Negative Specific Heat Research Articles

Transition of velocity distributions in collapsing self-gravitatingN-body systems

By means of N-body simulations, we study the evolution of gravity-dominated systems from an early relaxation to a collapse, focusing on the velocity distributions and thermodynamic properties. To simulate the dynamical evolution, we consider self-gravitating small N-body systems enclosed in a spherical container with adiabatic or semipermeable walls. It is demonstrated that in the early relaxation process, the velocity distribution is non-Gaussian and q-Gaussian, since the system is in quasiequilibrium states (here q is the Tsallis entropic parameter). Thereafter, the velocity distribution undergoes higher non-Gaussian distributions, especially when the core forms rapidly in the collapse process; i.e., q tends to be larger than that for the quasiequilibrium state, since the velocity distribution further deviates from Gaussian. However, after the core forms sufficiently, the velocity distribution gradually relaxes toward a Gaussian-like distribution. Accordingly, the velocity distribution evolves from a non-Gaussian distribution through a higher non-Gaussian distribution to a Gaussian-like distribution; i.e., the velocity distribution does not monotonically relax toward a Gaussian-like distribution in our collapse simulations. We clearly show such a transition of the velocity distribution, based not only on the Tsallis entropic parameter but also on the ratio of velocity moments. We also find that a negative specific heat occurs in a collapse process with mass and energy loss (such as the escape of stars from globular clusters), even if the velocity distribution is Gaussian-like.

Melting of Argon Cluster: Dependence of Caloric Curves on MD Simulation Parameters

We report on molecular dynamics simulations performed using microcanonical ensemble to predict the melting of argon particles in nanometer size range 10 nm and to investigate the effect of the time step integration. We use the Lennard- Jones potential functions to describe the interatomic interactions, and the results are evaluated by using caloric curves of the melting phenomenon. Thermodynamic properties, including the total energy, Lindemann parameter, kinetic and potential distribution’s functions, are used to characterize the melting process. The data shows bimodal behavior only in a certain interval of integration time step Δt, while the internal energy increases monotonically with the temperature. For the other time step values, the back bending disappears. We claim that negative specific heat is related to a possible decrease of entropy in an isolated system; this can be interpreted as a result of the internal interactions, especially attractive process and specific relaxation time.

Time-Reversibility, Instability and Thermodynamics in N-body Systems Interacting with Long-Range Potentials

Long-range attractive potentials cause self-gravitating N-body systems to exhibit not only chaotic behavior but also peculiar features such as gravothermal catastrophe, negative specific heat and nonextensive statistical mechanics. Especially when its potential energy is significantly dominated, a system should gradually evolve from quasi-equilibrium states through collapses to core-halo states. In dynamical evolution, velocity distributions are generally expected to monotonically relax from non-Gaussian towards Gaussian (Maxwell–Boltzmann) distributions. To clarify the velocity relaxation, we numerically examine the long-term evolution of a self-gravitating N-body system enclosed in a spherical container with adiabatic walls. We found that the velocity distribution non-monotonically relaxes from a non-Gaussian distribution to a Gaussian-like distribution when a core forms rapidly through the collapse process.

Thermodynamic anomaly of the free damped quantum particle: the bath perspective

A possible definition of the specific heat of open quantum systems is based on the reduced partition function of the system. For a free damped quantum particle, it has been found that under certain conditions, this specific heat can become negative at low temperatures. In contrast to the conventional approaches focusing on the system degree of freedom, here we concentrate on the changes induced in the environment when the system is coupled to it. Our analysis is carried out for an Ohmic environment consisting of harmonic oscillators and allows to identify the mechanism by which the specific heat becomes negative. Furthermore, the formal condition for the occurrence of a negative specific heat is given a physical interpretation in terms of the total mass of bath oscillators and the system mass.

Equilibrium statistical mechanics for self-gravitating systems: local ergodicity and extended Boltzmann-Gibbs/White-Narayan statistics

The long-standing puzzle surrounding the statistical mechanics of self-gravitating systems has not yet been solved successfully. We formulate a systematic theoretical framework of entropy-based statistical mechanics for spherically symmetric collisionless self-gravitating systems. We use an approach that is very different from that of the conventional statistical mechanics of short-range interaction systems. We demonstrate that the equilibrium states of self-gravitating systems consist of both mechanical and statistical equilibria, with the former characterized by a series of velocity-moment equations and the latter by statistical equilibrium equations, which should be derived from the entropy principle. The velocity-moment equations of all orders are derived from the steady-state collisionless Boltzmann equation. We point out that the ergodicity is invalid for the whole self-gravitating systems, but it can be re-established locally. Based on the local ergodicity, using Fermi-Dirac-like statistics, with the nondegenerate condition and the spatial independence of the local microstates, we rederive the Boltzmann-Gibbs entropy. This is consistent with the validity of the collisionless Boltzmann equation, and should be the correct entropy form for collisionless self-gravitating systems. Apart from the usual constraints of mass and energy conservation, we demonstrate that the series of moment or virialization equations must be included as additional constraints on the entropy functional when performing the variational calculus; this is an extension to the original prescription by White & Narayan. Any possible velocity distribution can be produced by the statistical-mechanical approach that we have developed with the extended Boltzmann-Gibbs/White-Narayan statistics. Finally, we discuss the questions of negative specific heat and ensemble inequivalence for self-gravitating systems.

Thermodynamics of phase transition in higher dimensional AdS black holes

We investigate the thermodynamics of phase transition for (n + 1) dimensional Reissner Nordstrom (RN)-AdS black holes using a grand canonical ensemble. This phase transition is characterized by a discontinuity in specific heat. The phase transition occurs from a lower mass black hole with negative specific heat to a higher mass black hole with positive specific heat. By exploring Ehrenfest’s scheme we show that this is a second order phase transition. Explicit expressions for the critical temperature and critical mass are derived. In appropriate limits the results for (n + 1) dimensional Schwarzschild AdS black holes are obtained.

Synthesis, structural and thermal characterization of metaphosphatenickel(II) salt

A new inorganically template metaphosphate of Ni(II) complex has been synthesized and characterized by different measurements such as DSC, FT-IR, C–H–N–S, X-RD and ICP-AES. Differential scanning calorimeter (DSC) elucidated negative specific heat of the system and has used to evaluate some thermodynamical constants like specific heat, enthalpy and entropy of that system. The specific heat capacity of the system is measured in atmospheric O2 at heating rate of 278 and 283 K min−1. The specific heat is found both positive and negative at 278 K min−1.

Synthesis and thermal characterization of sulfur containing methionine bridged cobalt(III) and copper(II) complex

The sulfur containing amino acid bridging polynuclear transition metal complex has been synthesized and characterized by different measurements such as UV- Vis, FT-IR, C-H-N-S, TG-DTA, ICP-AES, differential scanning calorimeter (DSC), and XRD. DSC has showed negative specific heat of this polynuclear system and has used to evaluate some thermodynamic constants like acti- vation energy (Ea), frequency factor (A), enthalpy, and entropy of that system. The specific heat capacity is mea- sured at heating rate of 10 � C min -1 in room atmosphere of this polynuclear complex. The characterization of this complex has showed five Co(III) and four Cu(II) atoms and this complex contained ten sulfur containing methionine amino acid units.

Nonanalyticities of thermodynamic functions in finite noninteracting Bose gases within an exact microcanonical ensemble

Within an exact microcanonical (MC) ensemble, we study the nonanalyticities of thermodynamic functions research in finite noninteracting Bose gases in traps. The results show that there exists a rich oscillatory behavior of MC thermodynamical quantities as a function of a system's total energy E (e.g., nonmonotonous temperature, nonanalytic and negative specific heats, and microscopic phase transitions). The origin of these nonanalyticities comes directly from the inverted curvature entropy S(E) with respect to E and the behaviors are different in different trap geometries, boundary conditions, and energy spectrum configurations. Contrary to the usual grandcanonical and canonical results, there exists Bose condensation and the nonanalyticities in the two-dimensional finite noninteracting Bose systems with different traps. We also discuss the critical temperature dependence on the particle number N with different ensembles, traps, and boundary conditions. In large enough N, almost all the results of the thermodynamical quantities become smooth, which are similar to the usual canonical behaviors. We emphasize the finite-size effects on the MC entropy change, which should, in principle, be observable in suitably designed experiments of the small systems.

Can the correlated stability conjecture be saved?

Correlated stability conjecture (CSC) proposed by Gubser and Mitra [1,2] linked the thermodynamic and classical (in)stabilities of black branes. In [3] it was shown that the thermodynamic instabilities, specifically the negative specific heat, indeed result in the instabilities in the hydrodynamic spectrum of holographically dual plasma excitations. Counter-examples of CSC were presented in the context of black branes with scalar hair undergoing a second-order phase transition [4,5]. The latter translationary invariant horizons have scalar hair, raising the question whether the asymptotic parameters of the scalar hair can be appropriately interpreted as additional charges leading to a generalization of the thermodynamic stability criterion. In this paper we show that the generalization of the thermodynamic stability criterion of this type can not save CSC. We further present a simple statistical model which makes it clear that thermodynamic and dynamical (in)stabilities generically are not correlated.

Thermodynamics of the HMF model with a magnetic field

We study the thermodynamics of the Hamiltonian Mean Field (HMF) model with an external potential playing the role of a "magnetic field". If we consider only fully stable states, this system does not present any phase transition. However, if we take into account metastable states (for a restricted class of perturbations), we find a very rich phenomenology. In particular, the system displays a region of negative specific heats in the microcanonical ensemble in which the temperature decreases as the energy increases. This leads to ensembles inequivalence and to zeroth order phase transitions similar to the "gravothermal catastrophe" and to the "isothermal collapse" of self-gravitating systems. In the present case, they correspond to the reorganization of the system from an "anti-aligned" phase (magnetization pointing in the direction opposite to the magnetic field) to an "aligned" phase (magnetization pointing in the same direction as the magnetic field). We also find that the magnetic susceptibility can be negative in the microcanonical ensemble so that the magnetization decreases as the magnetic field increases. The magnetic curves can take various shapes depending on the values of energy or temperature. We describe hysteretic cycles involving positive or negative susceptibilities. We also show that this model exhibits gaps in the magnetization at fixed energy, resulting in ergodicity breaking.

Microcanonical Description of (Micro) Black Holes

The microcanonical ensemble is the proper ensemble to describe black holes which are not in thermodynamic equilibrium, such as radiating black holes. This choice of ensemble eliminates the problems, e.g., negative specific heat (not allowed in the canonical ensemble) and loss of unitarity, encountered when the canonical ensemble is used. In this review we present an overview of the weaknesses of the standard thermodynamic description of black holes and show how the microcanonical approach can provide a consistent description of black holes and their Hawking radiation at all energy scales. Our approach is based on viewing the horizon area as yielding the ensemble density at fixed system energy. We then compare the decay rates of black holes in the two different pictures. Our description is particularly relevant for the analysis of micro-black holes whose existenceis predicted in models with extra-spatial dimensions.

Correlated stability conjecture revisited

Correlated stability conjecture (CSC) proposed by Gubser and Mitra (2000, 2001) [1,2] linked the thermodynamic and classical (in)stabilities of black branes. The classical instabilities, whenever occurring, were conjectured to arise as Gregory–Laflamme (GL) instabilities of translationally invariant horizons. In Buchel (2005) [3] it was shown that the thermodynamic instabilities, specifically the negative specific heat, indeed result in the instabilities in the hydrodynamic spectrum of holographically dual plasma excitations. A counter-example of CSC was presented in the context of black branes with scalar hair undergoing a second-order phase transition (Friess et al., 2005 [4]). In this Letter we discuss a related counter-example of CSC conjecture, where a thermodynamically stable translationally invariant horizon has a genuine tachyonic instability. We study the spectrum of quasinormal excitations of a black brane undergoing a continuous phase transition, and explicitly identify the instability. We compute the critical exponents of the critical momenta and the frequency of the unstable fluctuations and identify the dynamical critical exponent of the model.

Ensemble inequivalence in the ferromagneticp-spin model in random fields

We study the effect that randomness has on long-range interacting systems by using the ferromagnetic Isingmodel with p-body interactions in random fields. The case withp = 2 yields a phase diagram similar to that of previously studied models and shows knownfeatures that the inequivalence of the canonical and microcanonical ensembles brings withit, for example negative specific heat in a narrow region of the phase diagram. Whenp > 2, however, the canonical phase diagram is completely different from the microcanonical one.The temperature does not necessarily determine the microcanonical phases uniquely, andthus the ferromagnetic and paramagnetic phases are not separated in such aregion of a conventional phase diagram drawn with the temperature and fieldstrength as the axes. Below a certain value of the external field strength, partof the ferromagnetic phase has negative specific heat. For large values of theexternal field strength the ergodicity is broken before the phase transition occurs forp > 2. Moreover,for p > 2, the Maxwell construction cannot be derived in a consistent manner and therefore, incontrast to previous cases with negative specific heat, the Maxwell construction does notbridge the gap between the ensembles.

Statistical mechanics of two-dimensional Euler flows and minimum enstrophy states

A simplified thermodynamic approach of the incompressible 2D Euler equation is considered based on the conservation of energy, circulation and microscopic enstrophy. Statistical equilibrium states are obtained by maximizing the Miller-Robert-Sommeria (MRS) entropy under these sole constraints. The vorticity fluctuations are Gaussian while the mean flow is characterized by a linear $\bar{\omega}-\psi$ relationship. Furthermore, the maximization of entropy at fixed energy, circulation and microscopic enstrophy is equivalent to the minimization of macroscopic enstrophy at fixed energy and circulation. This provides a justification of the minimum enstrophy principle from statistical mechanics when only the microscopic enstrophy is conserved among the infinite class of Casimir constraints. A new class of relaxation equations towards the statistical equilibrium state is derived. These equations can provide an effective description of the dynamics towards equilibrium or serve as numerical algorithms to determine maximum entropy or minimum enstrophy states. We use these relaxation equations to study geometry induced phase transitions in rectangular domains. In particular, we illustrate with the relaxation equations the transition between monopoles and dipoles predicted by Chavanis and Sommeria [J. Fluid. Mech. 314, 267 (1996)]. We take into account stable as well as metastable states and show that metastable states are robust and have negative specific heats. This is the first evidence of negative specific heats in that context. We also argue that saddle points of entropy can be long-lived and play a role in the dynamics because the system may not spontaneously generate the perturbations that destabilize them.

The extended Gaussian ensemble and metastabilities in the Blume–Capel model

The Blume-Capel model with infinite-range interactions presents analytical solutions in both canonical and microcanonical ensembles and therefore, its phase diagram is known in both ensembles. This model exhibits nonequivalent solutions and the microcanonical thermodynamical features present peculiar behaviors like nonconcave entropy, negative specific heat, and a jump in the thermodynamical temperature. Examples of nonequivalent ensembles are in general related to systems with long-range interactions that undergo canonical first-order phase transitions. Recently, the extended gaussian ensemble (EGE) solution was obtained for this model. The gaussian ensemble and its extended version can be considered as a regularization of the microcanonical ensemble. They are known to play the role of an interpolating ensemble between the microcanonical and the canonical ones. Here, we explicitly show how the microcanonical energy equilibrium states related to the metastable and unstable canonical solutions for the Blume-Capel model are recovered from EGE, which presents a concave "extended" entropy as a function of energy.

Schwarzschild–anti-de Sitter black holes within isothermal cavity: Thermodynamics, phase transitions, and the Dirichlet problem

The thermodynamics of Schwarzschild black holes within an isothermal cavity and the associated Euclidean Dirichlet boundary-value problem are studied for four and higher dimensions in anti-de Sitter (AdS) space. For such boundary conditions classically there always exists a unique hot AdS solution and two or no Schwarzschild-AdS black-hole solutions depending on whether or not the temperature of the cavity-wall is above a minimum value, the latter being a function of the radius of the cavity. Assuming the standard area-law of black-hole entropy, it was known that larger and smaller holes have positive and negative specific heats and hence are locally thermodynamically stable and unstable respectively. In this paper we present the first derivation of this by showing that the standard area law of black-hole entropy holds in the semi-classical approximation of the Euclidean path integral for such boundary conditions. We further show that for wall-temperatures above a critical value a phase transition takes hot AdS to the larger Schwarzschild-AdS within the cavity. The larger hole thus can be globally thermodynamically stable above this temperature. The phase transition can occur for a cavity of arbitrary radius above a (corresponding) critical temperature. In the infinite cavity limit this picture reduces to that considered by Hawking and Page. The case of five dimensions is found to be rather special since exact analytic expressions can be obtained for the masses of the two holes as functions of cavity radius and temperature thus solving exactly the Euclidean Dirichlet problem. This makes it possible to compute the on-shell Euclidean action as functions of them from which other quantities of interest can be evaluated exactly.

Thermodynamic properties of an evaporation process in self-gravitatingN-body systems

By means of N -body simulations, we consider self-gravitating open systems enclosed in a spherical container with semipermeable reflecting walls, in order to investigate the thermodynamics of the evaporation process in self-gravitating N -body systems (such as the escape of stars from globular clusters). To simulate the evaporation process, when the energy of a particle exceeds a certain threshold value, the particle passes through the semipermeable reflecting wall freely. We show that the thermodynamic properties of the evaporation process, such as the dependence of the temperature on energy, agree well with those of stellar polytropes, if the system is in an approximate virial equilibrium state. However, in a lower-energy region or for a rapid evaporation process, the thermodynamic properties deviate from those for the stellar polytrope. Nevertheless, we found that a negative specific heat occurs even in the lower-energy region or for a rapid evaporation process.

Searching for Insight

Is space-time only brought into being by its energy content? The jury is still out, but other questions that have been with me for much of my life—giant black holes in galactic nuclei, the formation of the Galaxy, the connection between first-order phase transitions and negative specific heats, the cause of the large-scale flow of galaxies relative to the cosmic microwave background—have all received reasonable answers. I have found great fun in understanding the dynamical mechanisms underlying such phenomena as magnetohydrodynamic jets, relativistic disks, and the bars, spirals, and chemical evolution of galaxies. The great challenges for future astronomers will be the exploration of the 96% of the Universe now believed to be neither atomic nor baryonic but perhaps partially leptonic. However, most advances do not come via frontal attack but from “bread-and-butter” investigations in related areas where observation is possible today!

Inhomogeneous Tsallis distributions in the HMF model

We study the maximization of the Tsallis functional at fixed mass and energy in the HMF model. We give a thermodynamical and a dynamical interpretation of this variational principle. This leads to q-distributions known as stellar polytropes in astrophysics. We study phase transitions between spatially homogeneous and spatially inhomogeneous equilibrium states. We show that there exists a particular index q_c=3 playing the role of a canonical tricritical point separating first and second order phase transitions in the canonical ensemble and marking the occurence of a negative specific heat region in the microcanonical ensemble. We apply our results to the situation considered by Antoni & Ruffo [Phys. Rev. E 52, 2361 (1995)] and show that the anomaly displayed on their caloric curve can be explained naturally by assuming that, in this region, the QSSs are polytropes with critical index q_c=3. We qualitatively justify the occurrence of polytropic (Tsallis) distributions with compact support in terms of incomplete relaxation and inefficient mixing (non-ergodicity). Our paper provides an exhaustive study of polytropic distributions in the HMF model and the first plausible explanation of the surprising result observed numerically by Antoni & Ruffo (1995). In the course of our analysis, we also report an interesting situation where the caloric curve presents both microcanonical first and second order phase transitions.