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In the study of crystal nucleation via computer simulations, hard spheres are arguably the most extensively explored model system. Nonetheless, even in this simple model system, the complex thermodynamics of crystal nuclei can sometimes give rise to counterintuitive results, such as the recent observation that the pressure inside a critical nucleus is lower than that of the surrounding fluid, seemingly clashing with the strictly positive Young–Laplace pressure we would expect in liquid droplets. Here, we re-derive many of the founding equations associated with crystal nucleation and use the hard-sphere model to demonstrate how they give rise to this negative pressure difference. We exploit the fact that, in the canonical ensemble, a nucleus can be in a (meta)stable equilibrium with the fluid and measure the surface stress for both flat and curved interfaces. Additionally, we explain the effect of defects on the chemical potential inside the crystal nucleus. Finally, we present a simple, fitted thermodynamic model to capture the properties of the nucleus, including the work required to form critical nuclei. [ABSTRACT FROM AUTHOR]

FEASST is an open-source Monte Carlo software package for particle-based simulations. This software, which was released in 2017, has been used to study phase equilibrium, self-assembly, aggregation or gelation in biological materials, colloids, polymers, ionic liquids, and adsorption in porous networks. We highlight some of the unique features available in FEASST, such as flat-histogram grand canonical ensemble, Gibbs ensemble, and Mayer-sampling simulations with support for anisotropic models and parallelization with flat-histogram and prefetching. We also discuss how the challenges of supporting a variety of Monte Carlo algorithms were overcome by an object-oriented design. This also allows others to extend classes, which improves software interoperability, as inspired by LAMMPS classes and user packages. This article describes version 0.25.1 with benchmarks, compilation instructions, and introductory tutorials for running, restarting, and testing simulations, user guidelines, software design strategies, alternative interfaces, and the test-driven development strategy. [ABSTRACT FROM AUTHOR]

Conventional molecular dynamics (MD) simulations struggle when simulating particles with steeply varying interaction potentials due to the need to use a very short time step. Here, we demonstrate that an event-driven Monte Carlo (EDMC) approach was first introduced by Peters and de With [Phys. Rev. E 85, 026703 (2012)] and represents an excellent substitute for MD in the canonical ensemble. In addition to correctly reproducing the static thermodynamic properties of the system, the EDMC method closely mimics the dynamics of systems of particles interacting via the steeply repulsive Weeks–Chandler–Andersen (WCA) potential. In comparison to time-driven MD simulations, EDMC runs faster by over an order of magnitude at sufficiently low temperatures. Moreover, the lack of a finite time step in EDMC circumvents the need to trade accuracy against the simulation speed associated with the choice of time step in MD. We showcase the usefulness of this model to explore the phase behavior of the WCA model at extremely low temperatures and to demonstrate that spontaneous nucleation and growth of the Laves phases are possible at temperatures significantly lower than previously reported. [ABSTRACT FROM AUTHOR]

Determining fluid pressure in Monte Carlo simulations can be a challenging task due to the reduced arsenal of computational tools available to perform such measurements. In addition, none of these tools is general enough to calculate the equilibrium pressure for a wide variety of models. The Gibbs–Duhem method, for example, is a very useful option, but only for pure (one component) systems. To enrich this arsenal, we propose here a direct method to calculate pressure in the canonical ensemble, which could easily be extended to mixtures at low densities. In analogy with the Widom method, our approach is based on the free energy variation with volume, described in terms of the removal of an empty or particle-occupied lattice column. We tested our approach for the lattice gas model and compared the results with exact Onsager solutions. Furthermore, the appearance of thermodynamic instabilities (loops) in the pressure isotherms during the phase transition and their relationship with interface effects are discussed. Finally, a phase diagram is obtained from these isotherms using the Hill construction. [ABSTRACT FROM AUTHOR]

Programmable locomotion of molecules inside the carbon nanotube (CNT) has a significant role in controlling the reactions and delivery systems based on nanotubes. Using molecular dynamics (MD) simulations as well as the theoretical approach, we evaluate the oscillation of C60 inside the CNTs that are subjected to strain gradients from both sides. The molecular dynamics simulations are implemented by LAMMPS open-source software. Using this program, the van der Waals (vdW) interactions are established between C60 and nanotube, and the simulations are performed in canonical ensemble. The strain gradient applied on CNT provides the restoring force of the oscillation of C60. The potential energy of fullerene finds the minimum value at the unstrained region of CNT, which makes it the equilibrium point of oscillation. The amplitude of the oscillations is shown to be related to the thermal energy of C60. The frequency of the oscillations depends on the magnitude of the strain gradient applied on the nanotubes. At higher strain gradients of CNTs, we observe the increase in the frequency due to the increase in the restoring force acting on the fullerene molecule. We exploit the strained carbon nanotubes to control the position of C60 inside the nanotube. It has been shown that by changing the strain gradient of CNT, it is possible to steer the locomotion of C60 to different points inside the nanotube. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

In this paper, we present an exact spherically symmetric and Yang–Mills charged AdS black hole solution in the context of 4D Einstein–Gauss–Bonnet (EGB) gravity in the presence of a cloud of strings. The regularity of the solution is checked. Thermodynamics of this solution is studied. The critical behavior, the types of phase transitions in canonical ensemble, the Joule–Thomson expansion, the Clapeyron equation and the critical exponents shall be investigated. [ABSTRACT FROM AUTHOR]

The goal of this work is to produce single species of polygons in two dimensions using the self-assembly of a simple patchy molecular model. A molecule is defined as a central soft repulsive Weeks Chandler and Anderson (WCA) atom that is decorated with two or four patches, which describe the two models studied in this work. Model I has two Lennard–Jones (LJ) patches inside of its central atom separated by an angle θ that intends to define the number of polygon sides. The attractive potential of these patches, interacting as pairs, is used to bond molecules together to form polygons. Model II, having two additional soft repulsive WCA patches, tangent on the outside, produces single polygon species configurations. An example of Model I with an angle $ \theta = 120^\circ $ θ = 120 ∘ , forming a mixture of polygons, is included. Model II produces single polygon species with 3-8 sides, depending on the angle θ between the attractive patches. Molecular dynamics simulations in the canonical ensemble are used to calculate thermodynamic properties such as internal energy, percentage of polygons, pressure and contour plots for the percentage of polygons following isochoric processes. Different starting configurations are used to verify the consistency of the equilibrium results. [ABSTRACT FROM AUTHOR]

We present a simulation method that allows us to calculate the titration curves for systems undergoing protonation/deprotonation reactions—such as charged colloidal suspensions with acidic/basic surface groups, polyelectrolytes, polyampholytes, and proteins. The new approach allows us to simultaneously obtain titration curves both for systems in contact with salt and acid reservoir (semi-grand canonical ensemble) and for isolated suspensions (canonical ensemble). To treat the electrostatic interactions, we present a new method based on Ewald summation—which accounts for the existence of both Bethe and Donnan potentials within the simulation cell. We show that the Donnan potential dramatically affects the pH of a suspension. Counterintuitively, we find that in suspensions with a large volume fraction of nanoparticles and low ionic strength, the number of deprotonated groups can be 100% larger in an isolated system, compared to a system connected to a reservoir by a semi-permeable membrane—both systems being at exactly the same pH. [ABSTRACT FROM AUTHOR]

The complexity of electrochemical interfaces has led to the development of several approximate density functional theory (DFT)-based schemes to study reaction thermodynamics and kinetics as a function of electrode potential. While fixed electrode potential conditions can be simulated with grand canonical ensemble DFT (GCE-DFT), various electrostatic corrections on canonical, constant charge DFT are often applied instead. In this work, we present a systematic derivation and analysis of the different electrostatic corrections on canonical DFT to understand their physical validity, implicit assumptions, and scope of applicability. Our work highlights the need to carefully address the suitability of a given model for the problem under study, especially if physical or chemical insight in addition to reaction energetics is sought. In particular, we analytically show that the different corrections cannot differentiate between electrostatic interactions and covalent or charge-transfer interactions. By numerically testing different models for CO2 adsorption on a single-atom catalyst as a function of the electrode potential, we further show that computed capacitances, dipole moments, and the obtained physical insight depend sensitively on the chosen approximation. These features limit the scope, generality, and physical insight of these corrective schemes despite their proven practicality for specific systems and energetics. Finally, we suggest guidelines for choosing different electrostatic corrections and propose the use of conceptual DFT to develop more general approximations for electrochemical interfaces and reactions using canonical DFT. [ABSTRACT FROM AUTHOR]

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 → + ∞ . [ABSTRACT FROM AUTHOR]

We attempt to construct a (two-dimensional) statistical mechanics description of the scenario in which the experiment in [T. Westphal et al., Nature 591, 225 (2021)] was made. It includes two masses that, simultaneously, (1) interact classically via the Newton gravitation and (2) harmonically vibrate. Within the strictures of Gibbs canonical ensemble at the temperature T, our treatment is exact. [ABSTRACT FROM AUTHOR]

Structural phase transitions are accompanied by a movement of one nucleus (or a few) in the crystallographic unit cell. If the nucleus movement is continuous, a second‐order phase transition without latent heat results, whereas an abrupt nucleus displacement indicates a first‐order phase transition with accompanying latent heat. Herein, a Hamiltonian including electron–phonon coupling (EPC) as proposed by Kristoffel and Konsel is taken. Contrary to their treatment, both the kinetic energy of the nucleus and its position are treated. The interaction of the many‐electron system with the single nucleus is taken into account by the Born–Oppenheimer approximation and perturbative expressions for the free energies are derived. The nuclei corrections due to the entangled electrons are found to be minor, but highlight the importance of the symmetry breaking at low temperature. Furthermore the free energy for a canonical ensemble is computed, whereas Kristoffel and Konsel use a grand canonical ensemble, which allows to derive more stringent bounds on the free energy. For the zero‐order nucleus correction, the shift of the phase transition temperature by evaluating the free energy is deduced. [ABSTRACT FROM AUTHOR]

We screened 1651 carbon and boron nitride (BN) nanotubes to investigate methane release at 298 K and hydrogen release at 77 K through grand canonical ensemble Monte Carlo (GCMC) simulations. The effects of nanotube radius, density, and van der Waals spacing on the release quantities of these gases were explored. Our research shows that CH 4 release in C(40,40) and BN(40,40) nanotubes reaches a high of 0.15 g/g, below the DOE's 0.5 g/g target. Similarly, the best nanotubes for volumetric release, C(21,9) and BN(18,18), peak at 175 cm³/cm³, below the DOE target of 330 cm³/cm³. This suggests that internal adsorption within isolated nanotubes is insufficient to meet DOE targets, highlighting the need for optimization of inter-tube van der Waals spacing. For H 2 , C(40,39) and BN(40,40) nanotubes are optimal for weight release, achieving up to 12.19% at a pressure of 8 MPa. This surpasses the 2025 DOE benchmark of 5.5 wt%. Meanwhile, C(28,7) and BN(24,24) nanotubes are most suitable for volumetric release. The C(28,7) nanotube, recommended for H 2 storage, reaches a volumetric release quantity close to the DOE target of 0.04 kg/L at a van der Waals spacing of 1 nm. At pressures exceeding 2 MPa, the weight adsorption outside nanotubes accounts for 40–55% of the total, underscoring the pivotal role of extra-tube adsorption in gas storage capacity for both CH 4 and H 2. The results provide a foundation for the design of nanotube-based materials for efficient gas storage and release, with implications for clean energy applications. • C(40,39) and BN(40,40) excel in hydrogen release, exceeding 2025 DOE goals. • C(28,7) and BN(24,24) top for hydrogen volume; C(28,7) nears DOE target at 1 nm spacing. • Extra-tube adsorption accounts for over 40% of CH4 and H 2 storage at pressures above 2 MPa. [ABSTRACT FROM AUTHOR]

Utilizing the holographic dictionary from the proposal that treats Newton's constant as a thermodynamic variable, we establish a thermodynamic topological equivalence between the AdS black holes in the bulk and the thermal states in the dual CFT. The findings further reveal that the thermodynamic topological characteristics of the RegMax AdS black holes are strongly influenced by the characteristic parameter of the regularized Maxwell theory. Additionally, we investigate the phase transition between low and high entropy thermal states within a canonical ensemble in the dual CFT. Our observations indicate that the phase transition behavior of the thermal states mirrors that of the black holes. By modeling the phase transition process as a stochastic process, we are able to calculate the rates of phase transition between the thermal states. This result enhances our understanding of the dominant processes involved in the phase transition of the thermal states in the dual CFT. [ABSTRACT FROM AUTHOR]

A theoretical approach aimed at the quantum statistical mechanics of a molecular ensemble coupled to a lossless cavity mode is presented. A canonical ensemble is considered and an approximate formula is devised for the Helmholtz free energy correction due to cavity-molecule coupling, which enables the derivation of experimentally measurable thermodynamic quantities. The frequency of the cavity mode is assumed to lie in the infrared range. Therefore, the cavity couples to molecular vibrations and our treatment is restricted to the electronic ground state of the molecule. The method is tested for an analytically solvable model system of one-dimensional harmonic oscillators coupled to the cavity. The performance of the approximation and its range of validity are discussed in detail. It is shown that the leading-order correction to the Helmholtz free energy is proportional to the square of the collective coupling strength. We also demonstrate that the cavity mode does not have a significant impact on the thermodynamic properties of the system in the collective ultrastrong coupling regime (the collective coupling strength is comparable to the frequency of the cavity mode). [ABSTRACT FROM AUTHOR]

Ultracold-atom simulations of the Hubbard model provide insights into the character of charge and spin correlations in and out of equilibrium. The corresponding numerical simulations, on the other hand, remain a significant challenge. We build on recent progress in the quantum Monte Carlo (QMC) simulation of electrons in continuous space and apply similar ideas to the square-lattice Hubbard model. We devise and benchmark two discrete-time QMC methods, namely the fermionic-propagator QMC (FPQMC) and the alternating-basis QMC (ABQMC). In FPQMC, the time evolution is represented by snapshots in real space, whereas the snapshots in ABQMC alternate between real and reciprocal space. The methods may be applied to study equilibrium properties within the grand-canonical or canonical ensemble, external field quenches, and even the evolution of pure states. Various real-space/reciprocal-space correlation functions are also within their reach. Both methods deal with matrices of size equal to the number of particles (thus independent of the number of orbitals or time slices), which allows for cheap updates. We benchmark the methods in relevant setups. In equilibrium, the FPQMC method is found to have an excellent average sign and, in some cases, yields correct results even with poor imaginary-time discretization. ABQMC has a significantly worse average sign, but also produces good results. Out of equilibrium, FPQMC suffers from a strong dynamical sign problem. On the contrary, in ABQMC, the sign problem is not time-dependent. Using ABQMC, we compute survival probabilities for several experimentally relevant pure states. [ABSTRACT FROM AUTHOR]

We present a method to estimate the transition rates of molecular systems under different environmental conditions that cause the formation or the breaking of bonds and require the sampling of the Grand Canonical Ensemble. For this purpose, we model the molecular system in terms of probable "scenarios," governed by different potential energy functions, which are separately sampled by classical MD simulations. Reweighting the canonical distribution of each scenario according to specific environmental variables, we estimate the grand canonical distribution, then use the Square Root Approximation method to discretize the Fokker-Planck operator into a rate matrix and the robust Perron Cluster Cluster Analysis method to coarse-grain the kinetic model. This permits efficiently estimating the transition rates of conformational states as functions of environmental variables, for example, the local pH at a cell membrane. In this work, we formalize the theoretical framework of the procedure, and we present a numerical experiment comparing the results with those provided by a constant-pH method based on non-equilibrium Molecular Dynamics Monte Carlo simulations. The method is relevant for the development of new drug design strategies that take into account how the cellular environment influences biochemical processes. [ABSTRACT FROM AUTHOR]

A set of position-dependent noncommutative algebra in two dimension (2D) that describes the space near the Planck scale had been introduced [J. Phys. A: Math. Theor. 53 (2020) 115303]. This algebra predicted the existence of maximal length of graviton measurable at low energy. From this algebra, we deduce in the present paper, a new noncommutative algebra that is compatible with the deformed algebra proposed by Costa Filho et al. [Phys. Rev. A84 (2011) 050102] to describe the Position-Dependent Mass (PDM) system in 1D. To this aim, we derive from the momentum operators, the Schrödinger-like equation which describes PDM system in null quantum well potential. The spectrum of this system is asymetrically deformed and exhibits different behaviours from the one obtained by Costa Filho et al. Thus, we observe that by increasing the PDM, the energy decreases and this decrease is more pronounced as the quantum number increases. Finally, we evaluate the thermodynamic quantities within the canonical ensemble and we show that these results are consistent with the ones recently obtained by Bensalem and Bouaziz [Phys. A Stat. Mech. Appl.523 (2019) 583–592]. [ABSTRACT FROM AUTHOR]

Adsorption of 4 He on graphene substrates has been a topic of great interest due to the intriguing effects of graphene corrugation on the manifestation of commensurate solid and exotic phases in low-dimensional systems. In this study, we employ worm algorithm quantum Monte Carlo to study helium adsorbed on a graphene substrate to explore corrugation effects in the grand canonical ensemble. We utilized a Szalewicz potential for helium–helium interactions and a summation of isotropic interactions between helium and carbon atoms to construct a helium–graphene potential. We implement different levels of approximation to achieve a smooth potential, three partially corrugated potentials, and a fully ab initio potential to test the effects of corrugation on the first and second layers. We demonstrate that the omission of corrugation within the helium–graphene potential could lead to finite-size effects in both the first and second layers. Thus, a fully corrugated potential should be used when simulating helium in this low-dimensional regime. [ABSTRACT FROM AUTHOR]

Close to equilibrium, the excess heat governs the static fluctuations. We study the heat capacity in that McLennan regime, i.e. in linear order around equilibrium, using an expression in terms of the average energy that extends the equilibrium formula in the canonical ensemble. It is derivable from an entropy and it always vanishes at zero temperature. Any violation of an extended Third Law is therefore a nonlinear effect. [ABSTRACT FROM AUTHOR]

We numerically study the transport properties of a two-dimensional Fermi gas in a weakly and strongly interacting regimes, in the range of temperatures close to the transition to a superfluid phase. For that we excite sound waves in a fermionic mixture by using the phase imprinting technique, follow their evolution, and finally determine both their speed and attenuation. Our formalism, originated from a density-functional theory, incorporates thermal fluctuations via the grand canonical ensemble description and with the help of Metropolis algoritm. From numerical simulations we extract temperature dependence of the sound velocity and diffusivity as well as the dependence on the interaction strength. We emphasize the role of virtual vortex–antivortex pairs creation in the process of sound dissipation. [ABSTRACT FROM AUTHOR]

We analyze the general relation between canonical and grand canonical ensembles in the thermodynamic limit. We begin our discussion by deriving, with an alternative approach, some standard results first obtained by Kac and coworkers in the late 1970s. Then, motivated by the Bose–Einstein condensation (BEC) of trapped gases with a fixed number of atoms, which is well described by the canonical ensemble and by the recent groundbreaking experimental realization of BEC with photons in a dye-filled optical microcavity under genuine grand canonical conditions, we apply our formalism to a system of non-interacting Bose particles confined in a two-dimensional harmonic trap. We discuss in detail the mathematical origin of the inequivalence of ensembles observed in the condensed phase, giving place to the so-called grand canonical catastrophe of density fluctuations. We also provide explicit analytical expressions for the internal energy and specific heat and compare them with available experimental data. For these quantities, we show the equivalence of ensembles in the thermodynamic limit. [ABSTRACT FROM AUTHOR]

The random batch method provides an efficient algorithm for computing statistical properties of a canonical ensemble of interacting particles. In this work, we study the error estimates of the fully discrete random batch method, especially in terms of approximating the invariant distribution. The triangle inequality framework employed in this paper is a convenient approach to estimate the long-time sampling error of the numerical methods. Using the triangle inequality framework, we show that the long-time error of the discrete random batch method is |$O(\sqrt {\tau } + e^{-\lambda t})$|⁠ , where |$\tau $| is the time step and |$\lambda $| is the convergence rate, which does not depend on the time step |$\tau $| or the number of particles |$N$|⁠. Our results also apply to the McKean–Vlasov process, which is the mean-field limit of the interacting particle system as the number of particles |$N\rightarrow \infty $|⁠. [ABSTRACT FROM AUTHOR]

In this paper, we will systematically present the method of collective variables with a reference system for a classical many-particle interacting system in the grand canonical ensemble. The emphasis will be placed on the details of calculations. In particular, the usage of total correlation functions defined for the grand canonical ensemble allows us to investigate very accurately the cumulants of the grand partition function for the reference system. It is shown that any cumulant Mn can be expressed as a product of three components: the average particle number within the reference system. Kronecker's symbol for n wave vectors, and the n-particle structure factor. The functional expression for the grand partition function is derived, with all coefficients explicitly defined. The coordinates of the critical point are computed in the mean field approximation. [ABSTRACT FROM AUTHOR]

Recently, it was shown that the power-Maxwell (PM) theory could remove the singularity of the electric field (B. Eslam Panah, Europhys. Lett. 134, 20005 (2021)). Motivated by a great interest in three-dimensional black holes and a surge of success in studying massive gravity from both the cosmological and astrophysical points of view, we investigate three-dimensional black hole solutions in de Rham, Gabadadze, and Tolley massive theory of gravity in the presence of PM electrodynamics. First, we extract exact three-dimensional solutions in this theory of gravity. Then we study the geometrical properties of these solutions. Calculating conserved and thermodynamic quantities, we check the validity of the first law of thermodynamics for these black holes. We also examine the stability of these black holes in the context of the canonical ensemble. We continue calculating this kind of black hole's optical features, such as the photon orbit radius, the energy emission rate, and the deflection angle. Considering these optical quantities, finally, we analyze the effective role of the parameters of models on them. [ABSTRACT FROM AUTHOR]

In this paper, we study the relativistic correction to Bekenstein–Hawking entropy in the canonical ensemble and isothermal–isobaric ensemble and apply it to the cases of non-rotating BTZ and AdS-Schwarzschild black holes. This is realized by generalizing the equations obtained using Boltzmann–Gibbs (BG) statistics with its relativistic generalization, Kaniadakis statistics, or κ -statistics. The relativistic corrections are found to be logarithmic in nature and it is observed that their effect becomes appreciable in the high-temperature limit suggesting that the entropy corrections must include these relativistically corrected terms while taking the aforementioned limit. The non-relativistic corrections are recovered in the κ → 0 limit. [ABSTRACT FROM AUTHOR]

The exponential random graph model (ERGM) is a central object in the study of clustering properties in social networks as well as canonical ensembles in statistical physics. Despite some breakthrough works in the mathematical understanding of ERGM, most notably by Bhamidi, Bresler, and Sly [Ann. Appl. Probab. 21 (2011), pp. 2146–2170] through the analysis of a natural Heat-bath Glauber dynamics, and by Chatterjee and Diaconis [Ann. Statist. 41 (2013), pp. 2428–2461; Eldan [Geom. Funct. Anal. 28 (2018), pp. 1548–1596; and Eldan and Gross [Ann. Appl. Probab. 28 (2018), pp. 3698–3735] via a large deviation theoretic perspective, several basic questions have remained unanswered owing to the lack of exact solvability unlike the much studied Curie-Weiss model (Ising model on the complete graph). In this paper, we establish a series of new concentration of measure results for the ERGM throughout the entire sub-critical phase , including a Poincaré inequality, Gaussian concentration for Lipschitz functions, and a central limit theorem. In addition, a new proof of a quantitative bound on the W_1-Wasserstein distance to Erdős-Rényi graphs, previously obtained by Reinert and Ross [Ann. Appl. Probab. 29 (2019), pp. 3201–3229], is also presented. The arguments rely on translating temporal mixing properties of Glauber dynamics to static spatial mixing properties of the equilibrium measure and have the potential of being useful in proving similar functional inequalities for other Gibbsian systems beyond the perturbative regime. [ABSTRACT FROM AUTHOR]

Employing the formalism of the joint density functional theory (JDFT) in the grand canonical ensemble, this paper studies the electronic structure and relative reactivity of an hydroxyl radical adsorbed on the surface of three different metal oxides that share the rutile-type structure and the (110) crystal plane; these materials are electrodes used in electrochemical advanced oxidation processes which consist in the generation of hydroxyl radicals, highly reactive species, that are able to oxidize organic compounds to CO 2 , making these anodes efficient for wastewater treatment. By analyzing the changes in the average number of electrons as a function of the applied chemical potential at fixed external potential, we studied two reactivity indices from conceptual density functional theory: global and local softness, to help us understand the differences between these materials, additionally we propose a new scheme to approximate the redox properties of clean surfaces or surfaces in contact with different adsorbed molecules so that these values can be arranged in a relative potential scale which would allow direct comparison with experimental results. [ABSTRACT FROM AUTHOR]

We present a multilevel Monte Carlo simulation method for analyzing multi-scale physical systems via a hierarchy of coarse-grained representations, to obtain numerically exact results, at the most detailed level. We apply the method to a mixture of size-asymmetric hard spheres, in the grand canonical ensemble. A three-level version of the method is compared with a previously studied two-level version. The extra level interpolates between the full mixture and a coarse-grained description where only the large particles are present—this is achieved by restricting the small particles to regions close to the large ones. The three-level method improves the performance of the estimator, at fixed computational cost. We analyze the asymptotic variance of the estimator and discuss the mechanisms for the improved performance. [ABSTRACT FROM AUTHOR]

A neural network-assisted molecular dynamics method is developed to reduce the computational cost of open boundary simulations. Particle influxes and neural network-derived forces are applied at the boundaries of an open domain consisting of explicitly modeled Lennard-Jones atoms in order to represent the effects of the unmodeled surrounding fluid. Canonical ensemble simulations with periodic boundaries are used to train the neural network and to sample boundary fluxes. The method, as implemented in the LAMMPS, yields temperature, kinetic energy, potential energy, and pressure values within 2.5% of those calculated using periodic molecular dynamics and runs two orders of magnitude faster than a comparable grand canonical molecular dynamics system. [ABSTRACT FROM AUTHOR]

We study the monatomic water model of Molinero and Moore the grand canonical ensemble Monte Carlo simulation. Measurements of the probability distribution of the number density obtained via multicanonical sampling and histogram reweighting provide accurate estimates of the temperature dependence of both the liquid–vapor coexistence densities and the surface tension. Using finite-size scaling methods, we locate the liquid–vapor critical point at Tc = 917.6 K, ρc = 0.311 g cm−3. When plotted in scaled variables, in order to test the law of corresponding states, the coexistence curve of monatomic water is close to that of real water. In this respect, it performs better than extended simple point charge (SPC/E), TIP4P, and TIP4P/2005 water. [ABSTRACT FROM AUTHOR]

In the present study, using dynamic mean field theory complemented by grand canonical molecular dynamics simulations, we investigate the extent to which the density distributions encountered during the dynamics of capillary condensation are related to those distributions at equilibrium or metastable equilibrium in a system at fixed average density (canonical ensemble). We find that the states encountered can be categorized as out of equilibrium or quasi-equilibrium based on the magnitude of the driving force for mass transfer. More specifically, in open-ended slit pores, pore filling via double bridging is an out of equilibrium process, induced by the dynamics of the system, while pore filling by single bridge formation is connected to a series of configurations that are equilibrium configurations in the canonical ensemble and that cannot be observed experimentally by a standard adsorption process, corresponding to the grand canonical ensemble. Likewise, in closed cap slits, the formation of a liquid bridge near the pore opening and its subsequent growth while the initially detached meniscus from the capped end remains immobilized are out of equilibrium processes that occur at large driving forces. On the other hand, at small driving forces, there is a continuous acceleration of the detached meniscus from the capped end, which is associated with complete reversibility in the limit of an infinitesimally small driving force. [ABSTRACT FROM AUTHOR]

The surface of a material often undergoes dramatic structure evolution under a chemical environment, which, in turn, helps determine the different properties of the material. Here, we develop a general-purpose method for the automated search of optimal surface phases (ASOPs) in the grand canonical ensemble, which is facilitated by the stochastic surface walking (SSW) global optimization based on global neural network (G-NN) potential. The ASOP simulation starts by enumerating a series of composition grids, then utilizes SSW-NN to explore the configuration and composition spaces of surface phases, and relies on the Monte Carlo scheme to focus on energetically favorable compositions. The method is applied to silver surface oxide formation under the catalytic ethene epoxidation conditions. The known phases of surface oxides on Ag(111) are reproduced, and new phases on Ag(100) are revealed, which exhibit novel structure features that could be critical for understanding ethene epoxidation. Our results demonstrate that the ASOP method provides an automated and efficient way for probing complex surface structures that are beneficial for designing new functional materials under working conditions. [ABSTRACT FROM AUTHOR]

Kirkwood–Buff integrals (KBIs) connect the microscopic structure and thermodynamic properties of liquid solutions. KBIs are defined in the grand canonical ensemble and evaluated by assuming the thermodynamic limit (TL). In order to reconcile analytical and numerical approaches, finite-size KBIs have been proposed in the literature, resulting in two strategies to obtain their TL values from computer simulations. (i) The spatial block analysis method in which the simulation box is divided into subdomains of volume V to compute density fluctuations. (ii) A direct integration method where a corrected radial distribution function and a kernel that accounts for the geometry of the integration subvolumes are combined to obtain KBI as a function of V. In this work, we propose a method that connects both strategies into a single framework. We start from the definition of finite-size KBI, including the integration subdomain and an asymptotic correction to the radial distribution function, and solve them in Fourier space where periodic boundary conditions are trivially introduced. The limit q → 0, equivalent to the value of the KBI in the TL, is obtained via the spatial block-analysis method. When compared to the latter, our approach gives nearly identical results for all values of V. Moreover, all finite-size effect contributions (ensemble, finite-integration domains, and periodic boundary conditions) are easily identifiable in the calculation. This feature allows us to analyze finite-size effects independently and extrapolates the results of a single simulation to different box sizes. To validate our approach, we investigate prototypical systems, including SPC/E water and aqueous urea mixtures. [ABSTRACT FROM AUTHOR]

It has been argued that the existence of a zero-point length is the hallmark of quantum gravity. In this paper, we suggest a thermal mechanism whereby this quantum of length arises in flat, Euclidean spacetime ℝ d . For this, we consider the infinite sequence of all flat, Euclidean spacetimes ℝ d ′ with d ′ ≥ d , and postulate a probability distribution for each d ′ to occur. The distribution considered is that of a canonical ensemble at temperature T , the energy levels those of a 1-dimensional harmonic oscillator. Since both the harmonic energy levels and the spacetime dimensions are evenly spaced, one can identify the canonical distribution of harmonic-oscillator eigenvalues with that of dimensions d ′ . The state describing this statistical ensemble has a mean square deviation in the position operator, that can be interpreted as a quantum of length. Thus, placing an oscillator in thermal equilibrium with a bath provides a thermal mechanism whereby a zero-point length is generated. The quantum-gravitational implications of this construction are then discussed. In particular, a model is presented that realizes a conjectured duality between a weakly gravitational, strongly quantum system and a weakly quantum, strongly gravitational system. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

We investigate the microstructural evolution of electrochemically deposited poly-crystalline Cu films during subsequent thermal process cycles at mild maximum temperatures, compatible with the integration in advanced metallization schemes for electronic device manufacturing. The modifications induced by the thermal budget have been characterized at different scales (from the film-substrate interface to the wafer scale) with different complementary techniques: X-ray Diffraction (XRD), scanning electron microscopy (SEM), atomic force microscopy (AFM), and dynamical warpage measurements. Moreover, the film internal grains' evolution has been modelled by a tri-dimensional on-cell model, derived by the Pott-like multi-states configurational energy dependence, able to consider multiple orientation of the grains and densification kinetics in the canonical ensemble. Finally, a macroscopic model of the warpage dependence on the process conditions is discussed. The presented joint theoretical and experimental analysis provides a complete and consistent scenario of the grain densification phenomenon and its impact for the Cu film microstructure and the composite system morphology, indicating several strategies for the integration of the process in real device structures. [ABSTRACT FROM AUTHOR]

In this paper, we shall study the phase transition of nonminimal coupling of Einstein–Hilbert gravity and electric field of Yang–Mills type in AdS space–time. We couple the Ricci scalar to the Yang–Mills invariant to obtain a modified theory of gravity. A black brane solution is introduced up to the first order of the term R F μ α (a) F (a) μ α in this model. Then, the phase transition of this solution will be investigated in canonical ensemble. Our investigation shows that only the second-order phase transition behavior is seen in this model. Also, due to the coupling of the Yang–Mills field and Ricci scalar, there are differences with the phase transitions of the usual minimal models. We shall show that in the absence of nonminimal coupling there is not any phase transition. [ABSTRACT FROM AUTHOR]

In this paper, we study topological numbers for uncharged and charged static black holes obtained in z = 3 Hořava–Lifshitz gravity theory in different ensembles, where z measures the degree of anisotropy between space and time. We first calculate the topological numbers for the uncharged black holes by changing the value of the dynamic coupling constant, and find that the black holes with spherical and flat horizons have the same topological number. When the black hole's horizon is hyperbolic, different values of the coupling constant generate different topological numbers, which can be 1, 0 or - 1 . This shows that the coupling constant plays an important role in the topological classification. Then we study the topological numbers for the charged black holes in different ensembles. The black hole with a spherical horizon has the same topological number in canonical and grand canonical ensembles. When the horizons are flat or hyperbolic, they have different topological numbers in canonical and grand canonical ensembles. Therefore, the topological numbers for the uncharged black holes are parameter dependent, and those for the charged black holes are ensemble dependent. [ABSTRACT FROM AUTHOR]

Polymer-mediated colloidal interactions control the stability and phase properties of colloid–polymer mixtures that are critical for a wide range of important applications. In this work, we develop a versatile self-consistent field theory (SCFT) approach to study this type of interaction based on a continuum confined polymer solution model with explicit solvent and confining walls. The model is formulated in the grand canonical ensemble, and the potential of mean force for the polymer-mediated interaction is computed from grand potentials. We focus on the case of non-adsorbing linear polymers and present a systematic investigation on depletion effects using SCFT. The properties of confined polymer solutions are probed, and mean-field profiles of induced interactions are shown across different physical regimes. We expose a detailed parametric dependence of the interaction, concerning both attractive and repulsive parts, on polymer concentration, chain length, and solvent quality and explore the effect of wall surface roughness, demonstrating the versatility of the proposed approach. Our findings show good agreement with previous numerical studies and experiments, yet extend prior work to new regimes. Moreover, the mechanisms of depletion attraction and repulsion, along with the influence of individual control factors, are further discussed. We anticipate that this study will provide useful insights into depletion forces and can be readily extended to examine more complex colloid–polymer mixtures. [ABSTRACT FROM AUTHOR]

A recently proposed extended Hamiltonian approach to switching interaction potentials is generalized to enable adaptive partitioning molecular dynamics simulations. Switching is performed along a fictitious classical degree of freedom whose value determines the mixing ratio of the two potentials on a time scale determined by its associated mass. We propose to choose this associated fictitious mass adaptively so as to ensure a constant time scale for all switching processes. For different model systems, including a harmonic oscillator and a Lennard-Jones fluid, we investigate the window of switching time scales that guarantees the conservation of the extended Hamiltonian for a large number of switching events. The methodology is first applied in the microcanonical ensemble and then generalized to the canonical ensemble using a Nosé–Hoover chain thermostat. It is shown that the method is stable for thousands of consecutive switching events during a single simulation, with constant temperature and a conserved extended Hamiltonian. A slight modification of the original Hamiltonian is introduced to avoid accumulation of small numerical errors incurred after each switching process. [ABSTRACT FROM AUTHOR]

A comprehensive and detailed account is presented for the finite-temperature many-body perturbation theory for electrons that expands in power series all thermodynamic functions on an equal footing. Algebraic recursions in the style of the Rayleigh–Schrödinger perturbation theory are derived for the grand potential, chemical potential, internal energy, and entropy in the grand canonical ensemble and for the Helmholtz energy, internal energy, and entropy in the canonical ensemble, leading to their sum-over-states analytical formulas at any arbitrary order. For the grand canonical ensemble, these sum-over-states formulas are systematically transformed to sum-over-orbitals reduced analytical formulas by the quantum-field-theoretical techniques of normal-ordered second quantization and Feynman diagrams extended to finite temperature. It is found that the perturbation corrections to energies entering the recursions have to be treated as a nondiagonal matrix, whose off-diagonal elements are generally nonzero within a subspace spanned by degenerate Slater determinants. They give rise to a unique set of linked diagrams—renormalization diagrams—whose resolvent lines are displaced upward, which are distinct from the well-known anomalous diagrams of which one or more resolvent lines are erased. A linked-diagram theorem is introduced that proves the size-consistency of the finite-temperature many-body perturbation theory at any order. General-order algorithms implementing the recursions establish the convergence of the perturbation series toward the finite-temperature full-configuration-interaction limit unless the series diverges. The normal-ordered Hamiltonian at finite temperature sheds light on the relationship between the finite-temperature Hartree–Fock and first-order many-body perturbation theories. [ABSTRACT FROM AUTHOR]

Progress in electrochemical technologies, such as automotive batteries, supercapacitors, and fuel cells, depends greatly on developing improved charged interfaces between electrodes and electrolytes. The rational development of such interfaces can benefit from the atomistic understanding of the materials involved by first-principles quantum mechanical simulations with Density Functional Theory (DFT). However, such simulations are typically performed on the electrode surface in the absence of its electrolyte environment and at constant charge. We have developed a new hybrid computational method combining DFT and the Poisson–Boltzmann equation (P–BE) capable of simulating experimental electrochemistry under potential control in the presence of a solvent and an electrolyte. The charged electrode is represented quantum-mechanically via linear-scaling DFT, which can model nanoscale systems with thousands of atoms and is neutralized by a counter electrolyte charge via the solution of a modified P–BE. Our approach works with the total free energy of the combined multiscale system in a grand canonical ensemble of electrons subject to a constant electrochemical potential. It is calibrated with respect to the reduction potential of common reference electrodes, such as the standard hydrogen electrode and the Li metal electrode, which is used as a reference electrode in Li-ion batteries. Our new method can be used to predict electrochemical properties under constant potential, and we demonstrate this in exemplar simulations of the differential capacitance of few-layer graphene electrodes and the charging of a graphene electrode coupled to a Li metal electrode at different voltages. [ABSTRACT FROM AUTHOR]

Extracting 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. [ABSTRACT FROM AUTHOR]

On the capital market, price movements of stock corporations can be observed independent of overall market developments as a result of company-specific news, which suggests the occurrence of a sudden risk event. In recent years, numerous concepts from statistical physics have been transferred to econometrics to model these effects and other issues, e.g., in socioeconomics. Like other studies, we extend the approaches based on the "buy" and "sell" positions of agents (investors' stance) with a third "hold" position. We develop the corresponding theory within the framework of the microcanonical and canonical ensembles for an ideal agent system and apply it to a capital market example. We thereby design a procedure to estimate the required model parameters from time series on the capital market. The aim is the appropriate modeling and the one-step-ahead assessment of the effect of a sudden risk event. From a one-step-ahead performance comparison with selected benchmark approaches, we infer that the model is well-specified and the model parameters are well determined. [ABSTRACT FROM AUTHOR]

Two ensembles are frequently used to model random graphs subject to constraints: the microcanonical ensemble (= hard constraint) and the canonical ensemble (= soft constraint). It is said that breaking of ensemble equivalence (BEE) occurs when the specific relative entropy of the two ensembles does not vanish as the size of the graph tends to infinity. Various examples have been analysed in the literature. It was found that BEE is the rule rather than the exception for two classes of constraints: sparse random graphs when the number of constraints is of the order of the number of vertices, and dense random graphs when there are two or more constraints that are frustrated. We establish BEE for a third class: dense random graphs with a single constraint on the density of a given simple graph. We show that BEE occurs in a certain range of choices for the density and the number of edges of the simple graph, which we refer to as the BEE-phase. We also show that, in part of the BEE-phase, there is a gap between the scaling limits of the averages of the maximal eigenvalue of the adjacency matrix of the random graph under the two ensembles, a property that is referred to as the spectral signature of BEE. We further show that in the replica symmetric region of the BEE-phase, BEE is due to the coexistence of two densities in the canonical ensemble. [ABSTRACT FROM AUTHOR]

This work is devoted to study the behavior of massless particles within the context of curved spacetime. In essence, we investigate the consequences of the scale factor C (η) of the Friedmann–Robertson–Walker metric in the Einstein–aether formalism to study photon-like particles. To do so, we consider the system within the canonical ensemble formalism in order to derive the following thermodynamic state quantities: spectral radiance, Helmholtz free energy, pressure, entropy, mean energy and the heat capacity. Moreover, the correction to the Stefan–Boltzmann law and the equation of states are also provided. Particularly, we separate our study within three distinct cases, i.e. s = 0 , p = 0 ; s = 1 , p = 1 ; s = 2 , p = 1. In the first one, the results are derived numerically. Nevertheless, for the rest of the cases, all the calculations are accomplished analytically showing explicitly the dependence of the scale factor C (η) and the Riemann zeta function ξ (s). Furthermore, our analyses are accomplished in general taking into account three different regimes of temperature of the universe, i.e. the inflationary era (T = 1 0 1 3 GeV), the electroweak epoch (T = 1 0 3 GeV) and the cosmic microwave background (T = 1 0 − 1 3 GeV). [ABSTRACT FROM AUTHOR]

The atomic scale solid–liquid interfacial process dominates the macro‐dissolution of silica, yet its direct experimental observation is challenging. Here we employed the reactive molecular dynamics to model this process in alkaline condition. An elevated temperature strategy under canonical ensemble (NVT) was applied to accelerate the process for sufficient sampling at temporal domain. A stepwise transformation from fully linked SiO4 network (Q3 and Q4) to reduced linkage and eventually aqueous species (Q0) was revealed. The simulated dissolution rate agrees well with the value predicted by empirical model. By tracing hydrogen atoms, we found that the dehydroxylated silica surfaces in alkaline electrolyte solutions underwent a transition from Si–O–Si bond cleavage‐dominated (mainly reacted with OH−) to hydroxylation‐dominated surface reactions. The free‐energy reconstruction of well‐tempered metadynamics reveals that 1500 K accelerates dissolution without altering the reaction pathways. These findings offer novel insights into the evolution of atomic‐scale surface configurations during the dissolution kinetics of silica. [ABSTRACT FROM AUTHOR]