Grand Canonical Ensemble Research Articles

Equation of state of partially ionized hydrogen and deuterium plasma revisited.

We present improved first-principle fermionic path integral Monte Carlo (PIMC) simulation results for a dense partially ionized hydrogen (deuterium) plasma, for temperatures in the range 15000K≤T≤400000K and densities 7×10^{-7}g/cm^{3}≤ρ_{H}≤0.085g/cm^{3} (1.4×10^{-6}g/cm^{3}≤ρ_{D}≤0.17g/cm^{3}), corresponding to 100≥r_{s}≥2, where r_{s}=r[over ¯]/a_{B} is the ratio of the mean interparticle distance to the Bohr radius. These simulations are based on the fermionic propagator PIMC (FP-PIMC) approach in the grand canonical ensemble [Filinov et al., Contrib. Plasma Phys. 61, e202100112 (2021)0863-104210.1002/ctpp.202100112] and fully account for correlation and quantum degeneracy and spin effects. For the application to hydrogen and deuterium, we develop a combination of the fourth-order factorization and the pair product ansatz for the density matrix. Moreover, we avoid the fixed node approximation that may lead to uncontrolled errors in restricted PIMC (RPIMC). Our results allow us to critically reevaluate the accuracy of the RPIMC simulations for hydrogen by Hu etal. [Phys. Rev. B 84, 224109 (2011)1098-012110.1103/PhysRevB.84.224109] and of various chemical models. The deviations are generally found to be small, but for the lowest temperature, T=15640K they reach several percent. We present detailed tables with our first principles results for the pressure and energy isotherms. We expect our updated results will serve as a valuable benchmark for comparison with other methods.

Strongly non-additive symmetric mixtures in slit-like pores

Using Monte Carlo simulation methods in the grand canonical ensemble, we have studied the behavior of symmetric binary mixtures confined between attractive planes. We have elucidated the effects of high negative geometric and energetic non-additivity on the phase behavior of mixtures in nanoscopic slit-like pores. In such systems, the geometric non-additivity favors mixing, while the energetic non-additivity favors the formation demixed dense phases. It has been shown that the confinement considerably influences the phase behavior. In particular, phase separation has been found to be suppressed in narrow pores, while in wide pores the phase behavior becomes qualitatively the same as in bulk systems. It has also been demonstrated that the demixed solid is stable over a certain temperature range in pores, while the bulk mixtures characterized by the same non-additivity exhibit only the presence of mixed solid.

Measuring many-body distribution functions in fluids using test-particle insertion.

We derive a hierarchy of equations, which allow a general n-body distribution function to be measured by test-particle insertion of between 1 and n particles. We apply it to measure the pair and three-body distribution functions in a simple fluid using snapshots from Monte Carlo simulations in the grand canonical ensemble. The resulting distribution functions obtained from insertion methods are compared with the conventional distance-histogram method: the insertion approach is shown to overcome the drawbacks of the histogram method, offering enhanced structural resolution and a more straightforward normalization. At high particle densities, the insertion method starts breaking down, which can be delayed by utilizing the underlying hierarchical structure of the insertion method. Our method will be especially useful in characterizing the structure of inhomogeneous fluids and investigating closure approximations in liquid state theory.

Generalized free energy and dynamical state transition of the dyonic AdS black hole in the grand canonical ensemble

We study the generalized free energy of the dyonic AdS black hole in an ensemble with varying electric charge qE and fixed magnetic charge qM. When we adjust the temperature T and the electric potential ΦE of the ensemble, the Ricci scalar curvature R and electromagnetic potential Au usually diverge at the horizon. We regularize them and incorporate the off-shell corrections into the Einstein-Hilbert action. Alternatively, we find that the off-shell corrections can also be obtained by adding a boundary near the horizon to exclude the singularities. Ultimately, we derive the generalized free energy which is consistent with the definition of the thermodynamic relations. Based on the generalized free energy landscape, we can describe the dynamics of state transition as a stochastic process quantified by the Langevin equation. The path integral framework can be formulated to derive the time-dependent trajectory of the order parameter and the time evolution of the transition probability. By comparing the probability with the result of the classical master equation, we attribute the contribution to the probability of one pseudomolecule or antipseudomolecule (the instanton and anti-instanton pairs) to the rate of state transition. These results are consistent with the qualitative analysis of the free energy landscape.

Microscopic interpretation of generalized entropy

Generalized entropy, that has been recently proposed, puts all the known and apparently different entropies like The Tsallis, the Rényi, the Barrow, the Kaniadakis, the Sharma-Mittal and the loop quantum gravity entropy within a single umbrella. However, the microscopic origin of such generalized entropy as well as its relation to thermodynamic system(s) is not clear. In the present work, we will provide a microscopic thermodynamic explanation of generalized entropy(ies) from canonical and grand-canonical ensembles. It turns out that in both the canonical and grand-canonical descriptions, the generalized entropies can be interpreted as the statistical ensemble average of a series of microscopic quantity(ies) given by various powers of (−kln⁡ρ)n (with n being a positive integer and ρ symbolizes the phase space density of the respective ensemble), along with a term representing the fluctuation of Hamiltonian and number of particles of the system under consideration (in case of canonical ensemble, the fluctuation on the particle number vanishes).

Casimir versus Helmholtz forces: Exact results

Recently, attention has turned to the issue of the ensemble dependence of fluctuation induced forces. As a noteworthy example, in O(n) systems the statistical mechanics underlying such forces can be shown to differ in the constant M→ magnetic canonical ensemble (CE) from those in the widely-studied constant h→ grand canonical ensemble (GCE). Here, the counterpart of the Casimir force in the GCE is the Helmholtz force in the CE. Given the difference between the two ensembles for finite systems, it is reasonable to anticipate that these forces will have, in general, different behavior for the same geometry and boundary conditions. Here we present some exact results for both the Casimir and the Helmholtz force in the case of the one-dimensional Ising model subject to periodic and antiperiodic boundary conditions and compare their behavior. We note that the Ising model has recently being solved in Phys.Rev. E 106 L042103 (2022), using a combinatorial approach, for the case of fixed value M of its order parameter. Here we derive exact result for the partition function of the one-dimensional Ising model of N spins and fixed value M using the transfer matrix method (TMM); earlier results obtained via the TMM were limited to M=0 and N even. As a byproduct, we derive several specific integral representations of the hypergeometric function of Gauss. Using those results, we rigorously derive that the free energies of the CE and grand GCE are related to each other via Legendre transformation in the thermodynamic limit, and establish the leading finite-size corrections for the canonical case, which turn out to be much more pronounced than the corresponding ones in the case of the GCE.

Fragility of the Schrödinger Cat in thermal environments

We describe the decoherence instability of Schrödinger Cat states in the two-site Bose-Hubbard model with an attractive on-site interaction between particles. For N particles with onsite attractive energy U and hopping amplitude between sites t, Cat states exist for uequiv frac{UN}{2t}<-1 at zero temperature. However, they are increasingly unstable to small thermal fluctuations as the Cat itself is increasingly well-defined and its components become well-separated. For any given u<-1, the decoherence temperature becomes smaller for large N. The loss of off-diagonal coherence peaks in the equilibrium density matrix is dominated by the thermal admixture of the first excited state of the many-body system with its ground state. Particle number fluctuations, described in the grand canonical ensemble also reduce coherence, but to a lesser degree than thermal fluctuations. The full density matrix of the Schrödinger Cat is obtained by exact numerical diagonalization of the many-body Hamiltonian and a narrow regime in the parameter space of the particle number, temperature, and U/t is identified where small Cat states may survive decoherence in a physical environment.

Theoretical Investigation of Inorganic Particulate Matter: The Case of Water Adsorption on a NaCl Particle Model Studied Using Grand Canonical Monte Carlo Simulations

Sodium chloride (NaCl) represents the principal component of atmospheric particulates of marine origin. To gain a molecular-level understanding of the adsorption process of water vapor on the NaCl surface, Monte Carlo simulations performed in the Grand Canonical ensemble were carried out, considering the water adsorption at different water pressures on a NaCl(001) surface. The calculated adsorption isotherm shows four different regions, whose coverages correspond to those of the low-, transition-, high-, and pre-solution-coverage regions experimentally observed. Detailed analysis reveals how the structure of the adsorbed water molecules (islands, layer, and multi-layer) changes depending on water pressure, and how their orientation with respect to the surface varies with the distance from the surface. This detailed information further supports the picture coming from previous experimental IR absorption spectroscopy studies.

Distribution of ions between different dielectric media: Direct simulation of the Donnan equilibrium in the grand canonical ensemble

A modification of the original Grand Canonical Monte Carlo (GCMC) method to handle Donnan equilibrium is proposed that provides an equilibrium between two implicit-solvent bath electrolytes with different dielectric constants. A solvation energy penalty (described by the Born theory) and an electrical potential difference (Donnan potential) exist between the two baths, the ‘system’ under investigation and the ‘reservoir’. These terms are deducted from the chemical potential of the ‘reservoir’, and the resulting chemical potential is used in the ‘system’. The simulation performed with this chemical potential in the acceptance probabilities of the ion insertion/deletions is called Donnan GCMC and provides a thermodynamic state in the ‘system’ that is in equilibrium with the electrolyte in the ‘reservoir’. The simulation provides the distribution of ions between the two baths (concentrations in both media) from a single run instead of a numerical procedure that requires several GCMC runs. Using individual ion insertion/deletions, the Donnan potential can be determined.

What can lattice DFT teach us about real-space DFT?

In this paper we establish a connection between density functional theory (DFT) for lattice models and common real-space DFT. We consider the lattice DFT description of a two-level model subject to generic interactions in Mermin's DFT formulation in the grand canonical ensemble at finite temperature. The case of only density-density and Hund's rule interaction studied in earlier work is shown to be equivalent to an exact-exchange description of DFT in the real-space picture. In addition, we also include the so-called pair-hopping interaction which can be treated analytically and, crucially, leads to non-integer occupations of the Kohn-Sham (KS) levels even in the limit of zero temperature. Treating the hydrogen molecule in a minimal basis is shown to be equivalent to our two-level lattice DFT model. By means of the fractional occupations of the KS orbitals (which, in this case, are identical to the many-body ones) we reproduce the results of full configuration interaction, even in the dissociation limit and without breaking the spin symmetry. Beyond the minimal basis, we embed our HOMO-LUMO model into a standard DFT calculation and, again, obtain results in overall good agreement with exact ones without the need of breaking the spin symmetry.

Criticality and topological classes of neutral Gauss–Bonnet AdS black holes in 5D

This work investigates the thermodynamics, phase transition, and topological classes of Neutral Gauss–Bonnet (GB) AdS black holes (BHs) in 5D. We obtain equations for several thermodynamic quantities for the Neutral GB of AdS BHs by using the well-known properties of BH thermodynamics. We find thermal stability and other significant thermodynamical parameters that assist us study stability and criticality. Thermodynamic features of Neutral GB AdS BHs are graphically explored by utilizing (Cp), (T−S), (P−rh), and G, volume expansivity (β), isothermal compressibility (κ). Second-order phase transitions, like Cp−S illustrates the divergence behavior. In addition, the swallow tail behavior finds the stability and instability of small black holes (SBH), intermediate black holes (IBH), and large black holes (LBH). Also, our investigation focuses on analyzing the topology of thermodynamic critical points and calculating their respective topological charges within the canonical and grand canonical ensembles. This analysis allows us to discuss the ensemble dependence of the Neutral GB-AdS BH exists.

Conformal hairy black holes of quartic quasi-topological gravity with power-Yang–Mills source

We study higher dimensional hairy black holes of quartic quasi-topological gravity in the framework of non-abelian power-Yang–Mills theory. It is shown that real solutions of the gravitational field equations exist only for positive values of quartic quasi-topological coefficient. Depending on the values of the mass, conformal coupling constants, quasi-topological coefficients, Yang–Mills magnetic charge, and nonlinearity parameter, they can be interpreted as black holes with one horizon, at most three horizons and naked singularity. It is also shown that the solution associated with these black holes has an essential curvature singularity at the centre r=0. Thermodynamic and conserved quantities for these hairy black holes are computed and we show that the first law has been verified. We also check thermodynamic stability in both canonical and grand canonical ensembles. In addition to this, we also formulate new power-Yang–Mills hairy black hole solutions in pure quasi-topological gravity. The physical and thermodynamic properties of these black holes are discussed as well. It is concluded that unlike Yang–Mills black holes without scalar hair there exist stability regions for the power-Yang–Mills hairy black holes in grand canonical ensemble. Finally, we discuss the thermodynamics of horizon flat power-Yang–Mills rotating black branes and analyze their thermodynamic and conserved quantities by using the counter-term method inspired by AdS/CFT correspondence.

Efficient estimation of transition rates as functions of pH

AbstractExtracting the kinetic properties of a system whose dynamics depend on the pH of the environment with which it exchanges energy and atoms requires sampling the grand canonical ensemble. As an alternative, we present a novel strategy that requires simulating only the most recurrent canonical ensembles that compose the grand canonical ensemble. The simulations are used to estimate the grand canonical distribution for a specific pH value by reweighting and to construct the transition rate matrix by discretizing the Fokker–Planck equation by square root approximation and robust Perron cluster cluster analysis. As an application, we have studied the tripeptide Ala‐Asp‐Ala.

Physics-informed Bayesian inference of external potentials in classical density-functional theory.

The swift progression and expansion of machine learning (ML) have not gone unnoticed within the realm of statistical mechanics. In particular, ML techniques have attracted attention by the classical density-functional theory (DFT) community, as they enable automatic discovery of free-energy functionals to determine the equilibrium-density profile of a many-particle system. Within classical DFT, the external potential accounts for the interaction of the many-particle system with an external field, thus, affecting the density distribution. In this context, we introduce a statistical-learning framework to infer the external potential exerted on a classical many-particle system. We combine a Bayesian inference approach with the classical DFT apparatus to reconstruct the external potential, yielding a probabilistic description of the external-potential functional form with inherent uncertainty quantification. Our framework is exemplified with a grand-canonical one-dimensional classical particle ensemble with excluded volume interactions in a confined geometry. The required training dataset is generated using a Monte Carlo (MC) simulation where the external potential is applied to the grand-canonical ensemble. The resulting particle coordinates from the MC simulation are fed into the learning framework to uncover the external potential. This eventually allows us to characterize the equilibrium density profile of the system by using the tools of DFT. Our approach benchmarks the inferred density against the exact one calculated through the DFT formulation with the true external potential. The proposed Bayesian procedure accurately infers the external potential and the density profile. We also highlight the external-potential uncertainty quantification conditioned on the amount of available simulated data. The seemingly simple case study introduced in this work might serve as a prototype for studying a wide variety of applications, including adsorption, wetting, and capillarity, to name a few.

Topology of nonlinearly charged black hole chemistry via massive gravity

The classification of critical points of charged topological black holes (TBHs) in anti-de Sitter spacetime (AdS) under the Power Maxwell Invariant (PMI)-massive gravity is accomplished within the framework of black hole chemistry (BHC). Considering the grand canonical ensemble (GCE), we show that d=4 black hole have only one topological class, whereas dge 5 black holes belong to two different topology classes. Furthermore, the conventional critical point characterized by negative topological charge coincides with the maximum extreme point of temperature; and the novel critical point featuring opposite topological charge corresponds to the minimum extreme point of temperature. With increasing pressure, new phases emerge at the novel critical point while disappear from the conventional one. Moreover, a atypical van der Waals (vdW) behavior is found in dge 6 dimensions, and the anomaly disappears at the traditional critical point. In the limit of nonlinearity parameter srightarrow 1, different topology classes are only obtained in the GCE and they may not exist within the canonical ensemble. With the absence of electric potential Phi , the neutral TBHs share the same topological classification results as the charged TBHs in the GCE of Maxwell-massive gravity.

Influence of the multibody interactions in a lipidic membrane model

This work presents a new model of lipid membrane that introduces multibody lateral interactions in the cooperative parameter ω. The traditional and widely-used assumption of pairwise interactions between neighboring lipids is replaced by a more general approach. In the literature, most models consider ω as a single interaction parameter while ignoring multibody interactions, where van der Waals forces, hydrogen bonding, and ionic bonding between acyl chains and head groups are present. In this study, a two-state model is considered where the gel-fluid interaction energy is parameterized and depends on the number of gel-fluid pairs in their nearest neighbors. Monte Carlo simulations in the grand canonical ensemble are used to analyze order parameters, heat capacity, and other parameters in the vicinity of the gel-fluid phase transition. The behavior is analyzed and discussed in terms of domain formation and the emergence of new ordered structures in the system.

Thermodynamics and phase transition structure of charged black holes in f(R) gravity background from Rényi statistics perception

In the current study, we investigate the Rényi statistics as an extension of the Boltzmann-Gibbs theory, in order to characterize the non-Boltzmannian phase structure of charged black holes in the f(R) gravity. Through this research, we discover that a Hawking-Page phase transition and a Van-der-Waals phase portrait, which are missing from the standard Boltzmann-Gibbs' description, emerge respectively within the grand canonical ensemble and canonical ensemble via the Rényi formalism.The thermodynamics associated with charged black holes without cosmological constant black holes using Rényi statistics processes the same properties as those of asymptotically AdS charged black holes via Gibbs-Boltzmann formalism, as further evidenced by such phase structures found here. This is another example of a potential and substantial relationship between the cosmological constant and the nonextensive Rényi parameter.

Off-Stoichiometric Restructuring and Sliding Dynamics of Hexagonal Boron Nitride Edges in Conditions of Oxidative Dehydrogenation of Propane.

Boron-containing materials, such as hexagonal boron nitride (h-BN), recently shown to be active and selective catalysts for the oxidative dehydrogenation of propane (ODHP), have been shown to undergo significant surface oxyfunctionalization and restructuring. Although experimental ex situ studies have probed the change in chemical environment on the surface, the structural evolution of it under varying reaction conditions has not been established. Herein, we perform global optimization structure search with a grand canonical genetic algorithm to explore the chemical space of off-stoichiometric restructuring of the h-BN surface under ambient as well as ODHP-relevant conditions. A grand canonical ensemble representation of the surface is established, and the predicted 11B solid-state NMR spectra are consistent with previous experimental reports. In addition, we investigated the relative sliding of h-BN sheets and how it influences the surface chemistry with ab initio molecular dynamics simulations. The B-O linkages on the edges are found to be significantly strained during the sliding, causing the metastable sliding configurations to have higher reactivity toward the activation of propane and water.

Stability and Hawking-Page-like phase transition of phantom AdS black holes

In this work, we investigate the thermodynamic stability and phase structure of AdS black holes with either a Maxwell field (where we revisit past studies) or a phantom field. We conduct a comprehensive analysis of the free energy and temperature of these systems in both the canonical and grand canonical ensembles. Our findings reveal the occurrence of a phase transition in the grand canonical ensemble, resembling the Hawking-Page-like phase transition observed between the thermal radiation of AdS spacetime and thermodynamically stable large black holes. We present graphical representations of these phase transitions on free energy-temperature diagrams for the black holes. Completing our study, we obtain the transition temperature, minimum temperature and their dual relations.

An infiltration load calculation model of large-space buildings based on the grand canonical ensemble theory

An infiltration load calculation model of large-space buildings based on the grand canonical ensemble theory