the results of these two alternative approaches, it has been found that they lead to the same results provided that the thermodynamic metric of HPEM is used.

Nonequilibrium thermal transport and thermodynamic geometry in periodically driven systems

With the in-depth understanding of nano-/micro-scaled systems and the developing of the corresponding experimental techniques, the heat transport and energy conversion processes in these small systems have attracted much interest recently. In contrast to the static manipulation methods, which hinge on the steady nonequilibrium sources such as temperature bias, chemical potential difference, etc., the temporal driving methods can control small systems in nonequilibrium non-steady states with much more versatility and universality. The research on periodically driven small systems holds both fundamental and pragmatic promises. This review is based on the fundamental concept of geometry. By analyzing the geometric phase and thermodynamic length in the transport process and the energy conversion process, we provide a unified perspective for the recent researches on the thermodynamic properties of driven nonequilibrium quantum systems. Thermodynamic geometry not only is the intrinsic origin of the nontrivial transport and dissipation, but also provides us with an all-applicable theoretical framework. The discussion over the geometry would yield multiple thermodynamic constraints on the transport and energy conversion, and can naturally construct a general optimization method as well. This will conduce to a better understanding of functionality for nonequilibrium quantum many-body systems acting as thermal machines. Also, this will inspire people to design quantum thermal machines with simultaneously more ideal performance, i.e. higher efficiency, higher power and higher constancy.

Thermodynamic geometry of the novel 4-D Gauss–Bonnet AdS black hole

In this paper, the new formalism of thermodynamic geometry proposed in [1] is employed in investigating phase transition points and the critical behavior of a Gauss Bonnet-AdS black hole in four dimensional spacetime. In this regard, extrinsic and intrinsic curvatures of a certain kind of hypersurface immersed in the thermodynamic manifold contain information about stability/instability of heat capacities. We, therefore, calculate the intrinsic curvature of the $Q$-zero hypersurface for a four-dimensional neutral Gauss Bonnet black hole case in the extended phase space. Interestingly, intrinsic curvature can be positive for small black holes at low temperature, which indicates a repulsive interaction among black hole microstructures. This finding is in contrast with the five-dimensional neutral Gauss Bonnet black hole with only dominant attractive interaction between its microstructures.

Nonequilibrium thermal transport and thermodynamic geometry in periodically driven systems

Nonequilibrium thermal transport and thermodynamic geometry in periodically driven systems

Effect of methylene chain length of perovskite-type layered [NH3(CH2) n NH3]ZnCl4 (n = 2, 3, and 4) crystals on thermodynamic properties, structural geometry, and molecular dynamics.

The structure of organic–inorganic perovskite [NH3(CH2)4NH3]ZnCl4 was determined; the lattice constants with monoclinic structure were determined to be a = 7.2527 Å, b = 8.1101 Å, c = 10.3842 Å, and β = 80.3436°. The crystal was almost thermally stable up to approximately 560 K. The endothermic peaks at 481 K and 506 K were assigned to the phase transition of the material. In addition, the structural characteristics and molecular dynamics of the cation were studied via magic angle spinning nuclear magnetic resonance experiments. Based on the results, the effects of the length of the CH2 group in the cation of the [NH3(CH2)nNH3]ZnCl4 (n = 2, 3, and 4) crystals were considered. Regardless of whether n was even or odd, the differences in the thermal and physical properties were minimal. Moreover, a difference in molecular motion relative to the length of the cation was observed only at high temperatures. These results provide useful information about the thermal stability and molecular dynamics of [NH3(CH2)nNH3]ZnCl4 crystals and are expected to facilitate potential applications of such compounds in supercapacitors, batteries, and fuel cells.

Fluctuations and thermodynamic geometry of the chiral phase transition

We study the thermodynamic curvature, $R$, around the chiral phase transition at finite temperature and chemical potential, within the quark-meson model augmented with meson fluctuations. We study the effect of the fluctuations, pions and $\sigma$-meson, on the top of the mean field thermodynamics and how these affect $R$ around the crossover. We find that for small chemical potential the fluctuations enhance the magnitude of $R$, while they do not affect substantially the thermodynamic geometry in the proximity of the critical endpoint. Moreover, in agreement with previous studies we find that $R$ changes sign in the pseudocritical region, suggesting a change of the nature of interactions at the mesoscopic level from statistically repulsive to attractive. Finally, we find that in the critical region around the critical endpoint $|R|$ scales with the correlation volume, $|R| =K\;\xi^3$, with $K = O(1)$, as expected from hyperscaling; far from the critical endpoint the correspondence between $|R|$ and the correlation volume is not as good as the one we have found at large $\mu$, which is not surprising because at small $\mu$ the chiral crossover is quite smooth; nevertheless, we have found that $R$ develops a characteristic peak structure, suggesting that it is still capable to capture the pseudocritical behavior of the condensate.

Universal criticality of thermodynamic curvatures for charged AdS black holes

In this paper, we analytically study the critical exponents and universal amplitudes of the thermodynamic curvatures such as the intrinsic and extrinsic curvature at the critical point of the small-large black hole phase transition for the charged AdS black holes. At the critical point, it is found that the normalized intrinsic curvature $R_N$ and extrinsic curvature $K_N$ has critical exponents 2 and 1, respectively. Based on them, the universal amplitudes $R_Nt^2$ and $K_Nt$ are calculated with the temperature parameter $t=T/T_c-1$ where $T_c$ the critical value of the temperature. Near the critical point, we find that the critical amplitude of $R_Nt^2$ and $K_Nt$ is $-\frac{1}{2}$ when $t\rightarrow0^+$, whereas $R_Nt^2\approx -\frac{1}{8}$ and $K_Nt\approx-\frac{1}{4}$ in the limit $t\rightarrow0^-$. These results not only hold for the four dimensional charged AdS black hole, but also for the higher dimensional cases. Therefore, such universal properties will cast new insight into the thermodynamic geometries and black hole phase transitions.

Analytic phase structures and thermodynamic curvature for the charged AdS black hole in alternative phase space

In this paper, we visit the thermodynamic criticality and thermodynamic curvature of the charged AdS black hole in a new phase space. It is shown that when the square of the total charge of the charged black hole is considered as a thermodynamic quantity, the charged AdS black hole also admits an van der Waals-type critical behavior without the help of thermodynamic pressure and thermodynamic volume. Based on this, we study the fine phase structures of the charged AdS black hole with fixed AdS background in the new framework. On the one hand, we give the phase diagram structures of the charged AdS black hole accurately and analytically, which fills up the gap in dealing with the phase transition of the charged AdS black holes by taking the square of the charge as a thermodynamic quantity. On the other hand, we analyse the thermodynamic curvature of the black hole in two coordinate spaces. The thermodynamic curvatures obtained in two different coordinate spaces are equivalent to each other and are also positive. Based on an empirical conclusion under the framework of thermodynamic geometry, we speculate that when the square of charge is treated as an independent thermodynamic quantity, the charged AdS black hole is likely to present a repulsive between its molecules. More importantly, based on the thermodynamic curvature, we obtain an universal exponent at the critical point of phase transition.

Skewed thermodynamic geometry and optimal free energy estimation.

Free energy differences are a central quantity of interest in physics, chemistry, and biology. We develop design principles that improve the precision and accuracy of free energy estimators, which have potential applications to screening for targeted drug discovery. Specifically, by exploiting the connection between the work statistics of time-reversed protocol pairs, we develop near-equilibrium approximations for moments of the excess work and analyze the dominant contributions to the precision and accuracy of standard nonequilibrium free-energy estimators. Within linear response, minimum-dissipation protocols follow the geodesics of the Riemannian metric induced by the Stokes friction tensor. We find that the next-order contribution arises from the rank-3 supra-Stokes tensor that skews the geometric structure such that minimum-dissipation protocols follow the geodesics of a generalized cubic Finsler metric. Thus, near equilibrium, the supra-Stokes tensor determines the leading-order contribution to the bias of bidirectional free-energy estimators.

Geometry of Work Fluctuations versus Efficiency in Microscopic Thermal Machines.

When engineering microscopic machines, increasing efficiency can often come at a price of reduced reliability due to the impact of stochastic fluctuations. Here we develop a general method for performing multiobjective optimization of efficiency and work fluctuations in thermal machines operating close to equilibrium in either the classical or quantum regime. Our method utilizes techniques from thermodynamic geometry, whereby we match optimal solutions to protocols parametrized by their thermodynamic length. We characterize the optimal protocols for continuous-variable Gaussian machines, which form a crucial class in the study of thermodynamics for microscopic systems.

Microstructure and Continuous Phase Transition of the Einstein-Gauss-Bonnet AdS Black Hole

The phase transition of the Einstein-Gauss-Bonnet AdS black hole has the similar property with the van der Waals thermodynamic system. However, it is determined by the Gauss-Bonnet coefficient α , not only the horizon radius. Furthermore, the phase transition is not the pure one between a big black hole and a small black hole. With this issue, we introduce a new order parameter to investigate the critical phenomenon and to give the microstructure explanation of the Einstein-Gauss-Bonnet AdS black hole phase transition. And the critical exponents are also obtained. At the critical point of the Einstein-Gauss-Bonnet AdS black hole, we reveal the microstructure of the black hole by investigating the thermodynamic geometry. These results perhaps provide some certain help to deeply explore the black hole microscopic structure and to build the quantum gravity.

Thermodynamic geometry of normal (exotic) BTZ black hole regarding to the fluctuation of cosmological constant

We construct the thermodynamic geometry of ([Formula: see text])-dimensional normal (exotic) BTZ black hole regarding the fluctuation of cosmological constant. We argue that while the thermodynamic geometry of black hole without fluctuation of cosmological constant is a two dimensional flat space, the three-dimensional space of thermodynamics parameters including the cosmological constant as a fluctuating parameter is curved. Some consequences of the fluctuation of cosmological constant will be investigated. We show that such a fluctuation leads to a thermodynamic curvature which is singular at the critical surface. Also, we consider the validity of first thermodynamics law regarding the fluctuation of the cosmological constant.

Thermodynamic curvature of the Schwarzschild-AdS black hole and Bose condensation

In the AdS/CFT correspondence, a dynamical cosmological constant Λ in the bulk corresponds to varying the number of colors N in the boundary gauge theory with a chemical potential μ as its thermodynamic conjugate. In this work, within the context of Schwarzschild black holes in AdS5×S5 and its dual finite temperature N=4 superconformal Yang-Mills theory at large N, we investigate thermodynamic geometry through the behavior of the Ruppeiner scalar R. The sign of R is an empirical indicator of the nature of microscopic interactions and is found to be negative for the large black hole branch implying that its thermodynamic characteristics bear qualitative similarities with that of an attraction dominated system, such as an ideal gas of bosons. We find that as the system's fugacity approaches unity, R takes increasingly negative values signifying long range correlations and strong quantum fluctuations signaling the onset of Bose condensation. On the other hand, R for the small black hole branch is negative at low temperatures and positive at high temperatures with a second order critical point which roughly separates the two regions.