Extensivity and the thermodynamic limit: Why size really does matter

The thermodynamic limit and extensivity are central concepts in thermodynamics. In this paper, these are critically examined in light of systems for which they appear inadequate. It is found that their limitations lead to counterintuitive thermodynamic results involving heat flow, phase separations, thermostatistics of gravitating systems and the conversion efficiency of heat into work. Ultimately, these limitations are shown to bear on the utility of entropy and the universality of the second law of thermodynamics.     

Energy conditions of the f(T,B) gravity dark energy model with the validity of thermodynamics

In this article, the analysis of Tsallis holographic dark energy(which turns into holographic dark energy for a particular choice of positive non-additivity parameter δ) in modified f(T, B) gravity with the validity of thermodynamics and energy conditions for a homogeneous and isotropic FLRW Universe has been studied. The enlightenment of the field equation towards f(T,B)=αT~m+βB~n, made possible by the fact that the model is purely accelerating,corresponds to q=-0.54(Mamon and Das 2017 Eur. Phys.J.C 77 49). The generalized second law of thermodynamics is valid not only for the same temperature inside the horizon, but also for the apparent horizon for a change in temperature. The essential inspiration driving this article is to exhibit the applicability that the holographic dark energy achieved from standard Tsallis holographic dark energy and the components acquired from f(T, B) gravity are identical for the specific bounty of constants. The analysis of energy conditions confirms that the weak energy condition and the null energy condition are fulfilled throughout the expansion, while violation of the strong energy condition validates the accelerated expansion of the Universe.With the expansion, the model becomes a quintessence dominated model. The dominant energy condition is not observed initially when the model is filled with genuine baryonic matter,whereas it appears when the model is in the quintessence dominated era.     

Thermodynamics of the Apparent Horizon in FRW Universe with Massive Gravity

Applying Clausius relation with energy-supply defined by the unified first law of thermodynamics formalism to the apparent horizon of a massive gravity model in cosmology proposed lately, the corrected entropic formula of the apparent horizon is obtained with the help of the modified Friedmann equations. This entropy-area relation, together with the identified Misner-Sharp internal energy, verifies the first law of thermodynamics for the apparent horizon with a volume change term for consistency. On the other hand, by means of the corrected entropy-area formula and the Clausius relation δQ=T dS, where the heat flow δQ is the energy-supply of pure matter projecting on the vector ζ tangent to the apparent horizon and should be looked on as the amount of energy crossing the apparent horizon during the time interval dt and the temperature of the apparent horizon for energy crossing during the same interval is 1/2πrA, the modified Friedmann equations governing the dynamical evolution of the universe are reproduced with the known energy density and pressure of massive graviton. The integration constant is found to correspond to a cosmological term which could be absorbed into the energy density of matter. Having established the correspondence of massive cosmology with the unified first law of thermodynamics on the apparent horizon, the validity of the generalized second law of thermodynamics is also discussed by assuming the thermal equilibrium between the apparent horizon and the matter field bounded by the apparent horizon. It is found that, in the limit H_c→0, which recovers the Minkowski reference metric solution in the flat case, the generalized second law of thermodynamics holds if α₃+4α₄<0. Without this condition, even for the simplest model of dRGT massive cosmology with α₃=α₄=0, the generalized second law of thermodynamics could be violated.     

Two long-standing problems in Tsallis’ statistics

On the basis of Tsallis’ entropy and the joint probability factorization condition, two controversial problems existing in Tsallis’ statistics are investigated, where one is whether energy is extensive or not and the other is whether it is necessary to introduce the so-called generalized zeroth law of thermodynamics or not. The results obtained show clearly that like entropy, energy is also nonextensive in Tsallis’ statistics, and that the zeroth law of thermodynamics has been implicitly used in Tsallis’ statistics since 1988. Moreover, it is expounded that the standard energy additivity rule adopted by a great number of researchers is not suitable in Tsallis’ statistics, because its corollary is in contradiction with the zeroth law of thermodynamics.     

Interacting three fluid system and thermodynamics of the universe bounded by the event horizon

The work deals with the thermodynamics of the universe bounded by the event horizon. The matter in the universe has three constituents namely dark energy, dark matter and radiation in nature and interaction between then is assumed. The variation of entropy of the surface of the horizon is obtained from unified first law while matter entropy variation is calculated from the Gibbss’ law. Finally, validity of the generalized second law of thermodynamics is examined and conclusions are written point wise.     

Thermodynamics and phase transition of topological dS black holes with a nonlinear source

We discuss black hole solutions of Einstein-Λ gravity in the presence of nonlinear electrodynamics in d S spacetime. Considering the correlation of the thermodynamic quantities respectively corresponding to the black hole horizon and cosmological horizon of dS spacetime and taking the region between the two horizons as a thermodynamic system, we derive effective thermodynamic quantities of the system according to the first law of thermodynamics, and investigate the thermodynamic properties of the system under the influence of nonlinearity parameter α. It is shown that nonlinearity parameter α influences the position of the black hole horizon and the critical state of the system, and along with electric charge has an effect on the phase structure of the system,which is obvious, especially as the effective temperature is below the critical temperature. The critical phase transition is proved to be second-order equilibrium phase transition by using the Gibbs free energy criterion and Ehrenfest equations.     

Fisher information as the basis for Maxwell's equations

Maxwell's equations of classical electrodynamics may be derived on the following statistical basis. Consider a gedanken experiment whereby the mean space-time coordinate for photons in an electromagnetic field is to be determined by observation of one photon's space-time coordinate. An efficient (i.e. optimum) estimate obeys a condition of minimum Fisher information, or minimum precision, according to the second law of thermodynamics. The Fisher information I is a simple functional of the probability law governing space-time coordinates of the “particles” of the field. This probability law is modeled as the source-free Poynting energy flow density, i.e., the ordinary local intensity in the optical sense, or, the square of the four-vector potential. When the Fisher information is extremized subject to an additive constraint term in the total interaction energy, Maxwell's equations result.     

Thermodynamical Aspects of Modified Holographic Dark Energy Model

We investigate the unified first law and the generalized second law in a modified holographic dark energy model. The thermodynamical analysis on the apparent horizon can work and the corresponding entropy formula is extracted from the systematic algorithm. The entropy correction term depends on the extra-dimension number of the brane as expected, but the interplay between the correction term and the extra dimensions is more complicated. With the unified first law of thermodynamics well-founded, the generalized second law of thermodynamics is discussed and it is found that the second law can be violated in certain circumstances. Particularly, if the number of the extra dimensions is larger than one, the generalized law of thermodynamics is always satisfied; otherwise, the validity of the second law can only be guaranteed with the Hubble radius greatly smaller than the crossover scale r_c of the 5-dimensional DGP model.     

The phase diagram of oxygen ordering in YBa2Cu3O6+δ: The nine point cluster approximation

The thermodynamics of oxygen ordering in the basal plane of YBa₂Cu₃O_6+δ is studied within the nine point approximation of the cluster variation method. The phenomena is modeled by an Ising model with nearest-and next-nearest-neighbor interactions. The composition (δ) versus the temperature (T) phase diagram is calculated and compared with results obtained by using smaller clusters and the Monte Carlo method. The composition dependence of the order parameters for various temperatures is also presented.     

Thermodynamics of Chaplygin Gas Interacting with Cold Dark Matter

The main goal of the present work is to investigate the validity of the second law of gravitational thermodynamics in an expanding Gödel-type universe filled with generalized Chaplygin gas interacting with cold dark matter. By assuming the Universe as a thermodynamical system bounded by the apparent horizon, and calculating separately the entropy variation for generalized Chaplygin gas, cold dark matter and for the horizon itself, we obtained an expression for the time derivative of the total entropy. We conclude that the 2nd law of gravitational thermodynamics is conditionally valid in the cosmological scenario where the generalized Chaplygin gas interacts with cold dark matter.     

Supercritical fluid thermodynamics from equations of state

Supercritical multicomponent fluid thermodynamics are often built from equations of state. We investigate mathematically such a construction of a Gibbsian thermodynamics compatible at low density with that of ideal gas mixtures starting from a pressure law. We further study the structure of chemical production rates obtained from nonequilibrium statistical thermodynamics. As a typical application, we consider the Soave-Redlich-Kwong cubic equation of state and investigate mathematically the corresponding thermodynamics. This thermodynamics is then used to study the stability of H₂-O₂-N₂ mixtures at high pressure and low temperature as well as to illustrate the role of nonidealities in a transcritical H₂-O₂-N₂ flame.     

Surface morphology and growth kinetics of zinc single crystals as influenced by vapour diffusion

The effect of diffusion in the vapour phase on both the surface morphology and growth kinetics of zinc single crystals growing in the presence of argon has been investigated. The shift of the basal face in the normal direction and the growth of the edge of that face as a function of time have been determined at constant temperature and supersaturation and at argon pressures within the range 5–250 Torr as well as in high vacuum of 1 × 10⁻⁶ Torr. It has been found that the crystal size R and the edge length a change with time t following a linear law in the kinetic and a parabolic law in the diffusion regime respectively. The results obtained by kinetic investigations and morphological observations of the basal face are compared.     

Statistical thermodynamics from the point of view of statistical theory of estimation

An essentially statistical theory of thermodynamics is developed on the basis of additivity and conservation laws of fundamental entities of thermodynamics. The theory developed by Dutta (1953–59) stresses first the importance of the statistical theory of estimation in the statistical foundation of thermodynamics. The method of finding the distribution law is based on the principle similar to that of Bayes' rule in probability theory and the method of maximum-likelihood estimation of mathematical statistics. The model used is macroscopic in nature and the laws of microscopic distributions (M-B, B-E and F-D statistics) have been obtained by additional axiom regarding the constituents of the system. The object of the paper is to present an outline of the theory with some of its extensions to quantum macro-system and non-equilibrium thermodynamics.     

Thermodynamics in the statistical theory of a crystal

The thermodynamics of a crystal is developed on the basis of the Vlasov equation. From this equation a system of equations is derived for the Fourier coefficients of the density. The internal energy and stress tensor are found. It is shown that the results obtained satisfy the second law of thermodynamics and the entropy is calculated.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 12, pp. 85–88, December, 1978.     

Thermodynamics of Evolving Lorentzian Wormhole at Apparent and Event Horizons

We have investigated the non-static Lorentzian Wormhole model in presence of anisotropic pressure. We have presented some exact solutions of Einstein equations for anisotropic pressure case. Introducing two EoS parameters we have shown that these solutions give very rich dynamics of the universe yielding to the different expansion history of it in the r - direction and in the T - direction. The corresponding explicit forms of the shape function b(r) is presented.We have shown that the Einstein’s field equations and unified first law are equivalent for the dynamical wormhole model. The first law of thermodynamics has been derived by using the Unified first law. The physical quantities including surface gravity and the temperature are derived for the wormhole. Here we have obtained all the results without any choice of the shape function. The validity of generalized second law (GSL) of thermodynamics has been examined at apparent and event horizons for the evolving Lorentzian wormhole.     

Thermodynamics of the Early Universe

The relationship between relativistic thermodynamics of the early Universe with the Logunov metric and a gravitational analog of statistical mechanics is examined. An equation of state for gravitational atoms is derived. These atoms can be the medium that gave rise to the contents of our Universe or miniUniverses. A gravitational analog of the first law of thermodynamics is obtained. It is also found that the symmetrical in time Liouville equation can have a partial solution with a broken symmetry in time.__________Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 3, pp. 7–17, March, 2005.     

On the Hebb–Wagner polarisation of SrTiO3 doped with redox-active ions

Hebb–Wagner polarisation is analysed for the case that internal redox-reactions complicate the situation. In the general case current–voltage characteristics deviate significantly from the Hebb–Wagner equation. Analytical results are derived and relevant approximations given. Theoretical and experimental current–voltage relations obtained for a 2.15×10¹⁸ cm⁻³ Fe-doped SrTiO₃ single crystal at P_O2=10⁵ Pa, agree reasonably well, and the results of the partial conductivities match findings obtained from other experiments. In addition, the fact is emphasised that a Hebb–Wagner-type current–voltage relation only requires the fulfilment of a power law for the non-blocked carrier concentrations with respect to the component partial pressure and not a field-free situation. In this case the Faraday constant appearing in the Hebb–Wagner equation has to be replaced by F, where is the product of the number of electrons necessary to ionise the gaseous component (n) times the absolute characteristic exponent (|N|). The condition of zero field, i.e. negligible chemical potential gradient with respect to the non-blocked species is identical to |N|=1/n. If |N|=const≠1/n, the internal field is non-zero but is still proportional to the gradient of the component potential.     

Thermodynamics in Higher Dimensional Vaidya Space-Time

In this work, we have considered the Vaidya spacetime in null radiating fluid with perfect fluid in higher dimension and have found the solution for barotropic fluid. We have shown that the Einstein’s field equations can be obtained from Unified first law i.e., field equations and unified first law are equivalent. The first law of thermodynamics has also been constructed by Unified first law. From this, the variation of entropy function has been derived on the horizon. The variation of entropy function inside the horizon has been derived using Gibb’s law of thermodynamics. So the total variation of entropy function has been constructed at apparent and event horizons both. If we do not assume the first law, then the entropy on the both horizons can be considered by area law and the variation of total entropy has been found at both the horizons. Also the validity of generalized second law (GSL) of thermodynamics has been examined at both apparent and event horizons by using the first law and the area law separately. When we use first law of thermodynamics and Bekenstein-Hawking area law of thermodynamics, the GSL for apparent horizon in any dimensions are satisfied, but the GSL for event horizon can not be satisfied in any dimensions.     

Thermodynamical Aspects of Modified Holographic Dark Energy Model

We investigate the unified first law and the generalized second law in a modified holographic dark energy model. The thermodynamical analysis on the apparent horizon can work and the corresponding entropy formula is extracted from the systematic algorithm. The entropy correction term depends on the extra-dimension number of the brane as expected, but the interplay between the correction term and the extra dimensions is more complicated. With the unified first law of thermodynamics well-founded, the generalized second law of thermodynamics is discussed and it is found that the second law can be violated in certain circumstances. Particularly, if the number of the extra dimensions is larger than one, the generalized law of thermodynamics is always satisfied; otherwise, the validity of the second law can only be guaranteed with the Hubble radius greatly smaller than the crossover scale rcof the 5-dimensional DGP model.     

Fisher info and thermodynamics’ first law

We show, starting from first principles, that thermodynamics’ first law can be microscopically obtained for Fisher's information measure without need of invoking the adiabatic theorem. Further, it is proved that enforcing the Fisher-first law requirements in a process in which the probability distribution is infinitesimally varied is equivalent to minimizing Fisher's information measure subject to appropriate constraints.