Helmholtz Free Energy Research Articles

Strain gradient viscoelasticity theory of polymer networks

A physically-based strain gradient viscoelasticity theory is proposed specially for polymer networks, accounting for the strain gradient effect and the history-dependent behavior, where the microstructure-dependence and history-dependence of stress and hyperstress are interpreted physically and quantitatively. The chain representation of the Helmholtz free energy density is transferred to the strain gradient continuum field representation through the geometric (spatial) connection between the chain stretch ratio and the continuum deformation measure. The necessity of using strain gradients to characterize asymmetric deformation of the microstructure of polymer networks serves as the origin of microstructure-dependent hyperstress, which is work-conjugated to the strain gradient field. The free energy per chain is assumed history-dependent to account for chain-environment interactions (e.g., friction between segments), which ultimately results in the history-dependent behavior of polymeric solids. A general constitutive relation is provided, which together with the assumed network structure, gives a concrete one. It is shown that independent of the assumed network structure, the stress, and the hyperstress share the same dimensionless relaxation function. A strain gradient viscoelastic constitutive relation based on the eight-chain network is applied to the analysis of polymer nanocomposites, where the complex moduli are rationally defined and calculated numerically. It is found that the strain gradient effect can be viewed as an amplifier that transforms the strain-induced stress into the effective stress.

Thermodynamics of massless particles in curved spacetime

This work is devoted to study the behavior of massless particles within the context of curved spacetime. In essence, we investigate the consequences of the scale factor [Formula: see text] of the Friedmann–Robertson–Walker metric in the Einstein–aether formalism to study photon-like particles. To do so, we consider the system within the canonical ensemble formalism in order to derive the following thermodynamic state quantities: spectral radiance, Helmholtz free energy, pressure, entropy, mean energy and the heat capacity. Moreover, the correction to the Stefan–Boltzmann law and the equation of states are also provided. Particularly, we separate our study within three distinct cases, i.e. [Formula: see text] [Formula: see text]; [Formula: see text] [Formula: see text]; [Formula: see text] [Formula: see text]. In the first one, the results are derived numerically. Nevertheless, for the rest of the cases, all the calculations are accomplished analytically showing explicitly the dependence of the scale factor [Formula: see text] and the Riemann zeta function [Formula: see text]. Furthermore, our analyses are accomplished in general taking into account three different regimes of temperature of the universe, i.e. the inflationary era ([Formula: see text][Formula: see text]GeV), the electroweak epoch ([Formula: see text][Formula: see text]GeV) and the cosmic microwave background ([Formula: see text][Formula: see text]GeV).

Horizon thermodynamics and cosmological equations: a holographic-like connection between thermostatistical quantities on a cosmological horizon and in the bulk

Horizon thermodynamics is expected to be related to the effective energy based on the energy density calculated from the Friedmann equation for a Friedmann–Robertson–Walker (FRW) universe. In the present study, the effective energy and thermostatistical quantities on a cosmological horizon are examined to clarify the holographic-like connection between them, with a focus on a de Sitter universe. To this end, the Helmholtz free energy on the horizon is derived from horizon thermodynamics. The free energy is found to be equivalent to the effective energy calculated from the Friedmann equation. This consistency is interpreted as a kind of holographic-like connection. To examine this connection, Padmanabhan’s holographic equipartition law, which is related to the origin of spacetime dynamics, is applied to a de Sitter universe. It is found that the law should lead to a holographic-like connection. The holographic-like connection is considered to be a bridge between thermostatistical quantities on the horizon and in the bulk. For example, cosmological equations for a flat FRW universe can be derived from horizon thermodynamics by accepting the connection as a viable scenario. In addition, a thermal entropy equivalent to the Bekenstein–Hawking entropy is obtained from the Friedmann equation using the concept of a canonical ensemble in statistical physics. The present study should provide new insight into the discussion of horizon thermodynamics and cosmological equations.

Entire Life Theoretical Model of Limestone under Unequal Cyclic Loading Based on the Expanding Theory of Thermodynamic System Analysis

Breakage, Helmholtz free energy, and nonlinearity are involved in many fundamental phenomena of complex systems across natural sciences. However, a mathematical equation that can express the entire life cycle of complex natural objects (such as rock-like quasi-brittle materials) is lacking. We expand the material body system to an isolated compound thermodynamic system to establish an innovative theoretical model induced by coupling nonlinear separation of Helmholtz free energy and breakage evolution that can be used to express the extreme entire life model of rock under unequal amplitude loading and unloading cycle. We gain crucial insights into the life essence of limestone, which is termed “Negative Dissipation.” This study first shows that the change in the mechanical properties of quasi-brittle materials caused by the timely evolution of breakage can be represented by the nonlinear separation of Helmholtz free energy and negative dissipation. An analytical solution to the nonlinear separation variables of Helmholtz free energy is provided by combining the method of solving nonlinear partial differential equations in mathematics and thermodynamic law. An analytical solution of Helmholtz free energy considering nonlinearity and breakage is proposed, and an equation that can reflect the constitutive mechanics law of the entire life cycle of rock in the theoretical model is presented. Theoretical results are consistent with the experimental data obtained from limestone samples with different prefabricated cracks. This original study provides a theoretical foundation for the life model of complex natural objects for nonlinear breakage and an early warning investigation of rocks under various unprecedented conditions.Practical ApplicationsThis study first found that the change in the mechanical properties of complex natural objects caused by the timely evolution of breakage can be represented by the nonlinear separation of Helmholtz free energy. We establish an innovative theoretical model induced by coupling nonlinear separation of Helmholtz free energy and breakage evolution. The innovative theory includes three parts: (1) an analytical solution to the nonlinear separation variables of Helmholtz free energy, (2) an analytical solution of Helmholtz free energy considering nonlinearity and breakage, and (3) an equation of the theoretical model that can reflect the constitutive mechanics’ law of the entire life of quasi-brittle rocks is presented for the first time. This original research result provides the foundation for a more in-depth life cycle of quasi-brittle rocks to develop the theoretical basis for the nonlinear breakage and early warning research of materials under various unprecedented conditions. We report a route to material microstructure composition, which may open an alternative pathway to quasi-brittle materials, which may, in turn, open a door into the mysterious world of science.

Global well-posedness and convergence to equilibrium for the Abels-Garcke-Grün model with nonlocal free energy

We investigate the nonlocal version of the Abels-Garcke-Grün (AGG) system, which describes the motion of a mixture of two viscous incompressible fluids. This consists of the incompressible Navier-Stokes-Cahn-Hilliard system characterized by concentration-dependent density and viscosity, and an additional flux term due to interface diffusion. In particular, the Cahn-Hilliard dynamics of the concentration (phase-field) is governed by the aggregation/diffusion competition of the nonlocal Helmholtz free energy with singular (logarithmic) potential and constant mobility. We first prove the existence of global strong solutions in general two-dimensional bounded domains and their uniqueness when the initial datum is strictly separated from the pure phases. The key points are a novel well-posedness result of strong solutions to the nonlocal convective Cahn-Hilliard equation with singular potential and constant mobility under minimal integral assumption on the incompressible velocity field, and a new two-dimensional interpolation estimate for the L4(Ω) control of the pressure in the stationary Stokes problem. Secondly, we show that any weak solution, whose existence was already known, is globally defined, enjoys the propagation of regularity and converges towards an equilibrium (i.e., a stationary solution) as t→∞. Furthermore, we demonstrate the uniqueness of strong solutions and their continuous dependence with respect to general (not necessarily separated) initial data in the case of matched densities and unmatched viscosities (i.e., the nonlocal model H with variable viscosity, singular potential and constant mobility). Finally, we provide a stability estimate between the strong solutions to the nonlocal AGG model and the nonlocal Model H in terms of the difference of densities.

A General Model for Thermodynamic Properties of Fluid Mixtures Based on Helmholtz Energy Formulations for the Components. Virial Expansion and Reduction to van der Waals Mixing Rules

Over the recent decades, Helmholtz energy formulations became available for a broad range of fluids. These multiparameter equations of state (R. Span, Springer 2000) allow computation of thermodynamic properties essentially within the experimental errorbars. Corresponding states-based model by Lemmon and Tillner-Roth (Fluid Phase Equilib 165:1, 1999) enabled construction of Helmholtz energy formulations for mixtures. However, we show that this model generates a non-physical dependence of virial coefficients on composition, which can be strong when the components are dissimilar. We propose a new mixture model that overcomes this deficiency. It has two main ingredients: (i) Quadratic mixing of “Helmholtz volumities”. This quantity with units of molar volume is introduced as a ratio of the molar residual Helmholtz energy to a product of gas constant, thermodynamic temperature, and molar density. It reduces to the second virial coefficient in the zero-density limit. Helmholtz volumities are considered for components and “cross-components”, hypothetical fluids representing the binary interactions. (ii) Replacing the variables—reduced reciprocal temperatures and reduced densities—with temperature and density scaling functions. Different scaling functions can be used for different components and cross-components, thus providing a highly flexible framework for representing the properties of mixtures. The scaling functions must be expandable into Taylor series in terms of molar concentrations in the zero-density limit. For the proposed mixture model, we develop formulas for computing virial coefficients up to the fourth order. Furthermore, we show that when the proposed mixture model is applied to a cubic equation of state, the conventional van der Waals mixing rules can be retrieved. These findings allow to consider the new model as a viable alternative to the corresponding states method of modeling thermodynamic properties of fluid mixtures.

Scaling theories for predicting the viscosity of binary and ternary refrigerant mixtures

Modeling the viscosity of potential replacements for refrigerants like hydrofluorocarbons (HFCs) and hydrofluoroolefins (HFOs) is important because this information can be used to determine which refrigerants are suitable for use in different refrigeration systems and to guide decisions about system design, operation, and maintenance. In this work, the viscosities of four selected refrigerants, two HFCs and two HFOs, and their mixtures were calculated using two similar approaches, namely entropy scaling and Helmholtz scaling. In both cases, the polar perturbed-chain statistical associating fluid theory (PPC-SAFT) EoS provides the molecular parameters and the entropy or Helmholtz energy dependency required. Results indicate good predictive capabilities for both models, with comparable error to the available experimental data. Even so, the Helmholtz scaling exhibits superior overall robustness, given its extra adjustable parameter. However, predictive results tend to be the same or less quality than those given by the entropy scaling approach.

A new class of Finitely Extensible Nonlinear Elastic FENE-P model obtained with a thermodynamical approach and the use of compressible natural configurations. Part I: Isothermal deformations

In a recent contribution to the fundamental understanding of polymer fluid dynamics, Khambhampati and Rajagopal (2021) established a connection between the natural configuration theory of Rajagopal and Srinivasa (2000) and the FENE-P model of Bird et al. (1987 [6]). In this paper we capitalize on the result in Khambhampati and Rajagopal (2021) and present a new class of FENE-P models using a more general Helmholtz potential within the conceptual framework of evolving natural configurations. To show its qualitative behavior, we exemplify with a classical Couette flow between infinite parallel plates. The model is capable of reproducing key features experimentally observed such as stress relaxation and the overshoot of the shear stress at the beginning of typical shear stress growth experiments. Comparison against the FENE-P type model obtained in Khambhampati and Rajagopal (2021) is used for comparison.

Phase Behavior of Ternary Polymer Mixtures in a Common Solvent.

The Edmond-Ogston model for phase separation is extended to ternary polymer mixtures in a common solvent (de facto a quaternary mixture). The model assumes a truncated virial expansion of the Helmholtz free energy up to the second-order terms in the concentration of the polymers, and the second virial coefficients (B11, B22, B33, B12, B13, B23) are the six parameters of the model. New results from this model are presented in relation to earlier work on binary mixtures: a necessary condition for the virial coefficients for the occurrence of phase separation in two or three phases, an analysis of the different regions of (local) thermodynamic instability using the Descartes sign rule, an expression for the critical curves, a relation between the tangents in points along the critical curve, a relation between the concentration of components in the different phases according to the so-called Lambert-W function, and a consistency check for the composition of coexisting phases in ternary mixtures. The obtained results are evaluated in the maximally symmetric version of the model, where (B11, B22, B33) are equal and (B12, B13, B23) are equal, which leads to two remarkable observations: the concentration range over which two phases are formed is relatively narrow; not all phase separation occurs within a Gibbs triangle, but also, "out-of-Gibbs-triangle" binodals are observed. These results lead to a deeper insight into the phase behavior of ternary mixtures and show promise as a stepping stone toward modeling phase separation in mixtures with many components.

Advanced modeling of diffusion and convection in multiphase compositional simulation for tight porous media

Small pore sizes influence the mass transport in gas enhanced oil recovery (EOR) in tight formations through complex phenomena such as diffusion and sorption. However, conventional simulators use relatively simple models for diffusion and sorption that do not properly represent non-ideal fluid interactions at high pressures. This paper presents a novel implementation of coupled diffusion and convection for multicomponent transport via the sorbed and bulk regions in tight porous media. Case studies highlight the important implication of convection in diffusion-dominant compositional transport in tight porous media.The simulator uses multicomponent diffusion based on the dusty-gas model (DGM), which uses the fugacity gradient as the driving force. The adsorption model used is an approximate solution based on the Multicomponent Potential Theory of Adsorption (MPTA) using the sorbed and central regions. The capillary pressure is included both in the flow equations and the flash calculations by minimizing the Helmholtz free energy using the Peng-Robinson equation of state at a fixed temperature, pore volume, and overall composition.A case study presents CO2 injection into a 1-D tight porous medium with a ternary mixture of methane, n-butane, and n-decane as the initial oil. Before CO2 injection, n-decane is most attracted to the pore wall and contained at a high concentration in the sorbed region. During CO2 injection, CO2 displaces n-decane from the pore wall, causing the counter-current transport of CO2 and n-decane.Simulation results show that the mixing of reservoir oil with CO2 in small pores results in local pressure changes, which tend to drive both diffusion and convection. In all cases, convection enhances multicomponent transport by dissipating these pressure changes. When CO2 is strongly attracted to the pore walls, a large pressure change on mixing accelerates the CO2 diffusion into the reservoir through the sorbed region and also the counter-current transport of oil components through the central region.

Investigations on mechanical, thermodynamic and surface properties of U–Si alloys

The investigation of U–Si alloys as advanced technology fuels or accident tolerance fuels (ATFs) has gained momentum due to their higher uranium density and exceptional thermal conductivity. Density functional theory (DFT) calculations were used to examine the mechanical, thermodynamic and surface properties of U–Si alloys. Voigt-Reuss-Hill model was used to evaluate the mechanical properties such as Young's modulus, shear modulus, bulk modulus and so on. The results suggest that U3Si2 possesses an inherent brittleness, as opposed to the remaining U–Si alloys which demonstrate reliable ductility. In comparison to the hardness of other U–Si alloys (40–80 GPa), U3Si2 exhibits a significantly lower hardness (∼10 GPa). The results of anisotropy show that U3Si2 exhibits a lower anisotropy compared to USi2 (I41/amd) and U3Si (Pm-3m). Also, based on these mechanical properties, the sound velocity and Debye temperature are determined. Additionally, the study calculates the internal energy, Helmholtz free energy, entropy and heat capacity as a function of temperature, which is beneficial for fuel design and other applications. The surface energies and work function are investigated comprehensively for 161 surface slab models and used to obtain other surface properties, including equilibrium morphology, dominant surface orientations, area-weighted surface energy and area-weighted work function.

A physically-based hydrostatic strain energy model for rubber-like materials inspired by Flory-Orwoll-Vrij equation of state theory

In a compressible hyperelastic model for rubber-like materials, the strain energy function is generally composed of the hydrostatic part corresponding to changes in interchain interactions and the compressible elastic part attributable to the network structure. It is well known that the hydrostatic part dominates the compressibility of the material. In this study, inspired by the Flory-Orwoll-Vrij (FOV) equation of state (EOS) theory for pure polymer liquids (note: a cell-like EOS theory), we assume that (i) the compressibility of rubber-like materials corresponds to changes in free volume of polymer chain segments; (ii) the hydrostatic strain energy of a rubber-like solid is attributable to changes in interchain interaction energy and chain segments’ external motion degrees of freedom (the latter solely depends on interchain forces). With a focus on the reversible isothermal deformation process, we construct a physically-based hydrostatic strain energy function based on the Helmholtz free energy formulation in FOV EOS theory. With a view towards applications, we provide a specific compressible hyperelastic model by incorporating the new hydrostatic strain energy function and compressible eight-chain model, where the latter is utilized as the elastic part of the strain energy function. Our model is capable of predicting various volume data of rubber-like materials from the literature, such as the nonlinear pressure-volume response at finite volume changes in hydrostatic compression (HC), the volume change-stretch and stress-stretch data in uniaxial tension (UT), and the stress-volume data in constrained uniaxial compression (CUC). Given the severely limited volume change data for finite stretches in UT and other modes of deformation, we simulate the volume change-volume modulus-stretch responses in UT, equibiaxial tension (ET), pure shear (PS), and uniaxial compression (UC) over their respective theoretical range of stretch and successfully predict some available (qualitative) experimental observations. Together with the simulations of the volume change-stretch responses in UT, ET, PS, and UC based on the Ogden’s and Bischoff et al.’s compressible models, we summarize the characteristics of these responses and analyze the possible deformation mechanisms. These simulations can provide some guidance for future corresponding experiments. Finally, we analyze the deformation mechanisms of HC, CUC, UT, ET, PS, and UC by simulating the responses for total strain energy and its components. This study provides a new physical picture and corresponding theoretical model for the hydrostatic strain energy function of rubber-like materials and finally proposes the general research strategy for constructing new hydrostatic strain energy functions based on the EOS theories for pure polymer liquids.

Thermal fluctuations of Torus-like charged AdS Black Hole

In this article, we analyze the thermodynamics, thermal fluctuations and quasi-normal modes of a torus-like AdS black hole. The parameters of a torus-like charged AdS black hole are its mass, electric charge and cosmological constant. We used a torus-like charge in an AdS black hole to evaluate thermodynamics through thermal fluctuations and found that it significantly affected the corrected thermodynamic potentials. Black holes of larger radii are more stable and significant, as determined by our investigation of the effect of thermal fluctuation on other thermodynamic potentials such as Helmholtz free energy, Gibbs free energy, enthalpy and specific heat with constant pressure and volume, using the concept of corrected entropy. Finally, we use null geodesics to deal with quasi-normal modes. The null geodesics provide the photon sphere’s angular velocity and Lyapunov exponent, which are the real and imaginary components of the quasi-normal modes in the eikonal limit. With rising charge and photon radius, the angular velocity reaches its maximum, similar to Lyapunov’s exponent.

A novel continuum mechanical framework for decoupled material behavior in thickness and in-plane directions

A novel continuum mechanical framework for decoupling the material behavior in in-plane and thickness directions is presented that is motivated by the macroscopic behavior of paper and paperboard. Due to the manufacturing process of these composite materials and the underlying microstructure resulting thereof, the material response is highly anisotropic and can be divided into in-plane and out-of-plane behavior that is independent of each other. This decoupling is mostly incorporated in existing material models by setting the corresponding material parameters to zero which limits the models to an extension to nonlinear problems and is restrictive from a continuum mechanical point of view.Therefore, the aim of this paper is to derive a continuum mechanical framework that captures the in-plane and out-of-plane decoupling in a thermodynamically consistent manner for large deformations. Thus, the right Cauchy–Green deformation tensor was split into in-plane and out-of-plane deformations by use of structural tensors that were aligned with the preferential material directions. Therefore, the Helmholtz free energy was formulated in terms of modified invariants of the deformation tensor for in-plane and out-of-plane deformations separately, which lead to a decoupled framework that could easily be modified due to the flexible formulation of the Helmholtz free energy terms. The extension to inelasticity, in particular viscoelasticity here, was straightforward due to the split of the deformation tensor. Finally, the decoupling of the in-plane and out-of-plane response of the presented framework was demonstrated by structural examples for elastic and inelastic material behavior.The presented model incorporates the split into in-plane and out-of-plane behavior in a continuum mechanical manner, and thus, is a flexible constitutive framework for decoupled material responses that can easily be adjusted to different material behavior.

A 3D Anisotropic Thermomechanical Model for Thermally Induced Woven-Fabric-Reinforced Shape Memory Polymer Composites.

Soft robotic grippers offer great advantages over traditional rigid grippers with respect to grabbing objects with irregular or fragile shapes. Shape memory polymer composites are widely used as actuators and holding elements in soft robotic grippers owing to their finite strain, high specific strength, and high driving force. In this paper, a general 3D anisotropic thermomechanical model for woven fabric-reinforced shape memory polymer composites (SMPCs) is proposed based on Helmholtz free energy decomposition and the second law of thermodynamics. Furthermore, the rule of mixtures is modified to describe the stress distribution in the SMPCs, and stress concentration factors are introduced to account for the shearing interaction between the fabric and matrix and warp yarns and weft yarns. The developed model is implemented with a user material subroutine (UMAT) to simulate the shape memory behaivors of SMPCs. The good consistency between the simulation results and experimental validated the proposed model. Furthermore, a numerical investigation of the effects of yarn orientation on the shape memory behavior of the SMPC soft gripper was also performed.

Bitangent planes of surfaces and applications to thermodynamics

The classical van der Waals equation, applied to one or two mixing fluids, and the Helmholtz (free) energy function A yield, for fixed temperature T, a curve in the plane R2 (one fluid) or a surface in 3-space R3 (binary fluid). A line tangent to this curve in two places (bitangent line), or a set of planes tangent to this surface in two places (bitangent planes) have a thermodynamic significance which is well documented in the classical literature. Points of contact of bitangent planes trace ‘binodal curves’ on the surface in R3. The study of these bitangents is also classical, starting with D.J. Korteweg and J.D. van der Waals at the end of the 19th century, but continuing into modern times. In this paper we give a summary of the thermodynamic background and of other mathematical investigations and then present a new mathematical approach which classifies a wide range of situations in R3 where bitangents occur. In particular, we are able to justify many of the details in diagrams of binodal curves observed by Korteweg and others, using techniques from singularity theory.

Consistent thermodynamics and topological classes for the four-dimensional Lorentzian charged Taub-NUT spacetimes

In this paper, we derive the consistent thermodynamics of the four-dimensional Lorentzian Reissner-Nordström-NUT (RN-NUT), Kerr–Newman-NUT (KN-NUT), and RN-NUT-AdS spacetimes in the framework of the (psi -mathscr {N})-pair formalism, and then investigate their topological numbers by using the uniformly modified form of the generalized off-shell Helmholtz free energy. We find that these solutions can be included into one of three categories of those well-known black hole solutions, which implies that these spacetimes should be viewed as generic black holes from the perspective of the topological thermodynamic defects. In addition, we demonstrate that although the existence of the NUT charge parameter seems to have no impact on the topological number of the charged asymptotically locally flat spacetimes, it has a remarkable effect on the topological number of the charged asymptotically locally AdS spacetime.

Pgm: A Python package for free energy calculations within the phonon gas model

The quasi-harmonic approximation (QHA) is a powerful method that uses the volume dependence of non-interacting phonons to compute the free energy of materials at high pressures (P) and temperatures (T). However, anharmonicity, electronic excitations in metals, or both, introduce an intrinsic T-dependence on phonon frequencies, rendering the QHA inadequate. Here we present a Python code, pgm, to compute the free energy and thermodynamic property within the phonon gas model (PGM) that uses T-dependent phonon quasiparticle frequencies. In this case, the vibrational contribution to the Helmholtz free energy is obtained by integrating the vibrational entropy, which can be readily calculated for a system of phonon quasiparticles. Other thermodynamic properties are then obtained from standard thermodynamic relations. We demonstrate the successful applications of pgm to two cases of geophysical significance: cubic CaSiO3-perovskite (cCaPv), a strongly anharmonic insulator and the third most abundant phase of the Earth's lower mantle, and NiAs-type (B8) FeO, a partially covalent-metallic system. This is the oxide endmember of a recently discovered iron-rich FenO alloy phase likely to exist in the Earth's inner core. Program summaryProgram Title:pgmCPC Library link to program files:https://doi.org/10.17632/8rfw6syvzp.1Developer's repository link:https://github.com/MineralsCloud/pgmLicensing provisions: GNU General Public License 3Programming language: Python3Nature of problem: The classic quasi-harmonic approximation (QHA) method to compute the vibrational free energy does not apply to physical systems when phonon frequencies have an intrinsic and non-negligible temperature (T) dependence. Examples are anharmonic systems or metals with abundant electronic thermal excitations. Both cases introduce an intrinsic T-dependence on phonon frequencies.Solution method: The method implemented in pgm is based on the phonon gas model where the entropy is well defined for T-dependent phonon quasiparticle. The free energy is calculated by integrating the entropy, making it suitable for anharmonic systems or systems with extensive thermal electronic excitations affecting phonon frequencies. The static free energy, the vibrational density of states (VDoS), and the entropy are first computed on sparse T- and P-grids. The entropy is then suitably interpolated on denser user-specified grids for integration.Additional comments, including restrictions and unusual features: The package allows it to be run directly in the command line. It can also be incorporated into other programs. We implemented Just-in-time (JIT) compiling and parallel computing techniques [1] in pgm Python code to speed up the numerical calculation.

Black holes in R2-dRGT massive gravity

Massive gravity offers an interesting modification of general relativity by considering a nonzero mass for the graviton. We present a de Rham, Gabadadze and Tolley (dRGT) massive gravity model in the presence of higher order curvature gravity. We obtain a spherical solution for the field equations in this theory. The spherical solution possesses an electric charge and a cosmological constant, and it reduces to the Schwarzschild solution in the limit of a negligible graviton mass and the absence of the higher order term. We study the thermodynamics of this black hole and find that the mass of the graviton and the higher order term of gravity have a significant effect on the thermodynamic properties of the black hole. We also show the entropy of a black hole is independent of the mass of the graviton. Using the observational constraints on the coefficient of the higher order term, we determine an upper limit for the graviton mass. We compute the other thermodynamic quantities, such as heat capacity and Helmholtz free energy. Assuming that the spherical solutions are the modified Schwarzschild, we consider the relativistic thin accretion disk and study the effects of higher order term on thermal properties of the disk at the infrared limit in an asymptotic safety scenario.

Thermodynamics, statistical mechanics and the vanishing pore width limit of confined fluids

Temperature, particle number and volume are the independent variables of the Helmholtz free energy for a bulk fluid. For a fluid confined in a slit pore between two walls, they are usually complemented by the surface area. However, an alternative choice is possible with the volume replaced by the pore width. Although the formulations with such two sets of independent variables are different, we show they are equivalent and present their relations. Corresponding general statistical-mechanics results are also presented. When the pore width becomes very small, the system behaves rather like a two-dimensional (2D) fluid and one can wonder if thermodynamics still holds. We find it remains valid even in the limit of vanishing pore width and show how to treat the divergences in the normal pressure and the chemical potential so that the corresponding 2D results can be obtained. Thus, we show that the Gibbs surface thermodynamics is perfectly capable of describing small systems.