Microcanonical Ensemble Research Articles

Microcanonical analysis of Boltzmann and Gibbs entropies in trapped cold atomic gases

We analyze a gas of noninteracting fermions confined to a one-dimensional harmonic oscillator potential, with the aim of distinguishing between two proposed definitions of the thermodynamic entropy in the microcanonical ensemble, namely the standard Boltzmann entropy and the Gibbs (or volume) entropy. The distinction between these two definitions is crucial for systems with an upper bound on allowed energy levels, where the Boltzmann definition can lead to the notion of negative absolute temperature. Although negative temperatures do not exist for the system of fermions studied here, we still find a significant difference between the Boltzmann and Gibbs entropies, and between the corresponding temperatures with the Gibbs temperature being closer (for small particle number) to the temperature based on a grand canonical picture.

A solvable problem in statistical mechanics: The dipole-type Hamiltonian mean field model

The present study documents a type of mean field approximation inspired by the dipole interaction model, which is analytically solved in the canonical and microcanonical ensembles. The current calculations were derived from the Hamiltonian mean field model developments. After describing the canonical partition function, the free and internal energies, magnetization, and specific heat are derived and graphically depicted. In the microcanonical ensemble, the entropy is calculated as well as other thermodynamic properties. The system shows a second-order phase transition emphasizing that both methods coincide, which is only valid only in equilibrium. In addition, the current model represents a nonsymmetric Hamiltonian mean field model that shows a phase transition.

Photon Gas Thermodynamics in Noncommutative Momentum Spaces

In this paper, we study thermodynamical properties of the extreme relativistic gases in canonical ensembles in non-commutative momentum spaces. In this regards, we exactly calculate the modified partition function, Helmholtz free energy, internal energy, entropy, heat capacity and the thermal pressure in the momentum spaces. Moreover, we conclude at high temperature limits due to the decreasing of the number of micro-states, these quantities reach to maximal bounds which do not exist in standard cases and it concludes that at the presence of gravity for both micro-canonical and canonical ensembles, the internal energy and the entropy tend to these upper bounds.

R\xe9nyi entropy at large energy density in 2D CFT

We investigate the Rényi entropy and entanglement entropy of an interval with an arbitrary length in the canonical ensemble, microcanonical ensemble and primary excited states at large energy density in the thermodynamic limit of a two-dimensional large central charge c conformal field theory. As a generalization of the recent work [17], the main purpose of the paper is to see whether one can distinguish these various large energy density states by the Rényi entropies of an interval at different size scales, namely, short, medium and long. Collecting earlier results and performing new calculations in order to compare with and fill gaps in the literature, we give a more complete and detailed analysis of the problem. Especially, we find some corrections to the recent results for the holographic Rényi entropy of a medium size interval, which enlarge the validity region of the results. Based on the Rényi entropies of the three interval scales, we find that Rényi entropy cannot distinguish the canonical and microcanonical ensemble states for a short interval, but can do the job for both medium and long intervals. At the leading order of large c the entanglement entropy cannot distinguish the canonical and microcanonical ensemble states for all interval lengths, but the difference of entanglement entropy for a long interval between the two states would appear with 1/c corrections. We also discuss Rényi entropy and entanglement entropy differences between the thermal states and primary excited state. Overall, our work provide an up-to-date picture of distinguishing different thermal or primary states at various length scales of the subsystem.

Black hole thermodynamics in Sharma–Mittal generalized entropy formalism

Using the Sharma-Mittal entropy, we study some properties of the Schwarzschild and Schwarzschild-de Sitter black holes. The results are compared with those obtained by attributing the Bekenstein entropy bound to the mentioned black holes. Our main results show that while the Schwarzschild black hole is always stable in the micro-canonical ensemble, it can be stable in the canonical ensemble if its mass is bigger than the mass of the coldest Schwarzschild black hole. A semi-classical analysis has also been used to find an approximate relation between the entropy free parameters. Throughout the paper, we use units $c=G=\hbar=k_B=1$, where $k_B$ denotes the Boltzmann constant.

General framework to study the extremal phase transition of black holes

We investigate the universality of some features for the extremal phase transition of black holes and unify all the approaches which have been applied in different spacetimes. Unlike the other existing approaches where the information of the spacetime and its dimension is directly used to get various results, we provide a general formulation in which those results are obtained for any arbitrary black hole spacetime having an extremal limit. Calculating the second order moments of fluctuations of some thermodynamic quantities we show that, the phase transition occurs only in the microcanonical ensemble. Without considering any specific black hole we calculate the values of critical exponents for this type of phase transition. These are shown to be in agreement with the values obtained earlier for metric specified cases. Finally we extend our analysis to the geometrothermodynamics (henceforth GTD) formulation. We show that for any black hole, if there is an extremal point, the Ricci scalar for the Ruppeiner metric must diverge at that point.

Generalized Wigner-von Neumann entropy and its typicality.

We propose a generalization of the quantum entropy introduced by Wigner and von Neumann [Z. Phys. 57, 30 (1929)10.1007/BF01339852]. Our generalization is applicable to both quantum pure states and mixed states. When the dimension N of the Hilbert space is large, this generalized Wigner-von Neumann (GWvN) entropy becomes independent of the choices of basis and is asymptotically equal to lnN in the sense of typicality. The dynamic evolution of our entropy is also typical, and is reminiscent of quantum H theorem proved by von Neumann. For a composite system, the GWvN entropy is typically additive; for the microcanonical ensemble, it is equivalent to the Boltzmann entropy; and for a system entangled with environment, it is consistent with the familiar von Neumann entropy, which is zero for pure states. In addition, the GWvN entropy can be used to derive the Gibbs ensemble.

Violation of the temperature-signifies-heat-flow rule in systems with long-range interactions.

For a long-range interacting spin chain model, the microcanonical ensemble predicts a region of negative specific heat and a temperature jump at the transition energy. After two similar long-range interacting subsystems of different size at different temperatures are weakly coupled, they exchange energy and the total microcanonical entropy of the full system increases irreversibly. The hot subsystem could spontaneously absorb heat from the cold subsystem via the thermal contact and the final equilibrium temperature could be lower than the initial temperatures of the cold subsystem. This result is confirmed by numerical simulations using the microcanonical Monte Carlo algorithm.

Diagrams of States of Single Flexible-Semiflexible Multi-Block Copolymer Chains: A Flat-Histogram Monte Carlo Study.

The combination of flexibility and semiflexibility in a single molecule is a powerful design principle both in nature and in materials science. We present results on the conformational behavior of a single multiblock-copolymer chain, consisting of equal amounts of Flexible (F) and Semiflexible (S) blocks with different affinity to an implicit solvent. We consider a manifold of macrostates defined by two terms in the total energy: intermonomer interaction energy and stiffness energy. To obtain diagrams of states (pseudo-phase diagrams), we performed flat-histogram Monte Carlo simulations using the Stochastic Approximation Monte Carlo algorithm (SAMC). We have accumulated two-Dimensional Density of States (2D DoS) functions (defined on the 2D manifold of macrostates) for a SF-multiblock-copolymer chain of length with block lengths b = 4, 8, 16, and 32 in two different selective solvents. In an analysis of the canonical ensemble, we calculated the heat capacity and determined its maxima and the most probable morphologies in different regions of the state diagrams. These are rich in various, non-trivial morphologies, which are formed without any specific interactions, and depend on the block length and the type of solvent selectivity (preferring S or F blocks, respectively). We compared the diagrams with those for the non-selective solvent and reveal essential changes in some cases. Additionally, we implemented microcanonical analysis in the “conformational” microcanonical (, where U is the potential energy) and the true microcanonical (, where E is the total energy) ensembles with the aim to reveal and classify pseudo-phase transitions, occurring under the change of temperature.

Statistical mechanics of systems with long-range interactions and negative absolute temperature.

A Hamiltonian model living in a bounded phase space and with long-range interactions is studied. It is shown, by analytical computations, that there exists an energy interval in which the microcanonical entropy is a decreasing convex function of the total energy, meaning that ensemble equivalence is violated in a negative-temperature regime. The equilibrium properties of the model are then investigated by molecular dynamics simulations: first, the caloric curve is reconstructed for the microcanonical ensemble and compared to the analytical prediction, and a generalized Maxwell-Boltzmann distribution for the momenta is observed; then the nonequivalence between the microcanonical and canonical descriptions is explicitly shown. Moreover, the validity of the Fluctuation-Dissipation Theorem is verified through a numerical study, also at negative temperature and in the region where the two ensembles are nonequivalent.

Kinked Entropy and Discontinuous Microcanonical Spontaneous Symmetry Breaking.

Spontaneous symmetry breaking (SSB) in statistical physics is a macroscopic collective phenomenon. For the paradigmatic Q-state Potts model it means a transition from the disordered color-symmetric phase to an ordered phase in which one color dominates. Existing mean field theories imply that SSB in the microcanonical statistical ensemble (with energy being the control parameter) should be a continuous process. Here we study microcanonical SSB on the random-graph Potts model and discover that the entropy is a kinked function of energy. This kink leads to a discontinuous phase transition at certain energy density value, characterized by a jump in the density of the dominant color and a jump in the microcanonical temperature. This discontinuous SSB in random graphs is confirmed by microcanonical MonteCarlo simulations, and it is also observed in bond-diluted finite-size lattice systems.

Ferromagnetism-induced phase separation in a two-dimensional spin fluid.

We study the liquid-gas phase separation observed in a system of repulsive particles dressed with ferromagnetically aligning spins, a so-called "spin fluid." Microcanonical ensemble numerical simulations of finite-size systems reveal that magnetization sets in and induces a liquid-gas phase separation between a disordered gas and a ferromagnetic dense phase at low enough energies and large enough densities. The dynamics after a quench into the coexistence region show that the order parameter associated with the liquid-vapor phase separation follows an algebraic law with an unusual exponent, as it is forced to synchronize with the growth of the magnetization: this suggests that for finite size systems the magnetization sets in along a Curie line, which is also the gas-side spinodal line, and that the coexistence region ends at a tricritical point. This picture is confirmed at the mean-field level with different approximation schemes, namely, a Bethe lattice resolution and a virial expansion complemented by the introduction of a self-consistent Weiss-like molecular field. However, a detailed finite-size scaling analysis shows that in two dimensions the ferromagnetic phase escapes the Berezinskii-Kosterlitz-Thouless scenario and that the long-range order is not destroyed by the unbinding of topological defects. The Curie line thus becomes a magnetic crossover in the thermodynamic limit. Finally, the effects of the magnetic interaction range and those of the interaction softness are characterized within a mean-field semianalytical low-density approach.

Nuclear level density and thermal properties of 270110, 278112, and 290116 superheavy isotopes in low excitation energies

The level density parameter α is calculated using the semiclassical approach for 270 110, 278 112, and 290 116 superheavy isotopes. The Woods-Saxon potential is considered for the interaction of nucleons inside a nucleus. Thermal properties of these isotopes, including the entropy, the nuclear temperature, and the heat capacity are calculated within the framework of microcanonical ensemble and the back-shifted Fermi gas model with one adjustable parameter E 1 . The process of breaking the first Cooper pair and cooling of the 270 110, 278 112, and 290 116 superheavy isotopes have also been observed.

Thermodynamic analysis of a long-range interacting spin system

A long-range interacting Fermi chain placed in the uniform and the staggered magnetic field is studied via the micro-canonical approach. The relation between the entropy and the energy of the system is obtained by counting the number of microscopic states. We find that this system is non-ergodic and can exhibit first-order phase transition, second-order phase transition, or both. The microcanonical ensemble predicts negative specific heat regions and temperature jumps. Moreover, the global phase diagram of the system is constructed.

Microcanonical thermodynamics of three and four atoms.

Clusters are treated in the microcanonical ensemble. Two classical choices for the partition function are used in Boltzmann's assumption for entropy: Phase space volume and phase space density. For both definitions, exact statistical analogs are derived for energy derivatives of entropy to arbitrary order. The analogs are used in rigorous microcanonical Monte Carlo simulation to measure entropy derivatives up to order three for systems of three and four atoms. All simulation results are confirmed by direct numerical integration of the statistical analogs.

Statistical description of an ideal gas in maximum length quantum mechanics

We discuss the effects of the recent proposed Generalized Uncertainty Principle (GUP) (Perivolaropoulos, 2017), which includes a maximum length on the thermostatistics of an ideal gas. In the framework of this modified version of quantum mechanics, the deformed Schrödinger equation is solved analytically for a particle in an infinite square-well potential, then the corresponding energy spectrum is extracted. By computing the modified canonical partition function, the thermodynamic properties of the system are investigated within the canonical and microcanonical ensembles; the results show a complete consistency between both statistical descriptions. Furthermore, a comparison with the results obtained in the context of the minimal length GUP indicates that the maximum length induces fundamentally different effects, which become important at high temperatures and for large space confining the system. Especially, a modified equation of state, similar to that of real gases, emerges in the scope of this new formalism.

Analysis of the parametrically periodically driven classical and quantum linear oscillator.

We study theoretically and computationally the behavior of the classical and quantum parametrically periodically driven linear oscillator. As a basic paradigm of such a Floquet system we consider the case of the harmonic oscillation of the oscillator frequency, which is convenient to handle theoretically and computationally, while keeping the general features. We derive an explicit analytic formula for the quantum propagator in terms of the classical propagator. Using this, we derive the explicit exact formula for the evolution of the expectation value of the energy starting from an arbitrary normalizable initial state. In the case of the starting pure stationary eigenstate the evolution is exactly the same as for the classical microcanonical ensemble of initial conditions of the same starting energy. We perform a rather complete computational analysis of the system's behavior inside the instability regions (lacunae), where the energy of the oscillator increases exponentially, as well as in the stability regions, and in particular in the vicinity of the (in)stability borders. We confirm also numerically with absolute certainty that the borders of (in)stability regions classically and quantally coincide exactly, in accordance with the theory, which is an important check of the numerical accuracy of computations, and we find a number of important empirical results, especially an equation of the elliptic type describing the rate of exponential energy growth inside the lacunae in terms of other systems' quantities. We believe that our approach and findings are of generic linear type, i.e., applicable in most such linear Floquet systems, and we present a strong motivation for a general theory, classically and quantally.

Holographic Rényi Entropy at High Energy Density.

We show that Rényi entropies of subregions can be used to distinguish when the entire system is in a microcanonical ensemble from when it is in a canonical ensemble, at least in theories holographically dual to gravity. Simple expressions are provided for these Rényi entropies in a particular thermodynamic limit with high energy density and fixed fractional size of the subregion. Holographically, the Rényi entropies are determined by the areas of cosmic branes inserted into the bulk spacetime. They differ between a microcanonical and a canonical ensemble because the two ensembles provide different boundary conditions for the gravitational theory under which cosmic branes lead to different backreacted geometries. This is in contrast to the von Neumann entropy which is more coarse-grained and does not differentiate microcanonical ensembles from canonical ensembles.

Dynamical structure factor of complex plasmas for varying wave vectors

The dynamical structure factor has been reported for three dimensional strongly coupled Yukawa liquids (SCYLs) through state-of-the-art equilibrium molecular dynamics (EMD) simulations in a microcanonical ensemble (NVE). The effects of varying wave vectors (k = 2π/L) have been computed along with different arrangements of Coulomb coupling (Γ) and the Debye screening parameter (κ) on the dynamical structure-factor S(k,ω) using EMD simulations. Our new investigations of S(k,ω) show that the amplitude of oscillation decreases and the frequency increases with increasing Γ, respectively, for the SCYLs. Our simulations show that the decreasing behavior is noted for the frequency of plasma S(k,ω) with increasing κ and system size (N). The obtained EMD results are found to be more efficient and accurate than that of various previous simulation data, and the present EMD approach gives more satisfactory results with appropriate system sizes at high values of k, for a wide range of plasma states (Γ, κ). It has been shown that the present density S(k,ω) of SCYLs fluctuates more at intermediate to high Coulomb couplings (average to lower system temperature ≡ 1/Γ) and low values of Debye screening; however, it less fluctuates at higher N and κ.

Canonical-Ensemble Calculations of the Magnetic Susceptibility for a Spin-1/2 Spherical Kagome Cluster With Dzyaloshinskii–Moriya Interactions by Using Microcanonical Thermal Pure Quantum States

For the spherical kagome system {W <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">72</sub> V <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">30</sub> }, which is a magnetic cluster with 30 V <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">4+</sup> ions, recent experimental and theoretical studies on the magnetization process at low temperatures have indicated that Dzyaloshinskii-Moriya interaction is an important ingredient in this material. In this paper, we use microcanonical thermal pure quantum (mTPQ) states to calculate the temperature dependence of the magnetic susceptibility and compare our calculated results with an existing experimental susceptibility. These mTPQ states are generated to reproduce the smooth microcanonical ensemble, which is difficult from the canonical ensemble. We show that adequate choice of a parameter in the generation procedure of mTPQ states leads to the same temperature dependence as the canonical ensemble.