Caloric Curve Research Articles

Random Transitions of a Binary Star in the Canonical Ensemble.

After reviewing the peculiar thermodynamics and statistical mechanics of self-gravitating systems, we consider the case of a "binary star" consisting of two particles of size a in gravitational interaction in a box of radius R. The caloric curve of this system displays a region of negative specific heat in the microcanonical ensemble, which is replaced by a first-order phase transition in the canonical ensemble. The free energy viewed as a thermodynamic potential exhibits two local minima that correspond to two metastable states separated by an unstable maximum forming a barrier of potential. By introducing a Langevin equation to model the interaction of the particles with the thermal bath, we study the random transitions of the system between a "dilute" state, where the particles are well separated, and a "condensed" state, where the particles are bound together. We show that the evolution of the system is given by a Fokker-Planck equation in energy space and that the lifetime of a metastable state is given by the Kramers formula involving the barrier of free energy. This is a particular case of the theory developed in a previous paper (Chavanis, 2005) for N Brownian particles in gravitational interaction associated with the canonical ensemble. In the case of a binary star (N=2), all the quantities can be calculated exactly analytically. We compare these results with those obtained in the mean field limit N→+∞.

Glass transition temperatures of binary oxides from ab initio simulations

The glass transition temperatures of common binary oxides, including those with low glass-forming ability, are estimated using pair distribution functions (PDFs) from ab initio molecular dynamics simulations. The computed glass transition temperatures for good glass-formers such as silica (SiO2), germania (GeO2), and boron oxide (B2O3) are in agreement with measured values. These calculations are then used to compute the glass transition temperatures of alumina (Al2O3), tantala (Ta2O5), and telluria (TeO2), which are known to exhibit low glass-forming ability. For Al2O3 and Ta2O5, we also compute the simulated caloric curve from molecular dynamics simulations using two-body empirical force fields. Finally, we discuss the possibility of extracting the glass transition temperature by measuring the thermal broadening of the PDFs from scattering measurements.

Apparent temperatures of neutron-poor and neutron-rich compound nuclei

The possibility of the dependence of the nuclear caloric curve on neutron excess sets a limit on the accuracy of our knowledge of the nuclear equation of state, and thus impacts predictive capabilities of nuclear reaction and nuclear astrophysics models. To date, theoretical models have not reached consensus on the magnitude or sign on the asymmetry dependence. To provide constraints, we have measured evaporated particles and heavy residues for complete and incomplete fusion-evaporation reactions in inverse kinematics. The temperatures extracted from the observed light charged particles tend to favor higher temperatures for the neutron-rich fused systems, though they are near the limits of the systematic uncertainty. The present measurement may be used as an upper limit to constrain the asymmetry dependence of the nuclear caloric curve.

The Thermal Stability of Asymmetric Separated Configurations inside Alloy Nanoparticles: Atomic-Scale Modeling of Pd-Ir Nanophase Diagrams.

Compared to alloy bulk phase diagrams, the experimental determination of phase diagrams for alloy nanoparticles (NPs), which are useful in various nanotechnological applications, involves significant technical difficulties, making theoretical modeling a feasible alternative. Yet, being quite challenging, modeling of separation nanophase diagrams is scarce in the literature. The task of predicting comprehensive nanophase diagrams for Pd-Ir face-centered cubic-based three cuboctahedra is facilitated in this study by combining the computationally efficient statistical-mechanical Free-energy Concentration Expansion Method, which includes short-range order (SRO) with coordination-dependent bond-energy variations as part of the input and with rotationally symmetric site grouping for extra efficiency. This nanosystem has been chosen mainly because of the very small atomic mismatch that simplifies the modeling, e.g., in the assessment of vibrational entropy contributions based in this work on fitting to the Pd-Ir experimental bulk critical temperature. This entropic effect, together with SRO, leads to significant destabilization of low-T Quasi-Janus (QJ) asymmetric configurations of the NP core, which transform to symmetric partially mixed nanophases. First-order and second-order intracore transitions are predicted for dilute and intermediate-range compositions, respectively. Caloric curves computed for the former case yield the NP-size dependent transition latent heat, and in the latter case critical temperatures exhibit a specific scaling behavior. The computed separation diagrams and intracore solubility diagrams reflect enhanced elemental mixing in smaller QJ nanophases. In addition to these diagrams, the revealed near-surface compositional variations are likely to be pertinent to the utilization of Pd-Ir NPs, e.g., in catalysis.

Melting behavior of Ir-Ag-Au nanoalloys: a molecular dynamic study

ABSTRACT MD simulations have been employed to investigate the melting behaviour of Ir-Ag-Au nanoalloys with two different systems: Ir43AgnAu12-n (n = 0-12) and some specific nanoalloys (Ir1Ag54, Ir1Au54, Ir13Ag42, Ir13Au42, Ir1Ag42Au12, and Ir13Ag12Au30). Energetically optimal homotops obtained with the local optimizations were taken as the initial configurations in this simulation. The local optimization results of the Ir-rich nanoalloys demonstrate that Ag and Au atoms occupy the vertex position on the surface. Caloric curve, heat capacity, and Lindemann index methods were used for examining the thermal stability of the Ag-Au surface structure. It was obtained that the melting temperature of Ir-rich nanoalloys depends on the composition. For melting transitions of Ir-rich nanoalloys, two types were identified: structural deformations around the melting temperature and sudden jump. It was discovered that the melting mechanism of some specific nanoalloys generally has surface and inner shell melting.

Self-gravitating clusters of Bose-Einstein gas with planar, cylindrical, or spherical symmetry: Gaseous density profiles and onset of condensation.

We calculate density profiles for self-gravitating clusters of an ideal Bose-Einstein gas with nonrelativistic energy-momentum relation and macroscopic mass at thermal equilibrium. Our study includes clusters with planar symmetry in dimensions D=1,2,3, clusters with cylindrical symmetry in D=2,3, and clusters with spherical symmetry in D=3. Wall confinement is imposed where needed to prevent escape. The length scale and energy scale in use for the gaseous phase render density profiles for gaseous macrostates independent of total mass. Density profiles for mixed-phase macrostates have a condensed core surrounded by a gaseous halo. The spatial extension of the core is negligibly small on the length scale tailored for the halo. The mechanical stability conditions as evident in caloric curves permit multiple macrostates to coexist. Their status regarding thermal equilibrium is examined by a comparison of free energies. The onset of condensation takes place at a nonzero temperature in all cases. The critical singularities and the nature of the phase transition vary with the symmetry of the cluster and the dimensionality of the space.

Investigation of the chemical ordering and thermal properties of the Rh–Ag–Au nanoalloys

In this paper, the melting behaviors of Rh–Ag–Au nanoalloys are investigated with MD simulation. For Rh–Ag–Au nanoalloys, icosahedron structure was considered. The local optimizations of Rh–Ag–Au nanoalloys were carried out with the BH algorithm. The interatomic interactions were modeled with the Gupta potential. The local optimization results of Rh–Ag–Au nanoalloys show that Au and Ag atoms prefer to locate on the surface, and Rh atoms prefer to locate in the inner shells. The bond order parameter result is compatible with the excess energy analysis. It is noted that structures with more Ag–Au bonds are more energetically stable. Caloric curve, heat capacity, Lindemann index, and RMSD methods were used for estimating the melting temperatures of Rh–Ag–Au nanoalloys. According to the simulation results, melting temperatures depend on the composition. Also, it is discovered that nanoalloys are generally melting in two stages. Surface melting of the third shell is occupied by Ag and Au atoms, and then homogeneous melting of the inner shells is occupied by Rh atoms. It is found that the difference between surface melting temperatures and homogeneous melting temperatures in Ag-poor compositions is more significant than that of Ag-rich nanoalloys. In addition, the melting temperatures of the nanoalloys are found to be increased as the size of nanoalloys increases.

Molecular dynamics and DFT study of 38-atom coinage metal clusters

The thermal behavior of 38-atom mono-, bi-, and trimetallic clusters consisting of Cu, Ag, and Au atoms, is analyzed employing molecular dynamics simulations and DFT calculations for selected cluster compositions. Low-energy structures were singled out to perform NVT molecular dynamics simulations at several temperatures, using the Andersen thermostat for temperature control. The caloric curve is used to estimate the melting temperature and the specific heat. The pair distribution function g(r) of the solid and liquid-phase clusters is examined at different temperatures. When comparing the estimated melting points (Tm) among the monatomic clusters, the order becomes TmCu38>TmAg38>TmAu38. For bimetallic clusters, an increase of Tm is observed for Cu-Au compared to their monatomic counterparts, while the opposite occurs for Cu-Ag clusters. For trimetallic clusters, two low-energy isomers of the Cu36Ag1Au1 cluster are investigated. In this case, Tm is estimated to be 475 K, for the two isomers with the lowest-energy and second-to-lowest energy, respectively. For all the clusters studied, the pair distribution function g(r) shows that the first peak position is not shifted as an effect of temperature and its maximum value varies with composition, while the second peak essentially vanishes upon melting. The common-neighbor analysis (CNA) technique is used to analyze the local structural changes for the trimetallic clusters, again demonstrating a clear structural change upon melting. The HOMO-LUMO energy gap indicates that the trimetallic isomers' behavior is metallic, while the average binding energy show these clusters' energetic stability to be similar.

Multiple metastable states in an off-lattice Potts model

The interactions between a group of components are commonly studied in several areas of science (social science, biology, material science, complex dynamical systems, among others) using the methods of thermodynamics and statistical mechanics. In this work we study the properties of the recently proposed off-lattice, two-dimensional Potts model (Davis et al., 2014), originally motivated by the dynamics of agent opinions, and which is described by a Hamiltonian obtained by a maximum entropy inference procedure. We performed microcanonical and canonical Monte Carlo simulations of the first-order phase transition in the model, revealing a caloric curve with metastable regions. Furthermore, we report a “switching” behavior between multiple metastable states. We also note that the thermodynamics of the model has striking similarities with systems having long-range interactions, even though the interactions are short-ranged.

Investigation of the nuclear liquid-gas phase transition in the static AMD

Nuclear liquid-gas phase transitions are investigated in the framework of static antisymmetrized molecular dynamics (static AMD) model under either a constant volume or a constant pressure. A deuteron quadrupole momentum fluctuation thermometer is applied to extract the temperature of fragmenting systems of 36Ar and 100Sn. A plateau structure of caloric curves is observed under a constant volume for those system with a density ρ ⩽ 0.03 fm−3. A clear backbending in the caloric curves, which indicates a first order phase transition, is observed under a constant pressure with all pressures studied. The similar behavior of caloric curves of 36Ar and 100Sn systems indicates that there is no strong system size effect under a constant volume or a constant pressure. Both the mass distributions and the light particle multiplicities show a strong α clusterization at low excitation energies in the static AMD simulations. The liquid-gas phase transition measures of the multiplicity derivative (dM/dT) and the normalized variance of Z max (NVZ) are applied. The experimental caloric curves are also compared with those of 100Sn of the static AMD simulations under both the constant volume and the constant pressure conditions. Discussions are presented with the available experimental results and those from the static AMD simulations. Large errors in the experimental temperature measurements and those in the reconstruction technique for the primary fragmenting source hinder to draw a conclusion whether the phase transition occurs under either a constant volume or a constant pressure. This study suggests that different measures for the liquid-gas phase transitions should be examined besides the caloric curves in order to draw a conclusion.

Probing Dense QCD Matter: Muon Measurements with the CBM Experiment at FAIR

The Compressed Baryonic Matter (CBM) experiment at the future Facility for Antiproton and Ion Research (FAIR) in Darmstadt is designed to investigate the properties of high-density QCD matter with multi-differential measurements of hadrons and leptons, including rare probes such as multi-strange anti-hyperons and charmed particles. The research program covers the study of the high-density equation-of-state of nuclear matter and the exploration of the QCD phase diagram at large baryon chemical potentials, including the search for quark matter and the critical endpoint of a hypothetical 1st order phase transition. The CBM setup comprises detector systems for the identification of charged hadrons, electrons, and muons; for the determination of collision centrality and the orientation of the reaction plane; and a free-streaming data read-out and acquisition system, which allows online reconstruction and selection of events up to reaction rates of 10 MHz. In this article, emphasis is placed on the measurement of muon pairs in Au-Au collisions at FAIR beam energies, which are unique probes used to determine the temperature of the fireball, and hence to search for a caloric curve of QCD matter. Simultaneously, the subthreshold production of charmonium can be studied via its dimuon decay in order to shed light on the microscopic structure of QCD matter at high baryon densities. The CBM setup with focus on dimuon measurements and the results of the corresponding physics performance studies will be presented.

Heating rate effects for the melting transition of Pt–Ag–Au nanoalloys

The classical molecular dynamics simulations in canonical NVT ensemble conditions are used to investigate the melting transition in different heating rates of Pt–Ag–Au ternary nanoalloys. In order to obtain the initial configurations used in the molecular dynamics simulations, optimizing the chemical ordering of Pt13Ag n Au42 – n (n = 0–42) ternary nanoalloys was performed using the Basin–Hopping algorithm which would not allow changes in the icosahedron structure. The Gupta many-body potential was used to model interatomic interactions in both molecular dynamics simulations and optimization simulations. The melting transitions of selected Pt–Ag–Au nanoalloys were explored using caloric curves and Lindemann parameters. There have been two identified types of melting mechanisms, one includes sudden jump behavior in the caloric curve and the other is an isomerization while melting transition. The temperature range in which the isomerization takes place depends on the heating rate value.

Determining the temperature in heavy-ion collisions with multiplicity distribution

By relating the charge multiplicity distribution and the temperature of a de-exciting nucleus through a deep neural network, we propose that the charge multiplicity distribution can be used as a thermometer of heavy-ion collisions. Based on an isospin-dependent quantum molecular dynamics model, we study the caloric curve of reaction Pd103 + Be9 with the apparent temperature determined through the charge multiplicity distribution. The caloric curve shows a characteristic signature of nuclear liquid-gas phase transition around the apparent temperature Tap = 6.4MeV, which is consistent with that through a traditional heavy-ion collision thermometer, and indicates the viability of determining the temperature in heavy-ion collisions with multiplicity distribution.

Geometry and molecular dynamics of the Hamiltonian mean-field model in a magnetic field.

The Hamiltonian mean-field model is investigated in the presence of a field. The self-consistent equations for the magnetization and the energy per particle are derived, and the field effect on the caloric curve is presented. The analytical geometric approach to Hamiltonian dynamics, under the hypothesis of quasi-isotropy, allows us to calculate the field effect on the energy-dependent microcanonical mean Ricci curvature and its fluctuations. Notably, the method proved suitable to identify that stable and metastable solutions of the Lyapunov exponent exhibit intriguing distinct curvature behavior very close to the critical point at extremely low field values. In addition, finite-size molecular dynamics (MD) simulations are used to observe the evolution of the magnetization and their components, including the stability properties of the solutions. Most importantly, comparison of finite-size MD calculations of the Lyapunov exponent and related properties with those via the geometric approach unveil the sensible dependence of these microcanonical quantities on energy, number of particles, and field, before a quasisaturation behavior at high fields. Finally, relaxation properties from out-of-equilibrium initial conditions are discussed in light of MD simulations.

Properties of nuclear pastas

In this review we study the nuclear pastas as they are expected to be formed in neutron star crusts. We start with a study of the pastas formed in nuclear matter (composed of protons and neutrons), we follow with the role of the electron gas on the formation of pastas, and we then investigate the pastas in neutron star matter (nuclear matter embedded in an electron gas).Nuclear matter (NM) at intermediate temperatures (1 MeV ≲ T ≲ 15 MeV), at saturation and sub-saturation densities, and with proton content ranging from 30% to 50% was found to have liquid, gaseous and liquid-gas mixed phases. The isospin-dependent phase diagram was obtained along with the critical points, and the symmetry energy was calculated and compared to experimental data and other theories. At low temperatures (T ≲ 1 MeV) NM produces crystal-like structures around saturation densities, and pasta-like structures at sub-saturation densities. Properties of the pasta structures were studied with cluster-recognition algorithms, caloric curve, the radial distribution function, the Lindemann coefficient, Kolmogorov statistics, Minkowski functionals; the symmetry energy of the pasta showed a connection with its morphology.Neutron star matter (NSM) is nuclear matter embedded in an electron gas. The electron gas is included in the calculation by the inclusion of an screened Coulomb potential. To connect the NM pastas with those in neutron star matter (NSM), the role the strength and screening length of the Coulomb interaction have on the formation of the pastas in NM was investigated. Pasta was found to exist even without the presence of the electron gas, but the effect of the Coulomb interaction is to form more defined pasta structures, among other effects. Likewise, it was determined that there is a minimal screening length for the developed structures to be independent of the cell size.Neutron star matter was found to have similar phases as NM, phase transitions, symmetry energy, structure function and thermal conductivity. Like in NM, pasta forms at around T ≈ 1.5 MeV, and liquid-to-solid phase changes were detected at T ≈ 0.5 MeV. The structure function and the symmetry energy were also found to depend on the pasta structures.

Nuclear liquid-gas phase transition with machine learning

The machine-learning techniques have shown their capability for studying phase transitions in condensed matter physics. Here, we employ the machine-learning techniques to study the nuclear liquid-gas phase transition. We adopt an unsupervised learning and classify the liquid and gas phases of nuclei directly from the final state raw experimental data of heavy-ion reactions. Based on a confusion scheme which combines the supervised and unsupervised learning, we obtain the limiting temperature of the nuclear liquid-gas phase transition. Its value $9.24\pm0.04~\rm MeV$ is consistent with that obtained by the traditional caloric curve method. Our study explores the paradigm of combining the machine-learning techniques with heavy-ion experimental data, and it is also instructive for studying the phase transition of other uncontrollable systems, like QCD matter.

Caloric curves of self-gravitating fermions in general relativity

We study the nature of phase transitions between gaseous and condensed states in the self-gravitating Fermi gas at finite temperature in general relativity. The condensed states can represent compact objects such as white dwarfs, neutron stars, or dark matter fermion balls. The caloric curves depend on two parameters: the system size R and the particle number N. When N < NOV, where NOV is the Oppenheimer–Volkoff limit, there exists an equilibrium state for any value of the temperature T and energy E as in the nonrelativistic case [P.H. Chavanis, Int. J. Mod. Phys. B 20, 3113 (2006)]. Gravitational collapse is prevented by quantum mechanics (Pauli’s exclusion principle). When N > NOV, there is no equilibrium state below a critical energy and below a critical temperature. In that case, the system is expected to collapse toward a black hole. We plot the caloric curves of the general relativistic Fermi gas, study the different types of phase transitions that occur in the system, and determine the phase diagram in the (R, N) plane. The nonrelativistic results are recovered for N ≪ NOV and R ≫ ROV with NR3 fixed. The classical (non quantum) results are recovered for N ≫ NOV and R ≫ ROV with N∕R fixed. We discuss the commutation of the limits c → +∞ and ℏ → 0. We study the relativistic corrections to the nonrelativistic caloric curves and the quantum corrections to the classical caloric curves. We highlight a situation of physical interest where a self-gravitating Fermi gas, by cooling, first undergoes a phase transition toward a compact object (white dwarf, neutron star, dark matter fermion ball), then collapses into a black hole. This situation occurs in the microcanonical ensemble when NOV < N < 3.73 NOV. We also relate the phase transitions from a gaseous state to a core-halo state in the microcanonical ensemble to the onset of red-giant structure and to the supernova phenomenon.

Computational statistical mechanics of a confined, three-dimensional Coulomb gas.

The thermodynamic properties of systems with long-range interactions present an ongoing challenge, from the point of view of both theory as well as computer simulation. In this paper we study a model system, a Coulomb gas confined inside a sphere, by using the Wang-Landau algorithm. We have computed the configurational density of states, the thermodynamic entropy, and the caloric curve, and compared with microcanonical Metropolis simulations, while showing how concepts such as the configurational inverse temperature can be used to understand some aspects of thermodynamic behavior. A dynamical multistability behavior is seen at low energies in microcanonical Monte Carlo simulations, suggesting that flat-histogram methods can in fact be useful and complementary alternatives to traditional Metropolis simulation in complex systems.

Structure, energetics and thermodynamics of PLGA condensed phases from Molecular Dynamics

Poly-lactic-co-glycolic acid (PLGA) is a biodegradable co-polymer with common use in nanoparticle drug encapsulation. Although well studied experimentally, the mechanical behavior of PLGA is not well understood at the atomic level. Here, we develop atomic charges for the all-atom Generalized Amber Force Field (GAFF) and conduct all-atom molecular dynamics simulations of PLGA with a 50:50 ratio between its two constituent monomers for five samples of the polymer condensed phases that span 1579 u to 20183 u in molecular weight. We predict several PLGA properties that will improve the knowledge of its atomistic organization in the glassy solid, rubber, and liquid states. We report the impact of molecular weight on cohesive energy, solubility, thermodynamic response properties, structural properties related to chain entanglement, and glass transition temperatures. Properties are compared against known experimental values when available. We find that the restrained electrostatic potential atomic charges are better for simulating the caloric curve leading to the glass transition temperature, which agrees very well with experiments.

Caloric curves of classical self-gravitating systems in general relativity.

We determine the caloric curves of classical self-gravitating systems at statistical equilibrium in general relativity. In the classical limit, the caloric curves of a self-gravitating gas depend on a unique parameter ν=GNm/Rc^{2}, called the compactness parameter, where N is the particle number and R the system's size. Typically, the caloric curves have the form of a double spiral. The "cold spiral," corresponding to weakly relativistic configurations, is a generalization of the caloric curve of nonrelativistic classical self-gravitating systems. The "hot spiral," corresponding to strongly relativistic configurations, is similar (but not identical) to the caloric curve of the ultrarelativistic self-gravitating black-body radiation. We introduce two types of normalization of energy and temperature to obtain asymptotic caloric curves describing, respectively, the cold and the hot spirals in the limit ν→0. As the number of particles increases, the cold and the hot spirals approach each other, merge at ν_{S}^{'}=0.128, form a loop above ν_{S}=0.1415, reduce to a point at ν_{max}=0.1764, and finally disappear. Therefore, the double spiral shrinks when the compactness parameter ν increases, implying that general relativistic effects render the system more unstable. We discuss the nature of the gravitational collapse at low and high energies with respect to a dynamical (fast) or a thermodynamical (slow) instability. We also provide an historical account of the developments of the statistical mechanics of classical self-gravitating systems in Newtonian gravity and general relativity.