Microcanonical Ensemble Research Articles

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.

Quantum quenches in the Dicke model: Thermalization and failure of the generalized Gibbs ensemble* *Project supported by the National Natural Science Foundation of China (Grant No. 11147110) and the Natural Science Youth Foundation of Shanxi, China (Grant No. 2011021003).

Quantum quenches in the Dicke model were studied both in the thermodynamic limit and the finite systems. For the integrable situation in the thermodynamic limit, the generalized Gibbs ensemble can effectively describe the energy-level occupations for the quench within the normal phase, but it fails for the quench to the superradiant phase. For the finite systems which are considered non-integrable, the post quench systems were studied by comparing with the thermal ensembles. The canonical ensembles are directly available for the quench within the normal phase. With the increasing of the target coupling strength over the equilibrium phase transition critical point, sudden changes take place for the effective temperature and the distance to the thermal ensembles. The thermalization was also studied by comparing with the results of the microcanonical ensembles.

Weighted hypersoft configuration model

Maximum entropy null models of networks come in different flavors that depend on the type of constraints under which entropy is maximized. If the constraints are on degree sequences or distributions, we are dealing with configuration models. If the degree sequence is constrained exactly, the corresponding microcanonical ensemble of random graphs with a given degree sequence is the configuration model per se. If the degree sequence is constrained only on average, the corresponding grand-canonical ensemble of random graphs with a given expected degree sequence is the soft configuration model. If the degree sequence is not fixed at all but randomly drawn from a fixed distribution, the corresponding hypercanonical ensemble of random graphs with a given degree distribution is the hypersoft configuration model, a more adequate description of dynamic real-world networks in which degree sequences are never fixed but degree distributions often stay stable. Here, we introduce the hypersoft configuration model of weighted networks. The main contribution is a particular version of the model with power-law degree and strength distributions, and superlinear scaling of strengths with degrees, mimicking the properties of some real-world networks. As a byproduct, we generalize the notions of sparse graphons and their entropy to weighted networks.

Thermostatistical analysis for short-range interaction potentials

In this paper, we study the thermodynamics of short-range central potentials, namely, the Lee–Wick (LW) potential, and the Plasma potential. In the first part of the paper, we obtain the numerical solution for the orbits equation for these potentials. Posteriorly, we introduce the thermodynamics through the microcanonical and canonical ensembles formalism defined on the phase space of the system. We calculate the density of states associated with the LW and the Plasma potentials. From density of states, we obtain the thermodynamical physical quantities like entropy and temperature as functions of the energy. We also use the Boltzmann–Gibbs formalism to obtain the partition functions, the mean energy and the thermal capacity for these short-range potentials.

Microcanonical potential energy fluctuations and configurational density of states for nanoscale systems

In this work, we study the distribution of potential energies for the embedded atom model (EAM), commonly associated with monoatomic metals, in nanoscopic systems of the order of tens of particles. Classical molecular dynamics simulations in the microcanonical ensemble, both in solid and liquid phases, were used to produce samples of potential energy at different total energies. We characterize the shape of these fluctuations of potential energy and, in this context, we propose a model for the configurational density of states (CDOS) capable of describing the melting phase transition of a nanoscopic system. Our results show the shape of the fluctuations outside and within the range of the solid–liquid phase transition, and demonstrate the accuracy of the CDOS model for EAM systems.

Membrane permeability of small molecules from unbiased molecular dynamics simulations.

Permeation of many small molecules through lipid bilayers can be directly observed in molecular dynamics simulations on the nano- and microsecond timescale. While unbiased simulations provide an unobstructed view of the permeation process, their feasibility for computing permeability coefficients depends on various factors that differ for each permeant. The present work studies three small molecules for which unbiased simulations of permeation are feasible within less than a microsecond, one hydrophobic (oxygen), one hydrophilic (water), and one amphiphilic (ethanol). Permeabilities are computed using two approaches: counting methods and a maximum-likelihood estimation for the inhomogeneous solubility diffusion (ISD) model. Counting methods yield nearly model-free estimates of the permeability for all three permeants. While the ISD-based approach is reasonable for oxygen, it lacks precision for water due to insufficient sampling and results in misleading estimates for ethanol due to invalid model assumptions. It is also demonstrated that simulations using a Langevin thermostat with collision frequencies of 1/ps and 5/ps yield oxygen permeabilities and diffusion constants that are lower than those using Nosé-Hoover by statistically significant margins. In contrast, permeabilities from trajectories generated with Nosé-Hoover and the microcanonical ensemble do not show statistically significant differences. As molecular simulations become more affordable and accurate, calculation of permeability for an expanding range of molecules will be feasible using unbiased simulations. The present work summarizes theoretical underpinnings, identifies pitfalls, and develops best practices for such simulations.

Comment on "Partial equivalence of statistical ensembles in a simple spin model with discontinuous phase transitions".

In a recent paper by Fronczak etal. [Phys. Rev. E 101, 022111 (2020)2470-004510.1103/PhysRevE.101.022111], a simple spin model has been studied in full detail via microcanonical approaches. The authors stress that the range of microcanonical temperature β_{m}>1 is unattainable in this model and the system undergoes a phase transition when the external parameter a=1 in the microcanonical ensemble. The purpose of this comment is to state that the treatment of the microcanonical entropy in the commented paper is inappropriate since the fact that ergodicity is broken in the microcanonical dynamics is ignored by the authors. The phase transition in the microcanonical ensemble, considered in the commented paper, could occur only with a nonlocal dynamics which is often difficult to justify physically.

Large deviations and localization of the microcanonical ensembles given by multiple constraints

We develop a unified theory to analyze the microcanonical ensembles with several constraints given by unbounded observables. Several interesting phenomena that do not occur in the single constraint case can happen under the multiple constraints case. We systematically analyze the detailed structures of such microcanonical ensembles in two orthogonal directions using the theory of large deviations. First of all, we establish the equivalence of ensembles result, which exhibits an interesting phase transition phenomenon. Secondly, we study the localization and delocalization phenomena by obtaining large deviation results for the joint law of empirical distributions and the maximum component. Some concrete examples for which the theory applies will be given as well.

Gaussian concentration bound and Ensemble equivalence in generic quantum many-body systems including long-range interactions

This work explores fundamental statistical and thermodynamic properties of short-and long-range-interacting systems. The purpose of this study is twofold. Firstly, we rigorously prove that the probability distribution of arbitrary few-body observables is restricted by a Gaussian concentration bound (or Chernoff–Hoeffding inequality) above some threshold temperature. This bound is then derived for arbitrary Gibbs states of systems that include long-range interactions Secondly, we establish a quantitative relationship between the concentration bound of the Gibbs state and the equivalence of canonical and micro-canonical ensembles. We then evaluate the difference in the averages of thermodynamic properties between the canonical and the micro-canonical ensembles. Under the assumption of the Gaussian concentration bound on the canonical ensemble, the difference between the ensemble descriptions is upper-bounded by n−1log(n3∕2Δ−1)1∕2 with n being the system size and Δ being the width of the energy shell of the micro-canonical ensemble This limit gives a non-trivial upper bound exponentially small energy width with respect to the system size. By combining these two results, we prove the ensemble equivalence as well as the weak eigenstate thermalization in arbitrary long-range-interacting systems above a threshold temperature.

Notes on statistical ensembles in the Cell Model

Properties of basic statistical ensembles in the Cell Model are discussed. The simplest version of the model with a fixed total number of particles is considered. The microcanonical ensembles of distinguishable and indistinguishable particles, with and without a limit on the maximum number of particles in a single cell, are discussed. The joint probability distributions of particle multiplicities in cells for different ensembles are derived, and their second moments are calculated. The results for infinite volume limit are calculated. The obtained results resemble those in the statistical physics of bosons, fermions and boltzmanions.

Estimation of Local Microcanonical Averages in Two Lattice Mean-Field Models Using Coupling Techniques

We consider an application of probabilistic coupling techniques which provides explicit estimates for comparison of local expectation values between label permutation invariant states, for instance, between certain microcanonical, canonical, and grand canonical ensemble expectations. A particular goal is to obtain good bounds for how such errors will decay with increasing system size. As explicit examples, we focus on two well-studied mean-field models: the discrete model of a paramagnet and the mean-field spherical model of a continuum field, both of which are related to the Curie–Weiss model. The proof is based on a construction of suitable probabilistic couplings between the relevant states, using Wasserstein fluctuation distance to control the difference between the expectations in the thermodynamic limit.

Closeness of the reduced density matrix of an interacting small system to the Gibbs state.

I study the statistical description of a small quantum system, which is coupled to a large quantum environment in a generic form and with a generic interaction strength, when the total system lies in an equilibrium state described by a microcanonical ensemble. The focus is on the difference between the reduced density matrix (RDM) of the central system in this interacting case and the RDM obtained in the uncoupled case. In the eigenbasis of the central system's Hamiltonian, it is shown that the difference between diagonal elements is mainly confined by the ratio of the maximum width of the eigenfunctions of the total system in the uncoupled basis to the width of the microcanonical energy shell; meanwhile, the difference between off-diagonal elements is given by the ratio of certain property of the interaction Hamiltonian to the related level spacing of the central system. As an application, a sufficient condition is given, under which the RDM may have a canonical Gibbs form under system-environment interactions that are not necessarily weak; this Gibbs state usually includes certain averaged effect of the interaction. For central systems that interact locally with many-body quantum chaotic systems, it is shown that the RDM usually has a Gibbs form. I also study the RDM which is computed from a typical state of the total system within an energy shell.

Polarizable charges in a generalized Born reaction potential.

The generalized Born (GB) model is a fast implicit solvent model that is used as an approximation to the Poisson equation for solutes described by point charges. Due to the simple analytical form, GB models are widely used in molecular dynamics simulations to account for (implicit) solvation effects. In this work, we extend the application of the GB model to polarizable charges by coupling it to the bond capacity (BC) model. The resulting BC-GB model is a non-variational polarization model where the reaction potential is calculated from a GB expression and included in the polarization equation to account for solvation effects. Being non-variational, the BC-GB makes use of a Lagrange formulation for an efficient evaluation of energy gradients. The stability of the algorithm in molecular dynamics simulations is tested in the microcanonical ensemble, and the results show energy conservation as well as small fluctuations. The inclusion of implicit solvation increases the computational cost by only 15% compared to vacuum. Combined with a significant reduction in system size by describing the solvent as a continuum makes the BC-GB model an interesting model for applications requiring the combination of solute polarization and extensive conformational space sampling.

Eigenstate thermalization and ensemble equivalence in few-body fermionic systems.

We investigate eigenstate thermalization from the point of view of vanishing particle and heat currents between a few-body fermionic Hamiltonian prepared in one of its eigenstates and an external, weakly coupled Fermi-Dirac gas. The latter acts as a thermometric probe, with its temperature and chemical potential set so that there is neither particle nor heat current between the two subsystems. We argue that the probe temperature can be attributed to the few-fermion eigenstate in the sense that (i) it varies smoothly with energy from eigenstate to eigenstate, (ii) it is equal to the temperature obtained from a thermodynamic relation in a wide energy range, (iii) it is independent of details of the coupling between the two systems in a finite parameter range, (iv) it satisfies the transitivity condition underlying the zeroth law of thermodynamics, and (v) it is consistent with Carnot's theorem. For the spinless fermion model considered here, these conditions are essentially independent of the interaction strength. When the latter is weak, however, orbital occupancies in the few-fermion system differ from the Fermi-Dirac distribution so that partial currents from or to the probe will eventually change its state. We find that (vi) above a certain critical interaction strength, orbital occupancies become close to the Fermi-Dirac distribution, leading to a true equilibrium between the few-fermion system and the probe. In that case, the coupling between the Fermi-Dirac gas and few-fermion system does not modify the state of the latter, which justifies our approach a posteriori. From these results, we conjecture that for few-body systems with sufficiently strong interaction, the eigenstate thermalization hypothesis is complemented by ensemble equivalence: individual many-body eigenstates define a microcanonical ensemble that is equivalent to a canonical ensemble with grand canonical orbital occupancies.

Eigenstate Thermalization from the Clustering Property of Correlation.

The clustering property of an equilibrium bipartite correlation is one of the most general thermodynamic properties in noncritical many-body quantum systems. Herein, we consider the thermalization properties of a system class exhibiting the clustering property. We investigate two regimes, namely, regimes of high and low density of states corresponding to high- and low-energy regimes, respectively. We show that the clustering property is connected to several properties on the eigenstate thermalization through the density of states. Remarkably, the eigenstate thermalization is obtained in the low-energy regime with a sparse density of states, which is typically seen in gapped systems. For the high-energy regime, we demonstrate the ensemble equivalence between microcanonical and canonical ensembles even for a subexponentially small energy shell with respect to the system size, which eventually leads to the weak version of eigenstate thermalization.

Positive heat capacity in the microcanonical ensemble

The positivity of the heat capacity is the hallmark of thermal stability of systems in thermodynamic equilibrium. We show that this property remains valid for systems with negative derivative of energy with respect to temperature, as happens to some system described by the microcanonical ensemble. The demonstration rests on considering a trajectory on the Gibbs equilibrium surface, and its projection on the entropy-energy plane. The Gibbs equilibrium surface has the convexity property, but the projection might lack this property, leading to a negative derivative of energy with respect to temperature.

Critical dimension and negative specific heat in one-dimensional large- N reduced models

We investigate critical phenomena of the Yang-Mills (YM) type one-dimensional matrix model that is a large-$N$ reduction (or dimensional reduction) of the $D+1$ dimensional $U(N)$ pure YM theory (bosonic BFSS model). This model shows a large-$N$ phase transition at finite temperature, which is analogous to the confinement/deconfinement transition of the original YM theory. We study the matrix model at a three-loop calculation via the "principle of minimum sensitivity" and find that there is a critical dimension $D=35.5$: At $D \le 35$, the transition is of first order, while it is of second order at $D\ge 36$. Furthermore, we evaluate several observables in our method, and they nicely reproduce the existing Monte Carlo results. Through the gauge/gravity correspondence, the transition is expected to be related to a Gregory-Laflamme transition in gravity, and we argue that the existence of the critical dimension is qualitatively consistent with it. Besides, in the first order transition case, a stable phase having negative specific heat appears in the microcanonical ensemble, which is similar to Schwarzschild black holes. We study some properties of this phase.

From microcanonical ensemble to canonical ensemble: phase transitions of a spin chain with a long-range interaction

A long-range interacting spin chain placed in a staggered magnetic field can exhibit either first order phase transition or second order phase transition depending on the magnetic field intensity. The first order phase transition of this model is known to be accompanied with the temperature jump phenomenon in the microcanonical ensemble, while this anomalous temperature jump phenomenon can not be observed in the canonical ensemble. We obtain the crossover phase transition properties passing from a microcanonical to a canonical ensemble, by placing this previously isolated spin chain model in contact with a two-level system that acts as a thermal reservoir. The magnitude of the temperature jump monotonically decreases with the increase of the size of the thermal reservoir. And the microcanonical phase diagram can gradually turn into the canonical phase diagram by increasing the number of particles of the reservoir.

Nonstatistical Reaction Dynamics.

Nonstatistical dynamics is important for many chemical reactions. The Rice-Ramsperger-Kassel-Marcus (RRKM) theory of unimolecular kinetics assumes a reactant molecule maintains a statistical microcanonical ensemble of vibrational states during its dissociation so that its unimolecular dynamics are time independent. Such dynamics results when the reactant's atomic motion is chaotic or irregular. Intrinsic non-RRKM dynamics occurs when part of the reactant's phase space consists of quasiperiodic/regular motion and a bottleneck exists, so that the unimolecular rate constant is time dependent. Nonrandom excitation of a molecule may result in short-time apparent non-RRKM dynamics. For rotational activation, the 2J + 1 K levels for a particular J may be highly mixed, making K an active degree of freedom, or K may be a good quantum number and an adiabatic degree of freedom. Nonstatistical dynamics is often important for bimolecular reactions and their intermediates and for product-energy partitioning of bimolecular and unimolecular reactions. Post-transition state dynamics is often highly complex and nonstatistical.

Effect of electric field orientation on ferroelectric phase transition and electrocaloric effect

The influence of electric field orientation on phase transitions and electrocaloric effects (ECEs) in BaTiO3 single crystals is studied by performing molecular dynamics simulation of first-principles-based effective Hamiltonian. The ECE is directly characterized via the adiabatic temperature change (ΔT) under the microcanonical ensemble. The simulation results demonstrate that different orientation relationship between the electric field and the polarization direction of ferroelectric phase leads to abundant phase structures and ECE behaviors. The electric-field-induced phase transitions produce remarkable ECE peaks/valleys, whereas the polarization rotation and/or extension without phase transition induced by the electric field produces small positive ECEs, which are insensitive to temperature and electric field direction. For the tetragonal-cubic phase transition, ECEs are always positive regardless of the applied electric field direction, and the value and width of ECE peak both increase with the reduction of the angle between electric field and crystallographic orientation. For the orthorhombic-tetragonal and rhombohedral-orthorhombic phase transitions, the sign of ECE gradually changes from negative to positive during the field direction rotates in planes between crystallographic orientations, and the coexistence of positive and negative appears when the field along non-crystallographic directions. Our simulated results shed light on a comprehensive physical understanding of the electric-field-orientation dependent ECE and offer instruction for the design of the refrigeration cycle by combining positive and negative ECEs.