Microcanonical Ensemble Research Articles

Is ‘global’ warming concept consistent with thermodynamics according to ChatGPT?

Aim. Taking care of the environment on Planet Earth is still important and has be- come popular exceeding boundaries of science. Therefore description of climate changes has evolved into the concept of Global Warming since the 20th century. On the other hand, principles of thermodynamics where temperature is defined locally (not globally) are still valid. In my research I have applied tools like ChatGPT to detect possible contra- dictions and propose bridging gaps between statistical physics and the concept of global warming. Methods. The concept of temperature based on calculations of local particles’ kinetic energy in a microcanonical ensemble has been confronted with the concept of so called ‘average’ temperature in ‘global’ warming. The principles of Kinetic Theory and Statisti- cal Ensembles provided by statistical physics have been applied to ChatGPT based on natural language Results. Contradictions between the concept of average temperature in global warming and the concept of temperature based on local particle kinetic energy arise due to the different scales, heterogeneity, presence of feedback mechanisms, and boundary conditions. According to the chatGPT’s verdict, the concept of temperature based on calculations of local particles kinetic energy in the microcanonical ensemble is statistically more accurate. Conclusion. The concept of so called ‘global’ warming (confronted with the physics- based concept of temperature, providing a more accurate and nuanced understanding of temperature changes in different scales and contexts) should be replaced by the idea of local warming which is consistent with the concept of temperature based on calcula- tions of local particles kinetic energy in the microcanonical ensemble and experimental measurements provided by NASA. Therefore, talking about local (not global) warming is more reasonable when our goal is to prevent our coexistence with environment on Planet Earth from disasters.

The structural and electronic configuration of an amorphous silica surface, from its liquid to glassy state

Through classical molecular dynamics computational simulations, the surface of amorphous silica (vitreous silica) was generated for a system comprising 64 SiO2 molecules (192 atoms) via the classical three-body glass potential. Starting from the liquid state of silica at 3400 K, it was cooled in three different ways at intervals of 200, 400, and 2000 K, with a cooling rate of 4.4 K/ps and establishing relaxation at intermediate temperatures, yielding three groups of surfaces from 3400 to 1400 K. Subsequently, these systems underwent sudden cooling from 1400 to 300 K via the Andersen and Nosé‒Hoover thermostats, followed by relaxation at 300 K via a microcanonical ensemble (NVE constant). Each surface that underwent relaxation was analyzed through ab initio molecular dynamics for the following three physical properties, which are temperature dependent. The average electric charges of oxygen and silicon. Microstructures on the outermost part of the surface were observed and comprised nonbridging oxygen atoms, undercoordinated silicon, and N-membered rings (3 ≤ N ≤ 5). The radial distribution functions were analyzed for all surfaces. Below 2000 K, a type of freezing is observed in all the structures because of the glass transition. All analyses were performed at zero pressure.

Microcanonical Monte Carlo of Lennard-Jones microclusters.

A novel statistical mechanical methodology is applied to clusters of N ≤ 7 atoms. Exact statistical analogs for any energy derivative of entropy ∂mS/∂Em are used in rigorous microcanonical Monte Carlo simulations to vastly enlarge the pool of measurable thermodynamic properties relative to previous work. All analogs are given for two alternative partition functions of the microcanonical ensemble. Coarse grained energy distributions are used to establish the existence of melting transitions. LJ7, LJ5, and LJ4 are found to exhibit trimodal distributions, a feature not being reported before. Varieties of combinations of entropy derivatives are tested for a direct detection of the melting region. It is shown that for such a purpose, derivatives of at least fourth order are necessary.

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→+∞.

Chemical effect of H2 on NH3 combustion in an O2 environment via molecular dynamics simulations

Due to the imperative to reduce carbon emissions, NH3-H2 mixed combustion has garnered significant attention. However, the knowledge of the reaction kinetics of NH3-H2-O2 mixtures at the molecular level remains limited. In this paper, the reactive force-field (ReaxFF) method was employed to study the microscale reaction kinetics of the NH3-H2-O2 mixture with varying hydrogen ratios (HR). The results of the microcanonical ensemble simulations indicated that the addition of H2 significantly increased the combustion performance of NH3. However, H2 did not change the two primary reaction paths of NH3 oxidation (NH3 → N2H2 → N2 and NH3 → NO → N2). The addition H2 not only led to a shift in the initiation reactions from NH3-O2 to H2-O2 reactions, but also notably enhanced the concentration of HO2 and OH radicals (e.g., O2 + H2 → HO2 + H), which accelerated the extraction of H from NxHy and subsequently facilitated the immediate formation of N2. Furthermore, the ignition delay decreased from 120 ps to 40 ps, and the activation energies decreased from 295.46 kJ/mol to 162.12 kJ/mol as HR increased from 0 to 0.3 under the equivalence ratio of 0.4. These key findings can guide the development of clean NH₃ combustion technology.

Stochastic symplectic reduced-order modeling for model-form uncertainty quantification in molecular dynamics simulations in various statistical ensembles

This work focuses on the representation of model-form uncertainties in molecular dynamics simulations in various statistical ensembles. In prior contributions, the modeling of such uncertainties was formalized and applied to quantify the impact of, and the error generated by, pair-potential selection in the microcanonical ensemble (NVE). In this work, we extend this formulation and present a linear-subspace reduced-order model for the canonical (NVT) and isobaric (NPT) ensembles. The symplectic reduced-order basis is randomized on the tangent space of the Stiefel manifold to provide topological relationships and capture model-form uncertainty. Using the Large-scale Atomic/Molecular Massively Parallel Simulator (LAMMPS), we assess the relevance of these stochastic reduced-order atomistic models on canonical problems involving a Lennard-Jones fluid and an argon crystal melt.

Volumetric Flask with White and Blue Balls: Demonstration of Microcanonical Ensemble of Small Populations

Volumetric Flask with White and Blue Balls: Demonstration of Microcanonical Ensemble of Small Populations

On The Thermalization of One-Dimensional Lattices. I. Microcanonical Ensemble

In numerical simulation of biomacromolecule, the issues of thermalization, i.e., equal distribution of energy over the degrees of freedom, occupy an important place. In this paper we consider some mechanisms of lattice thermalization: Chirikov resonances, wave turbulence and some others. We consider thermalization in a microcanonical ensemble when the system is isolated from external fields and the total energy is conserved. Although microcanonical ensembles are rarely used in practical calculations, however, the basic ideas about the thermalization mechanisms are obtained for these systems. The main attention is paid to the consideration of the lattices thermalization with Fermi-Pasta-Ulam-Tsingou potentials, since the main efforts to understand the basis of thermalization have been made precisely for lattices of this type. The role of solitons and breathers in thermalization is discussed.

Canonical Typicality for Other Ensembles than Micro-canonical

AbstractWe generalize Lévy’s lemma, a concentration-of-measure result for the uniform probability distribution on high-dimensional spheres, to a much more general class of measures, so-called GAP measures. For any given density matrix $$\rho $$ ρ on a separable Hilbert space $${\mathcal {H}}$$ H , $${\textrm{GAP}}(\rho )$$ GAP ( ρ ) is the most spread-out probability measure on the unit sphere of $${\mathcal {H}}$$ H that has density matrix $$\rho $$ ρ and thus forms the natural generalization of the uniform distribution. We prove concentration-of-measure whenever the largest eigenvalue $$\Vert \rho \Vert $$ ‖ ρ ‖ of $$\rho $$ ρ is small. We use this fact to generalize and improve well-known and important typicality results of quantum statistical mechanics to GAP measures, namely canonical typicality and dynamical typicality. Canonical typicality is the statement that for “most” pure states $$\psi $$ ψ of a given ensemble, the reduced density matrix of a sufficiently small subsystem is very close to a $$\psi $$ ψ -independent matrix. Dynamical typicality is the statement that for any observable and any unitary time evolution, for “most” pure states $$\psi $$ ψ from a given ensemble the (coarse-grained) Born distribution of that observable in the time-evolved state $$\psi _t$$ ψ t is very close to a $$\psi $$ ψ -independent distribution. So far, canonical typicality and dynamical typicality were known for the uniform distribution on finite-dimensional spheres, corresponding to the micro-canonical ensemble, and for rather special mean-value ensembles. Our result shows that these typicality results hold also for $${\textrm{GAP}}(\rho )$$ GAP ( ρ ) , provided the density matrix $$\rho $$ ρ has small eigenvalues. Since certain GAP measures are quantum analogs of the canonical ensemble of classical mechanics, our results can also be regarded as a version of equivalence of ensembles.

Weak (non)conservation and stochastic dynamics of angular momentum

Angular momentum conservation influences equilibrium statistical mechanics, leading to a generalized microcanonical density for an isolated system and a generalized Gibbs density for a weakly coupled system. We study the stochastic decay of angular momentum due to the weakly imperfect rotational symmetry of the external potential that confines the isolated many-particle system. We present a mesoscopic description of the system, deriving Langevin and Fokker–Planck equations, which are consistent with equilibrium statistical mechanics when rotational symmetry is maintained. When the symmetry is weakly violated, we formulate a coarse-grained stochastic differential equation governing the decay of total angular momentum over time. To validate our analytical predictions, we conduct numerical simulations of the microcanonical ensemble, an isolated system undergoing thermalization due to weak two-body interactions. Our coarse-grained Langevin equation accurately characterizes both the decay of the angular momentum and its fluctuations in a steady state. Furthermore, we estimate the parameters of our mesoscopic model directly from simulations, providing insights into the dissipative phenomenological coefficients, such as friction. More generally, this study contributes to a deeper understanding of the behavior of the integrals of motion when the corresponding symmetry is weakly violated.

Order-disorder on cluster dynamics in the Q2R-Potts cellular automaton

Cellular automata (CA) are mathematical models that allow the study of emergent behavior from a bottom-up point of view, while the Potts model is renowned for its rich dynamics capable of developing both first and second order phase transitions. Here, we study the Q2R-Potts cellular automaton, which is a model that merges cellular automata and the Potts model through a microcanonical ensemble characterized by a conservative energy-like function. Our study, conducted via numerical simulations, focuses on the one-dimensional Q2R-Potts CA with three (q = 3) states, examining its dynamics through both macroscopic and microscopic quantities. We discover that the model exhibits a first-order phase transition from order to disorder around of a critical energy density, characterized by a discontinuity in a phase diagram and self-organizing clusters that follow a power-law behavior.

Microscopic ensemble bootstrap in phase space

The bootstrap method which has been studied under many quantum mechanical models turns out to be feasible in microcanonical ensembles as well. While the approach of Nakayama (2022 Mod. Phys. Lett. A 37 2250054) produces a sector when energy is negative, in this paper we report a method that has stronger constraints and results in a smaller region. We also study other models to demonstrate the effectiveness of our method.

Born-Oppenheimer Molecular Dynamics with a Linear Combination of Atomic Orbitals and Hybrid Functionals for Condensed Matter Simulations Made Possible. Theory and Performance for the Microcanonical and Canonical Ensembles.

The implementation of an original Born-Oppenheimer molecular dynamics module is presented, which is able to perform simulations of large and complex condensed phase systems for sufficiently long time scales at the level of density functional theory with hybrid functionals, in the microcanonical (NVE) and canonical (NVT) ensembles. The algorithm is fully integrated in the Crystal code, a program for quantum mechanical simulations of materials, whose peculiarity stems from the use of atom-centered basis functions within a linear combination of atomic orbitals to describe the wave function. The corresponding efficiency in the evaluation of the exact Fock exchange series has led to the implementation of a rich variety of hybrid density functionals at a low computational cost. In addition, the molecular dynamics implementation benefits also from the effective MPI parallelization of the code, suited to exploit high-performance computing resources available on current generation supercomputer architectures. Furthermore, the information contained in the trajectory of the dynamics is extracted through a series of postprocessing algorithms that provide the radial distribution function, the diffusion coefficient and the vibrational density of states. In this work, we present a detailed description of the theoretical framework and the algorithmic implementation, followed by a critical evaluation of the accuracy and parallel performance (e.g., strong and weak scaling) of this approach, when ice and liquid water simulations are performed in the microcanonical and canonical ensembles.

Derivation of the Langevin Equation from the Microcanonical Ensemble.

When writing down a Langevin equation for the time evolution of a "system" in contact with a thermal bath, one typically makes the implicit (and often tacit) assumption that the thermal environment is in equilibrium at all times. Here, we take this assumption as a starting point to formulate the problem of a system evolving in contact with a thermal bath from the perspective of the bath, which, since it is in equilibrium, can be described by the microcanonical ensemble. We show that the microcanonical ensemble of the bath, together with the Hamiltonian equations of motion for all the constituents of the bath and system together, give rise to a Langevin equation for the system evolution alone. The friction coefficient turns out to be given in terms of auto-correlation functions of the interaction forces between the bath particles and the system, and the Einstein relation is recovered. Moreover, the connection to the Fokker-Planck equation is established.

Ensemble inequivalence in an extended spin-1 Nagle-Kardar model

We conduct a comprehensive investigation of an extended spin-1 Nagle-Kardar model in the microcanonical ensemble, focusing on exploring the inequivalence between the canonical and microcanonical ensembles in this generalized model. Our results reveal that the general trends of the microcanonical phase diagram align with those in the canonical ensemble. The continuous phase transition lines of these two ensembles largely coincide, yet the first-order ones in the microcanonical ensemble exhibit greater complexity, showing various examples of ensemble inequivalence, particularly in cases involving the triple points.

Improved Precision Scaling for Simulating Coupled Quantum-Classical Dynamics

In this paper, we present a superpolynomial improvement in the precision scaling of quantum simulations for coupled quantum-classical systems. Such systems are found in, e.g., molecular-dynamics simulations within the Born-Oppenheimer approximation. By employing a framework based on the Koopman–von Neumann formulation of classical mechanics, we express the Liouville equation of motion as unitary dynamics and utilize phase kickback from a dynamical quantum simulation to calculate the quantum forces acting on classical particles. This approach allows us to simulate the dynamics of these classical particles without the overheads associated with measuring gradients and solving the equations of motion on a classical computer, resulting in a superpolynomial advantage at the price of increased space complexity. We demonstrate that these simulations can be performed in both microcanonical and canonical ensembles, enabling the estimation of thermodynamic properties from the prepared probability density. Published by the American Physical Society 2024

Tunnelling to holographic traversable wormholes

We study nonperturbative effects of quantum gravity in a system consisting of a coupled pair of holographic CFTs. The AdS_44/CFT_33 system has three possible ground states: two copies of empty AdS, a pair of extremal AdS black holes, and an eternal AdS traversable wormhole. We give a recipe for calculating transition rates via gravitational instantons and test it by calculating the emission rate of radiation shells from a black hole. We calculate the nucleation rate of a traversable wormhole between a pair of AdS-RN black holes in the canonical and microcanonical ensembles. Our results give predictions of nonpertubative quantum gravity that can be tested in a holographic simulation.

Nonextensive entropies impact onto thermodynamics and phase structure of Kerr–Newman black holes

Taking the nonextensive Tsallis and Rényi entropies into account, we explore thermodynamic properties and phase transitions of the Kerr–Newman black holes (KNBH) in the microcanonical and canonical ensembles. We also compare our results with those obtained by attributing the Bekenstein–Hawking entropy bound to the mentioned black holes. Our analysis indicates that, similarly to the standard Boltzmann picture, isolated KNBH in the microcanonical approach are stable against axisymmetric perturbations in both Tsallis and Rényi models. On the other hand, in considering the case when the black holes are enveloped by a bath of thermal radiation in the canonical treatment, the KNBH based on the Tsallis and Rényi entropies can be stable for some values of the entropy parameters, in contrast to the traditional Boltzmann framework. For the case of Rényi entropy, we find that a Hawking-Page transition and a first order small black hole/large black hole transition can occur in a similar fashion as in rotating black holes in an anti-de Sitter space. Finally, we employ the Ruppeneir geometrothermodynamic technique to provide a new perspective on studying the nature of interactions between black hole microstructures, revealing a non-trivial impact of nonextensive entropies.

Nano-friction behavior and deformation study of hydroxyapatite in ultra-precision polishing process.

In order to study the effect of ultra-precision machining on the surface quality of hydroxyapatite semiconductor materials as well as the material removal mechanism of hydroxyapatite, the mechanical polishing behaviors of hydroxyapatite at different polishing depths were studied by molecular dynamics method. The results show that the subsurface damage of hydroxyapatite increases with increasing polishing depth. The polishing temperature and the polishing force showed a positive correlation with the polishing depth, and the variation of the polishing force was related to the accumulation-release effect of the potential energy of hydroxyapatite material. In addition, the variation of stresses in hydroxyapatite during polishing is mainly influenced by the thermal softening effect. With a smaller polishing depth, the hydroxyapatite semiconductor material has fewer structural defects, fewer atoms undergoing phase transitions, lower surface roughness, and better surface quality. Therefore, to ensure the long-lasting service life of hydroxyapatite semiconductor materials, a small polishing depth should be used in ultra-precision machining. Additionally, this study also provides a theoretical reference for future research on the mechanical properties of hydroxyapatite-based composites. A Large-Scale Atomic/Molecular Parallel Simulator (LAMMPS) was utilized to perform molecular dynamics simulations. The output was visualized and analyzed by the Open Visualization Tools (OVITO) software. The intermolecular interactions were described by the polymer consistent force-field and the 12/6 Lennard-Jones potential functions. The workpiece was polished under a micro-canonical ensemble with the temperature settled at 300 K. Periodic boundary conditions were adopted and the velocity-Verlet algorithm was used to integrate the atomic motion with a timestep of 0.1 femtoseconds (fs).

Implementation and Validation of an OpenMM Plugin for the Deep Potential Representation of Potential Energy.

Machine learning potentials, particularly the deep potential (DP) model, have revolutionized molecular dynamics (MD) simulations, striking a balance between accuracy and computational efficiency. To facilitate the DP model's integration with the popular MD engine OpenMM, we have developed a versatile OpenMM plugin. This plugin supports a range of applications, from conventional MD simulations to alchemical free energy calculations and hybrid DP/MM simulations. Our extensive validation tests encompassed energy conservation in microcanonical ensemble simulations, fidelity in canonical ensemble generation, and the evaluation of the structural, transport, and thermodynamic properties of bulk water. The introduction of this plugin is expected to significantly expand the application scope of DP models within the MD simulation community, representing a major advancement in the field.