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Combining classical density functional theory (cDFT) with quantum mechanics (QM) methods offers a computationally efficient alternative to traditional QM/molecular mechanics (MM) approaches for modeling mixed quantum-classical systems at finite temperatures. However, both QM/MM and QM/cDFT rely on approximations that introduce ambiguities. This series of papers aims to establish a rigorous theoretical framework to guide the development of a QM/cDFT formulation that clarifies the approximations involved. In this first paper, we establish a comprehensive density functional theory (DFT) framework for mixed quantum (QM)-classical (MM) systems within the canonical ensemble. We derive the equilibrium density matrix for a system comprising N qm QM and N mm MM particles and, based on this derivation, propose a variational formulation of the Helmholtz free energy in terms of the full QM-MM density matrix. By leveraging permutational symmetry and thanks to constrained-search methods, we reformulate the variational problem using functionals of objects having reduced dimensionalities. Three different formulations are presented, each using quantities with progressively lower dimensionalities. This ultimately enables computation of the Helmholtz free energy using only the QM and MM one-body densities, generalizing both cDFT and electronic DFT (eDFT) for QM-MM systems. In a subsequent paper, this formalism will be extended to the grand canonical ensemble, allowing classical particle number fluctuations. This extension, in which cDFT is most naturally formulated, provides a framework for studying solvation problems.

We establish the connection between the Gibbons-Hawking Euclidean path integral approach applied to the canonical ensemble of a Reissner-Nordstr\"om black hole and the thermodynamic theory of black holes of Davies. We build the ensemble, characterized by a reservoir at infinity at temperature $T$ and electric charge $Q$, in $d$ dimensions. The Euclidean path integral yields the action and partition function. In zero loop, we uncover two solutions, one with horizon radius $r_{+1}$ the least massive, the other with $r_{+2}$. We find a saddle point separating the solutions at $T_s$ and $Q_s$ with radius $r_{+s}$. For $T > T_s$ there is only hot flat space with charge at infinity. We derive the thermodynamics. The heat capacity gives that $T_s$ and $Q_s$ separate stable, $r_{+1}$, from unstable, $r_{+2}$ , solutions, the phase transition being second order. The free energy of the stable solution is positive, so if the system is a black hole it makes a first order transition to hot space. An interpretation of the results as energy wavelengths is attempted. For $d = 4$, the thermodynamics from the path integral applied to the canonical ensemble is precisely the Davies thermodynamics theory of black holes, with $T_s$ being the Davies point. We sketch the case $d = 5$., Comment: 15 pages, 2 figures

This study investigates quantum-corrected black hole solutions derived from f(R) gravity and explores their thermodynamic properties using the canonical ensemble framework. By incorporating higher-order f(R) corrections into classical black hole metrics, we construct regular black hole solutions that eliminate classical singularities. Advanced canonical ensemble techniques, including path integral formulations and stability analyses, are employed to examine the thermodynamic stability, phase transitions, and critical phenomena of these f(R)-corrected black holes. The results indicate that f(R) corrections significantly alter the thermodynamic landscape, introducing novel phase structures and stability conditions. Additionally, numerical simulations are conducted to visualize the behavior of thermodynamic quantities under varying f(R) correction parameters. This work provides deeper insights into the interplay between modified gravity effects and black hole thermodynamics, contributing to the broader understanding of gravitational phenomena in strong gravitational fields.

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 micro-cavity 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., Comment: 24 pages, 2 figures, to be published in the journal Entropy, invited paper for a special issue on "Matter-Aggregating Systems at a Classical vs. Quantum Interface"

We propose a variational modelling method with differentiable temperature for canonical ensembles. Using a deep generative model, the free energy is estimated and minimized simultaneously in a continuous temperature range. At optimal, this generative model is a Boltzmann distribution with temperature dependence. The training process requires no dataset, and works with arbitrary explicit density generative models. We applied our method to study the phase transitions (PT) in the Ising and XY models, and showed that the direct-sampling simulation of our model is as accurate as the Markov Chain Monte Carlo (MCMC) simulation, but more efficient. Moreover, our method can give thermodynamic quantities as differentiable functions of temperature akin to an analytical solution. The free energy aligns closely with the exact one to the second-order derivative, so this inclusion of temperature dependence enables the otherwise biased variational model to capture the subtle thermal effects at the PTs. These findings shed light on the direct simulation of physical systems using deep generative models, Comment: Main text: 4.5 pages, 2 figures. Supplement: 9 pages

The Legendre transformation is a crucial tool in theoretical physics, known for its symmetry, especially when applied to multivariate functions. In statistical mechanics, ensembles represent the central focus. Leveraging the dimensionless aspect of Legendre transformation, this paper explores the transformation process from the entropy characteristic function of microcanonical ensembles to the analogous definition of partition function transformation. Additionally, it derives characteristic functions, partition functions, and establishes their interrelations, along with deriving corresponding thermodynamic formulas for various ensembles. This streamlined approach sheds light on the fundamental principles of statistical mechanics and underscores the symmetry inherent in Legendre transformation.

We consider Bose-Einstein condensation of noninteracting homogeneous three-dimensional gas in canonical ensemble when both particle number $N$ and total momentum $\mathbf{P}$ of all particles are fixed. Using the saddle point method, we derive the large-$N$ analytical approximations for partition function, free energy, and statistical distributions of occupation numbers of different single-particle energy levels. At temperatures below the critical point of phase transition, we predict, in some ranges of $\mathbf{P}$, fragmentation of the condensate, when more than one single-particle level is macroscopically occupied. The occupation number distributions have approximately Gaussian shapes for the levels hosting the condensate, and exponential shapes for other, noncondensate levels. Our analysis demonstrates breaking of Galilean invariance of moving finite-temperature many-particle system in the presence of Bose-Einstein condensation and extends the theory of moving and rotating quantum systems to the finite-temperature large-$N$ limit., Comment: 15 pages, 8 figures; Section IV.B and Supplemental Material were added

In a space with positive cosmological constant $\Lambda$, we consider a black hole surrounded by a heat reservoir at radius $R$ and temperature $T$, i.e., we analyze the Schwarzschild-de Sitter black hole in a cavity. We use the Euclidean path integral approach to quantum gravity to study its canonical ensemble and thermodynamics. We give the action, energy, entropy, temperature, and heat capacity. $T$, $\Lambda$, the black hole radius $r_+$, and the cosmological horizon radius $r_{\rm c}$, are gauged in $R$ units to $RT$, $\Lambda R^2$, $\frac{r_+}{R}$, and $\frac{r_{\rm c}}{R}$. The whole extension of $\Lambda R^2$, $0\leq\Lambda R^2\leq 3$, is divided into three ranges. The first, $0\leq\Lambda R^2<1$, includes York's Schwarzschild black holes. The second range, $\Lambda R^2=1$, opens up a folder of Nariai universes. The third range, $1<\Lambda R^2\leq 3$, is unusual. One feature here is that it interchanges the cosmological horizon with the black hole horizon. The end point, $\Lambda R^2=3$, only existing for infinite $RT$, is a cavity filled with de Sitter space, except for a singularity, with the cosmological horizon coinciding with the reservoir. For the three ranges, for low temperatures, there are no black holes and no Nariai universes, the space is hot de Sitter. The value of $RT$ that divides the nonexistence from existence of black holes or Nariai universes, depends on $\Lambda R^2$. For each $\Lambda R^2\neq1$, for high temperatures, there is one small and thermodynamically unstable black hole, and one large and stable. For $\Lambda R^2=1$, for high temperatures, there is the unstable black hole, and the neutrally stable Nariai universe. Phase transitions can be analyzed. The transitions are between the black hole and hot de Sitter and between Nariai and hot de Sitter. The Buchdahl radius, the radius for collapse, plays an interesting role in the analysis., Comment: 34 pages, 8 figures

Legendre transform between thermodynamic quantities such as the Helmholtz free energy and entropy plays a key role in the formulation of the canonical ensemble. The transform helps to exchange the independent variable from the system's internal energy to its conjugate variable -- the inverse temperature of the reservoir. In this article, we study the Legendre transform between the free energy and Shannon entropy, denoted as the microscopic Legendre transform ($\mathscr{L}_{\!\mathscr{M}}^{}$), where the conjugate variables are the microstate probabilities and the energies (scaled by the inverse temperature). We formulate the exact differential property of the Shannon entropy and utilize it to derive central relations within canonical ensemble. Thermodynamics of a system in contact with a heat reservoir is discussed from this perspective. Other approaches, in particular, Jaynes' maximum entropy principle is compared with the present approach., Comment: 14 pages, RevTex4-2, presentation and discussion modified, main results unaffected

We calculate the corrections due to noncommutativity of space on the Hamiltonian and then partition function of the canonical ensemble. We study some basic features of statistical mechanics and thermodynamics including equipartition and virial theorem and energy fluctuations: correspondence with microcanonical ensemble, in the framework of non-commutative canonical ensemble. The corrections imposed by noncommutativity of space are derived and the results are discussed., Comment: 6 pages, no figures

The thermodynamics of a system of interacting bosonic particles and antiparticles in the presence of the Bose-Einstein condensate is studied in the framework of the Skyrme-like mean-field model. It is assumed that the total charge density (isospin density) is conserved at all temperatures. Two cases are explicitly considered: zero and nonzero isospin charge of the system. A comparative analysis is carried out using Canonical Ensemble and Grand Canonical Ensemble. It is shown that the Grand Canonical Ensemble is not suitable for describing bosonic systems of particles and antiparticles in the presence of condensate., Comment: 9 pages, 2 figures. Several typos have been corrected. The introduction and the list of used literature have been expanded

Adsorption of 4He on atomically flat substrates such as graphene provides a route towards the engineering of low dimensional quantum phases including superfluids and strongly interacting insulators. In this study, we explore the effects of graphene corrugation on the helium adsorption process via quantum Monte Carlo simulations in the grand canonical ensemble. We utilize an empirical adsorption potential based on the superposition of individual helium-carbon interactions and systematically control corrugation, from a smooth membrane to the fully rough potential, via the implementation of a cutoff in reciprocal space. The results highlight the importance of using a fully corrugated potential to understand the plethora of commensurate and incommensurate phases in the first adsorbed layer. Surprisingly, some residual effects of corrugation are still present before and during the promotion and onset of a second adsorbed layer, where a smooth adsorption can lead to enhanced particle fluctuations in a helium-interaction dominated regime., Comment: 12 pages, 8 figures (added additional system sizes). Submitted to the JLTP Special Issue: Quantum Fluids and Solids (QFS2023). For associated data and code repository see: https://github.com/DelMaestroGroup/papers-code-GrapheneCorrugation

The grand canonical ensemble of a $d$-dimensional Reissner-Nordstr\"om black hole space in a cavity is analyzed. The realization of this ensemble is made through the Euclidean path integral approach by giving the Euclidean action for the black hole with the correct topology, and boundary conditions corresponding to a cavity, where the fixed quantities are the temperature and the electric potential. One performs a zero loop approximation to find and analyze the stationary points of the reduced action. This yields two solutions for the electrically charged black hole, $r_{+1}$, which is the smaller and unstable, and $r_{+2}$, which is the larger and stable. One also analyzes the most probable configurations, which are either a stable charged black hole or hot flat space, mimicked by a nongravitating charged shell. Making the correspondence between the action and the grand potential, one can get the black hole thermodynamic quantities, such as the entropy, the mean charge, the mean energy, and the thermodynamic pressure, as well as the Smarr formula, shown to be valid only for the unstable black hole. We find that thermodynamic stability is related to the positivity of the heat capacity at constant electric potential and area of the cavity. We also comment on the most favorable thermodynamic phases and phase transitions. We then choose $d = 5$, which is singled out naturally from the other higher dimensions as it provides an exact solution for the problem, and apply all the results previously found. The case $d = 4$ is mentioned. We compare thermodynamic radii with the photonic orbit radius and the Buchdahl-Andr\'easson-Wright bound radius in $d$-dimensional Reissner-Nordstr\"om spacetimes and find they are unconnected, showing that the connections displayed in the Schwarzschild case are not generic, rather they are very restricted holding only in the pure gravitational situation., Comment: 28 pages, 5 figures

We studied the escort averages in microcanonical and canonical ensembles in the Tsallis statistics of entropic parameter $q>1$. The quantity $(q-1)$ is the measure of the deviation from the Boltzmann-Gibbs statistics. We derived the relation between the escort average in the microcanonical ensemble and the escort average in the canonical ensemble. Conditions arise by requiring that the integrals appeared in the canonical ensemble do not diverge. A condition is the relation between the heat capacity $C_V^{\mathrm{CE}}$ at constant volume in the canonical ensemble and the entropic parameter $q$: $0 < (q-1) C_V^{\mathrm{CE}} < 1$. This condition gives the known condition when $C_V^{\mathrm{CE}}$ equals the number of ingredients $N$. With the derived relation, we calculated the energy, the energy fluctuation, and the difference between the canonical ensemble and the microcanonical ensemble in the expectation value of the square of Hamiltonian. The difference between the microcanonical ensemble and the canonical ensemble in energy is small because of the condition. The heat capacity $C_V^{\mathrm{CE}}$ and the quantity $(q-1)$ are related to the energy fluctuation and the difference. It was shown that the magnitude of the relative difference $|(S^{\mathrm{CE}}_{\mathrm{R}q}-S^{\mathrm{ME}}_{\mathrm{R}q})/S^{\mathrm{ME}}_{\mathrm{R}q}|$ is small when the number of free particles is large, where $S^{\mathrm{ME}}_{\mathrm{R}q}$ is the R\'enyi entropy in the microcanonical ensemble and $S^{\mathrm{CE}}_{\mathrm{R}q}$ is the R\'enyi entropy in the canonical ensemble. The similar result was also obtained for the Tsallis entropy., Comment: 12 pages

We study the generalized free energy of the dyonic AdS black hole in an ensemble with varying electric charge $q_E$ and fixed magnetic charge $q_M$. When we adjust the temperature $T$ and the electric potential $\Phi_E$ of the ensemble, the Ricci scalar curvature $R$ and electromagnetic potential $A_u$ 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 pair) to the rate of state transition. These results are consistent with the qualitative analysis of the free energy landscape.

The Weyl-Wigner representation of quantum mechanics allows one to map the density operator in a function in phase space - the Wigner function - which acts like a probability distribution. In the context of statistical mechanics, this mapping makes the transition from the classical to the quantum regimes very clear, because the thermal Wigner function tends to the Boltzmann distribution in the high temperature limit. We approximate this quantum phase space representation of the canonical density operator for general temperatures in terms of classical trajectories, which are obtained through a Wick rotation of the semiclassical approximation for the Weyl propagator. A numerical scheme which allows us to apply the approximation for a broad class of systems is also developed. The approximation is assessed by testing it against systems with one and two degrees of freedom, which shows that, for a considerable range of parameters, the thermodynamic averages are well reproduced.

We calculate the susceptibility of a canonical ensemble of quantum oscillators to the singular random metric. If the covariance of the metric is $\vert {\bf x}-{\bf x}^{\prime}\vert^{-4\alpha}$ $0< \alpha<\frac{1}{2}$ then the expansion of the partition function in powers of the temperature involves non-integer indices., Comment: 10 pages

Many experimentally-accessible, finite-sized interacting quantum systems are most appropriately described by the canonical ensemble of statistical mechanics. Conventional numerical simulation methods either approximate them as being coupled to a particle bath, or use projective algorithms which may suffer from non-optimal scaling with system size or large algorithmic prefactors. In this paper, we introduce a highly stable, recursive Auxiliary Field Quantum Monte Carlo approach that can directly simulate systems in the canonical ensemble. We apply the method to the fermion Hubbard model in one and two spatial dimensions in a regime known to exhibit a significant "sign" problem and find improved performance over existing approaches including rapid convergence to ground state expectation values. The effects of excitations above the ground state are quantified using an estimator-agnostic approach including studying the temperature dependence of the purity and overlap fidelity of the canonical and grand canonical density matrices. As an important application, we show that thermometry approaches often exploited in ultra-cold atoms that employ an analysis of the velocity distribution in the grand canonical ensemble may be subject to errors leading to an under-estimation of extracted temperatures with respect to the Fermi temperature., Comment: 14 pages, 7 figures, and Supplemental Information with key derivations. For associated data and code repository see: https://github.com/DelMaestroGroup/papers-code-CanEnsAFQMC

We obtain the conditions constituting a thermal equilibrium between two energy levels: (i) the total energy is equal in both levels; (ii) the temperature is equal for all particles. Exploiting these conditions, we derive a differential equation of thermal equilibrium that holds for all particles. Integration delivers the Boltzmann distribution, suggesting that it is the general distribution of thermal equilibrium. With excited-state and ground-state population numbers n2 and n1, respectively, Pauli's exclusion principle is formalized as n1=1-n2. Exploiting Einstein's rate-equation approach to Planck's law of blackbody radiation, we derive the equivalent quantum principle for bosons, n1=1+n2. Utilizing either quantum principle, as a boundary condition when integrating the differential equation of thermal equilibrium or by entering it into the Boltzmann distribution, delivers the Fermi-Dirac or Bose-Einstein distribution. The fermionic or bosonic ratios n2/n1 follow the Boltzmann distribution. These results suggest that the Fermi-Dirac and Bose-Einstein distributions are special cases of the Boltzmann distribution, ruled by Pauli's exclusion principle or the quantum principle for bosons. The grand canonical ensemble equals the Boltzmann distribution. The relating principle and the inhibition/enhancement factors for fermions/bosons confirm the quantum principles. The quantum principle for bosons comprises a vacuum state. Consequently, a populated vacuum state should exist for all bosons. Vacuum states are not detectable, therefore, they are dark states. The vacuum states of all bosons contribute to dark energy, while only the matter-boson vacuum states contribute to dark matter, supporting the fact that the amount of dark energy exceeds that of dark matter in the universe., Comment: 17 pages, 2 figures

The increase in yields of light nuclei with charged particle multiplicity, as reported by the ALICE collaboration at CERN in p-p and p-Pb collisions at the LHC energy is investigated in the thermal hadron resonance gas model. The model is extended to account for exact baryon number conservation. The focus is on the production of protons, deuterons, $^3$He, and $^3_\Lambda$H. A very good description of proton and deuteron yields is obtained as a function of charged particle multiplicity in the mid-rapidity region using the same fixed temperature as in central Pb-Pb collisions. The yields of light nuclei $^3$He and $^3_\Lambda$H though qualitatively explained as a function of multiplicity, are lower than the model expectation. One of the possible reasons could be that for $^{3}$He and $^{3}_{\Lambda}$H, the chemical equilibrium is not yet reached at small multiplicities., Comment: 12 pages, 3 captioned figures

We establish one-body reduced density-matrix functional theory for the canonical ensemble in a finite basis set at an elevated temperature. Including temperature guarantees differentiability of the universal functional by occupying all states and additionally not fully occupying the states in a fermionic system. We use convexity of the universal functional and invertibility of the potential-to-1RDM map to show that the subgradient contains only one element which is equivalent to differentiability. This allows us to show that all 1RDMs with a purely fractional occupation number spectrum ($0 < n_i < 1 \; \forall_i$) are uniquely $v$-representable up to a constant., Comment: 7 pages, 1 figure

We introduce 1-RDMFT in the canonical ensemble and then proceed to approximate the interacting ensemble by a non-interacting ensemble that maximizes the entropy, independently of temperature. Bosonic and Fermionic Sinkhorn algorithms are derived and used to invert the relationship between the Natural Orbital Occupation Numbers (NOONs) and the effective orbital energies of the non-interacting ensemble. Both the Bosonic and Fermionic Sinkhorn algorithms are shown to perform well in reproducing the NOONs of simulated distributions and the ground-state NOONs of H$_2$O and H$_2$. In the case of H$_2$ we also study the resulting non-interacting entropy and non-interacting approximation to the interaction energy within several wavefunction subspaces as the bond length varies. This provides several new starting points for approximations of the interaction energy, also at zero-temperature. Connections to entropically-regularized Multi-Marginal Optimal Transport (MMOT) are highlighted that may prove interesting for future research., Comment: 14 pages, 6 figures. Supplementary material: 3 pages, 4 figures

We study fluctuations in the canonical ensemble, where the net baryon number is exactly conserved. The focus is on cumulants and factorial cumulants linked to the baryon and antibaryon multiplicities and their sum or difference in full phase-space as well as in subsystems. In particular, we connect the fluctuations of the net baryon number in a subsystem, relevant for fluctuation studies in nucleus-nucleus collisions, with fluctuations of the baryon and antibaryon numbers of the total system. We derive analytic expressions for factorial cumulants of arbitrary order. Compact results are obtained in terms of cumulants of the baryon number of the total system. Moreover, we derive the asymptotic forms of the factorial cumulants of baryon and antibaryon multiplicities in the high- and low-temperature limits and discuss the results in the context of heavy-ion collision experiments.

The infinite set of coupled integral nonlinear equations for correlation functions in the case of classical canonical ensemble is considered. Some kind of graph expansions of correlation functions in the density parameter are constructed. Existing of unique solutions for small value of density and high temperature is discussed.

Superstatistics is a generalization of equilibrium statistical mechanics that describes systems in nonequilibrium steady states. Among the possible superstatistical distributions, the $q$-canonical ensemble (also known as Tsallis' statistics, and in plasma physics as Kappa distributions) is probably the most widely used, however the current explanations of its origin are not completely consistent. In this work it is shown that, under a Bayesian interpretation of superstatistics, the origin of the $q$-canonical ensemble can be explained as the superstatistical distribution with maximum Shannon-Jaynes entropy under noninformative constraints. The $q$-canonical distributions are singled out by the mathematical structure of superstatistics itself, and thus no assumptions about the physics of the systems of interest, or regarding their complexity or range of interactions, are needed. These results support the thesis that the success of the $q$-canonical ensemble is information-theoretical in nature, and explainable in terms of the original maximum entropy principle by Jaynes.

We revisit the stability of black hole saddles for the Euclidean path integral describing the canonical partition function $Z(\beta)$ for gravity inside a spherical reflecting cavity. The boundary condition at the cavity wall couples the transverse-traceless (TT) and pure-trace modes that are traditionally used to describe fluctuations about Euclidean Schwarzschild black holes in infinite-volume asymptotically flat and asymototically AdS spacetimes. This coupling obstructs the familiar Gibbons-Hawking-Perry treatment of the conformal factor problem, as Wick rotation of the pure-trace modes would require that the TT modes be rotated as well. The coupling also leads to complex eigenvalues for the \L operator. We nevertheless find that the \L operator can be diagonalized in the space of coupled modes. This observation allows the eigenmodes to define a natural generalization of the pure-trace Wick-rotation recipe used in infinite volume, with the result that a mode with eigenvalue $\lambda$ is stable when ${\rm Re}\,\lambda > 0$. In any cavity, and with any cosmological constant $\Lambda \le 0$, we show this recipe to reproduce the expectation from black hole thermodynamics that large Euclidean black holes define stable saddles while the saddles defined by small Euclidean black holes are unstable., Comment: 35 pages, 6 figures

Classical density functional theory for finite temperatures is usually formulated in the grand-canonical ensemble where arbitrary variations of the local density are possible. However, in many cases the systems of interest are closed with respect to mass, e.g. canonical systems with fixed temperature and particle number. Although the tools of standard, grand-canonical density functional theory are often used in an ad hoc manner to study closed systems, their formulation directly in the canonical ensemble has so far not been known. In this work, the fundamental theorems underlying classical DFT are revisited and carefully compared in the two ensembles showing that there are only trivial formal differences. The practicality of DFT in the canonical ensemble is then illustrated by deriving the exact Helmholtz functional for several systems: the ideal gas, certain restricted geometries in arbitrary numbers of dimensions and finally a system of two hard-spheres in one dimension (hard rods) in a small cavity. Some remarkable similarities between the ensembles are apparent even for small systems with the latter showing strong echoes of the famous exact of result of Percus in the grand-canonical ensemble., Comment: Added references to some prior work

We report the first multi-differential measurements of strange hadrons of $K^{-}$, $\phi$ and $\Xi^{-}$ yields as well as the ratios of $\phi/K^-$ and $\phi/\Xi^-$ in Au+Au collisions at ${\sqrt{s_{\rm NN}} = \rm{3\,GeV}}$ with the STAR experiment fixed target configuration at RHIC. The $\phi$ mesons and $\Xi^{-}$ hyperons are measured through hadronic decay channels, $\phi\rightarrow K^+K^-$ and $\Xi^-\rightarrow \Lambda\pi^-$. Collision centrality and rapidity dependence of the transverse momentum spectra for these strange hadrons are presented. The $4\pi$ yields and ratios are compared to thermal model and hadronic transport model predictions. At this collision energy, thermal model with grand canonical ensemble (GCE) under-predicts the $\phi/K^-$ and $\phi/\Xi^-$ ratios while the result of canonical ensemble (CE) calculations reproduce $\phi/K^-$, with the correlation length $r_c \sim 2.7$\,fm, and $\phi/\Xi^-$, $r_c \sim 4.2$\,fm, for the 0-10\% central collisions. Hadronic transport models including high mass resonance decays could also describe the ratios. While thermal calculations with GCE work well for strangeness production in high energy collisions, the change to CE at $\rm{3\,GeV}$ implies a rather different medium property at high baryon density., Comment: 9 pages, 5 figures, 2 tables

For classical discrete systems under constant composition, we re-examine how linear-nonlinear boundary in canonical ensemble, connecting a set of potential energy surface and that of microscopic configuration in thermodynamic equilibrium, is characterized by underlying lattice, from tropical geometry. We here show that by applying suitable tropical limit and multiple coordinate transform to time evolution of discrete dynamical system, reflecting geometric aspect of the nonlinearity, we successfully construct tropical diagram caputuring the universal character of linear/nonlinear region on configuration space for f = 2 structural degree of freedoms (SDFs). The diagram indicates that the boundary near disordered state is mainly dominated by constraints to individual SDF, while quasi linear-nonlinear boundary apart from disordered state is dominated by local non-separability in SDFs., Comment: 7pages, 2 figures

We present a practical analysis of the fermion sign problem in fermionic path integral Monte Carlo (PIMC) simulations in the grand-canonical ensemble (GCE). As a representative model system, we consider electrons in a $2D$ harmonic trap. We find that the sign problem in the GCE is even more severe than in the canonical ensemble at the same conditions, which, in general, makes the latter the preferred option. Despite these difficulties, we show that fermionic PIMC simulations in the GCE are still feasible in many cases, which potentially gives access to important quantities like the compressiblity or the Matsubara Greens function. This has important implications for contemporary fields of research such as warm dense matter, ultracold atoms, and electrons in quantum dots.

Calculating thermodynamic potentials and observables efficiently and accurately is key for the application of statistical mechanics simulations to materials science. However, naive Monte Carlo approaches, on which such calculations are often dependent, struggle to scale to complex materials in many state-of-the-art disciplines such as the design of high entropy alloys or multicomponent catalysts. To address this issue, we adapt sampling tools built upon machine-learning based generative modeling to the materials space by transforming them into the semi-grand canonical ensemble. Furthermore, we show that the resulting models are transferable across wide-ranges of thermodynamic conditions and can be implemented with any internal energy model U, allowing integration into many existing materials workflows. We demonstrate the applicability of this approach to the simulation of benchmark systems (AgPd, CuAu) that exhibit diverse thermodynamic behavior in their phase diagrams. Finally, we discuss remaining challenges in model development and promising research directions for future improvements., Comment: 29 pages, 28 figures