Canonical Ensemble Research Articles

Partition function approach to non-Gaussian likelihoods: macrocanonical partitions and replicating Markov-chains

Monte-Carlo techniques are standard numerical tools for exploring non-Gaussian and multivariate likelihoods. Many variants of the original Metropolis-Hastings algorithm have been proposed to increase the sampling efficiency. Motivated by Ensemble Monte Carlo we allow the number of Markov chains to vary by exchanging particles with a reservoir, controlled by a parameter analogous to a chemical potential μ, which effectively establishes a random process that samples microstates from a macrocanonical instead of a canonical ensemble. In this paper, we develop the theory of macrocanonical sampling for statistical inference on the basis of Bayesian macrocanonical partition functions, thereby bringing to light the relations between information-theoretical quantities and thermodynamic properties. Furthermore, we propose an algorithm for macrocanonical sampling, 𝙰𝚟𝚊𝚕𝚊𝚗𝚌𝚑𝚎 𝚂𝚊𝚖𝚙𝚕𝚒𝚗𝚐, and apply it to various toy problems as well as the likelihood on the cosmological parameters Ωm and w on the basis of data from the supernova distance redshift relation.

Generalized hydrodynamics and approach to generalized Gibbs equilibrium for a classical harmonic chain

Abstract We study the evolution of a classical harmonic chain with nearest-neighbor interactions starting from domain wall initial conditions. The initial state is taken to be either a product of two Gibbs ensembles (GEs) with unequal temperatures on the two halves of the chain or a product of two generalized Gibbs ensembles (GGEs) with different parameters in the two halves. For this system, we construct the Wigner function and demonstrate that its evolution defines the generalized hydrodynamics (GHD) describing the evolution of the conserved quantities. We solve the GHD for both finite and infinite chains and compute the evolution of conserved densities and currents. For a finite chain with fixed boundaries, we show that these quantities relax as ∼ 1 / t to their respective steady-state values given by the final expected GE or GGE state, depending on the initial conditions. Exact expressions for the Lagrange multipliers of the final expected GGE state are obtained in terms of the steady-state densities. In the case of an infinite chain, we find that the conserved densities and currents at any finite time exhibit ballistic scaling while, at infinite time, any finite segment of the system can be described by a current-carrying non-equilibrium steady state (NESS). We compute the scaling functions analytically and show that the relaxation to the NESS occurs as ∼ 1 / t for the densities and as ∼ 1 / t 2 for the currents. We compare the analytical results from hydrodynamics with those from exact microscopic numerics and find excellent agreement.

Phase behavior analysis of methane confined in nanopores using molecular simulation

Interest in the phase behavior of hydrocarbons in shale reservoirs has grown in recent years. Petroleum fluid phase behavior has been observed to differ significantly between conventional reservoirs and shale reservoirs. Within shale reservoirs, notable surface-fluid interactions can lead to non-uniform molecule distribution and an alteration in fluid phase behavior, primarily caused by the existence of nano-scale porous materials. In this work, we study the phase behavior of methane in single cylindrical pore models. We apply the gauge Gibbs ensemble Monte Carlo (gauge-GEMC) simulation technique to investigate the phase behavior of methane in 4–10 nm single nanopores and calculate the saturation pressures at various temperatures using the grand canonical Monte Carlo (GCMC) simulation technique. A shift in the phase diagram has been found for methane in nanopores. As pore size decreases, the shift becomes more significant.

MEDOC: A fast, scalable and mathematically exact algorithm for the site-specific prediction of the protonation degree in large disordered proteins.

Intrinsically disordered regions are found in most eukaryotic proteins and are enriched in positively and negatively charged residues. While it is often convenient to assume these residues follow their model-compound pK a values, recent work has shown that local charge effects (charge regulation) can upshift or downshift sidechain pK a values with major consequences for molecular function. Despite this, charge regulation is rarely considered when investigating disordered regions. The number of potential charge microstates that can be populated through acid/base regulation of a given number of ionizable residues in a sequence, N , scales as ∼2 N . This exponential scaling makes the assessment of the full charge landscape of most proteins computationally intractable. To address this problem, we developed MEDOC (Multisite Extent of Deprotonation Originating from Context) to determine the degree of protonation of a protein based on the local sequence context of each ionizable residue. We show that we can drastically reduce the number of parameters necessary to determine the full, analytic, Boltzmann partition function of the charge landscape at both global and site-specific levels. Our algorithm applies the structure of the q -canonical ensemble, combined with novel strategies to rapidly obtain the minimal set of parameters, thereby circumventing the combinatorial explosion of the number of charge microstates even for proteins containing a large number of ionizable amino acids. We apply MEDOC to several sequences, including a global analysis of the distribution of pK a values across the entire DisProt database. Our results show differences in the distribution of predicted pK a values for different amino acids, in agreement with NMR-measured distributions in proteins.

Dephasing dynamics of a central spin interacting with a classical Ising spin bath

We study the effect of interaction between bath spins on central spin dynamics. For this end, we consider various types of interactions among the bath fluctuators. We give an analytical relation for the dephasing dynamics of a central spin homogeneously interacting with a bath of classically interacting spins in thermal equilibrium. The obtained exact relation for the central spin dynamics depends on the average of the number of bath spins N in the canonical ensemble. We find the relation of dephasing time with bath's temperature and strength of interaction between bath spins. We show that if the classically interacting bath is large enough, off-diagonal coherence will decay into Gaussian form. This Gaussian decay is valid for temperatures higher than critical temperature. Moreover, we analytically prove the equivalence of dephasing dynamics for central spin models with classical Ising spin bath and quantum flip-flop spin bath interactions. Therefore, the dephasing dynamics for central spin models with quantum spin bath interactions can be reproduced with much less expensive classical spin bath interaction model instead. Our results can be applied in a number of quantum systems used in quantum sensing such as color centers in solids or quantum dots manuscript.

Robust finite-temperature many-body scarring on a quantum computer

Mechanisms for suppressing thermalization in disorder-free many-body systems, such as Hilbert space fragmentation and quantum many-body scars, have recently attracted much interest in foundations of quantum statistical physics and potential quantum information processing applications. However, their sensitivity to realistic effects such as finite temperature remains largely unexplored. Here, we have utilized IBM's Kolkata quantum processor to demonstrate an unexpected robustness of quantum many-body scars at finite temperatures when the system is prepared in a thermal Gibbs ensemble. We identify such robustness in the PXP model, which describes quantum many-body scars in experimental systems of Rydberg atom arrays and ultracold atoms in tilted Bose-Hubbard optical lattices. By contrast, other theoretical models which host exact quantum many-body scars are found to lack such robustness and their scarring properties quickly decay with temperature. Our study sheds light on the important differences between scarred models in terms of their algebraic structures, which impacts their resilience to finite temperature. Published by the American Physical Society 2024

Li diffusion in oxygen-chlorine mixed anion borosilicate glasses using a machine-learning simulation.

Lithium-ion conducting borate glasses are suitable for solid-state batteries as an interfacial material between a crystalline electrolyte and an electrode, thanks to their superior formability. Chlorine has been known to improve the electron conductivity of borate glasses as a secondary anion. To examine the impact of chlorine on lithium dynamics, molecular dynamics (MD) simulations were performed with a machine-learning interatomic potential (MLIP). The accuracy of the MLIP in modeling chlorine-doped lithium borate (LBCl) and borosilicate (LBSCl) glasses was verified by comparing with available experimental data on density, neutron diffraction S(q), and glass transition temperatures (Tg). While the MLIP-MD simulations underestimated the density when an isobaric-isothermal (NPT) ensemble was used, the glass models relaxed using the NPT ensemble after a melt-quench simulation employing a canonical (NVT) ensemble possessed reasonable density. The LBCl and LBSCl glass models exhibited increased lithium ion diffusion, and the ions were found to travel longer distances with an increase in the chlorine content. According to the structural analyses, it was observed that chlorine ions primarily interacted with lithium ions rather than the network formers. Consequently, lithium ions that interacted with a higher amount of chlorine showed a moderate increase in mobility. In summary, the MLIP demonstrated reasonable accuracy in modeling chlorine-containing borate glasses and enabled the investigation of the effect of chlorine on electron conductivity. In contrast, the first sharp diffraction peaks in S(q) deviated from the experimental diffractions, suggesting that additional efforts are required to accurately model the middle-range structure of the glasses.

Thermo-Magnetic properties of non-relativistic particles under the effect of energy-dependent Hellmann potential

This manuscript presents an analysis of the thermal and magnetic properties of particles moving in two dimensions under the influence of an external magnetic field and Aharanov-Bohm flux, considering an energy-dependent Hellmann potential. The energy eigenvalue is obtained using the asymptotic iteration method. Next, we model the system as a canonical ensemble and derive the partition function to acquire analytical expressions for the Helmholtz free energy, entropy, mean energy, and specific heat. We observe that an increase in the external magnetic field and AB flux generally increases the energy eigenvalues, with a stronger effect at lower flux values, while energy eigenvalues decrease with an increase in the screening parameter. Additionally, Helmholtz free energy and entropy functions are sensitive to temperature, particularly in the low-temperature regime, while specific heat exhibits a Schottky anomaly. We also compute the magnetisation and susceptibility functions, finding that both magnetisation and susceptibility increase with the external magnetic field strength and the AB flux.

Pseudogap Effects in the Strongly Correlated Regime of the Two-Dimensional Fermi Gas.

The two-species Fermi gas with attractive short-range interactions in two spatial dimensions provides a paradigmatic system for the understanding of strongly correlated Fermi superfluids in two dimensions. It is known to exhibit a BEC to BCS crossover as a function of ln(k_{F}a), where a is the scattering length, and to undergo a Berezinskii-Kosterlitz-Thouless superfluid transition below a critical temperature T_{c}. However, the extent of a pseudogap regime in the strongly correlated regime of ln(k_{F}a)∼1, in which pairing correlations persist above T_{c}, remains largely unexplored with controlled theoretical methods. Here, we use finite-temperature auxiliary-field quantum MonteCarlo methods on discrete lattices in the canonical ensemble formalism to calculate thermodynamical observables in the strongly correlated regime. We extrapolate to continuous time and the continuum limit to eliminate systematic errors and present results for particle numbers ranging from N=42 to N=162. We estimate T_{c} by a finite-size scaling analysis, and observe clear pseudogap signatures above T_{c} and below a temperature T^{*} in both the spin susceptibility and free-energy gap. We also present results for the contact, a fundamental thermodynamic property of quantum many-body systems with short-range interactions.

Emergence of the Gibbs ensemble as a steady state in Lindbladian dynamics

Emergence of the Gibbs ensemble as a steady state in Lindbladian dynamics

Semi-empirical supported, Ab initio derived thermodynamic properties for ClO2 and its sub and extended species, applied in water treatment cycles

This paper describes a group of sixty (60) sub and extended chlorine oxide species with the general formulae of ClxOy (with x ≤ 2, y ≤ 8). Their role in water treatment cycles, behaving as key reactive species, is represented by a complex sequence of chemical inter-dependencies, exposed as a cohesive set of chemical reactions to demonstrate their cyclic role in aqueous media. An empirical/semi-empirical computational approach, supported by Ab Initio simulations, in accordance with open-shell character, has been followed to determine their optimum molecular geometries, to obtain their thermochemical properties. Besides a single molecular analysis, Grand Canonical Ensemble simulations, supported by a revised library of force field parameters, constituted a core component of the computational approach and proved to be invaluable in confirming thermochemical properties. This approach also offered finite estimates of optimum model sizes, a benefit with wider modelling application. Extended molecular species of ClO2 display a complex sequence of bonding character, with a variable charge dissipation (reported as partial charges), which complicates selection of basis sets in optimizing molecular geometries, during Ab Initio analyses.Optimum molecular geometries were obtained using Gaussian and MOPAC, which in turn resulted in reliable Heats of Formation. These correlated well with energies extracted from the open literature. Thermodynamic Analysis of the reaction of selected chlorine oxides with water using FactSage, predicted the production of known and two previously undetected chlorine species, [ClO4]- and [ClOH2]+.

Temperature and pressure effects on the decomposition mechanisms of 2,6-diamino-3,5-dinitropyrazine-1-oxide crystal: ab initio molecular dynamics study.

The decomposition process of 2,6-diamino-3,5-dinitropyrazine-1-oxide (LLM-105) crystal at high temperatures (2500 and 3390K) and detonation pressure of 33.4 GPa coupled with temperatures were studied by ab initio molecular dynamics simulations. The results show that the initial decomposition mechanism of LLM-105 is the same under different conditions. The product analysis indicates that high temperature is conducive to the formation of N2 and CO2, but inhibited the formation of H2O. It is found that the formation mechanism of H2O is the same under different conditions, which involves the reaction between OH radical and H radical. Although the detailed processes of the formation of N2 are different, they all involve the reaction between nitrogen-containing fragments, and its core is the formation of intermediates with R1-NN-R2 structure. The core of the formation of CO2 under different conditions is to form the intermediate R1-CO-R2 with carbonyl structure, and then generate the fragment with -OCO- structure, and finally generate CO2. This research may provide new insights into the initiation and subsequent decomposition mechanisms of energetic materials under extreme conditions. The LLM-105 supercell was constructed using the Materials Studio 7.0 package. AIMD simulations were performed in the CASTEP package. AIMD simulations adopted NVT and NPT ensemble, and the temperature was controlled by Nosé thermostat, while the pressure was controlled by Andersen barostat. Besides, DFT calculations were carried out at the B3LYP/6-311 + G(d,p) level using the Gaussian 09 package.

Statistical Thermodynamic Description of Self-Assembly of Large Inclusions in Biological Membranes.

Recent studies revealed anomalous underscreening in concentrated electrolytes, and we suggest that the underscreened electrostatic forces between membrane proteins play a significant role in the process of self-assembly. In this work, we assumed that the underscreened electrostatic forces compete with the thermodynamic Casimir forces induced by concentration fluctuations in the lipid bilayer, and developed a simplified model for a binary mixture of oppositely charged membrane proteins with different preference to liquid-ordered and liquid-disordered domains in the membrane. In the model, like macromolecules interact with short-range Casimir attraction and long-range electrostatic repulsion, and the cross-interaction is of the opposite sign. We determine energetically favored patterns in a system in equilibrium with a bulk reservoir of the macromolecules. Different patterns consisting of clusters and stripes of the two components and of vacancies are energetically favorable for different values of the chemical potentials. Effects of thermal flutuations at low temperature are studied using Monte Carlo simulations in grand canonical and canonical ensembles. For fixed numbers of the macromolecules, a single two-component cluster with a regular pattern coexists with dispersed small one-component clusters, and the number of small clusters depends on the ratio of the numbers of the molecules of the two components. Our results show that the pattern formation is controlled by the shape of the interactions, the density of the proteins, and the proportion of the components.

Ab initio molecular dynamics investigation on hydrogen diffusion behavior in liquid aluminum alloy melts

In this work, ab initio molecular dynamics (AIMD) simulations under both NVT and NPT ensembles were performed to study the influence of alloying elements of Mg, Fe, Si, Ti, Cu, Zn, F, and Cl on the H diffusion behavior in liquid aluminum melt. It is found that the diffusion coefficient of H simulated by AIMD under the NVT model is highly consistent with the reported experimental results. Specifically, the addition of Cu, Cl, Si, and Zn is inclined to increase the diffusion coefficient of H in the melt, whereas the opposite result is observed with the addition of F, Fe, and Ti. Moreover, it is proved that the H diffusion coefficients are positively correlated with the H-Al bond lengths and the coordination number of H in a constant volume system. The obtained results deepen the understanding of the diffusion of H atoms in Al melts and contribute to the optimization of existing melt purification technologies.

Discharging of Ramsdellite MnO2 Cathode in a Lithium-Ion Battery.

We report an application of our unbiased Monte Carlo approach to investigate thermodynamic and electrochemical properties of lithiated manganese oxide in the ramsdellite phase (R-MnO2) to uncover the mechanism of lithium intercalation and understand charging/discharging of R-MnO2 as a cathode material in lithium-ion batteries. The lithium intercalation reaction was computationally explored by modeling thermodynamically significant distributions of lithium and reduced manganese in the R-MnO2 framework for a realistic range of lithium molar fractions 0 < x < 1 in Li x MnO2. We employed interatomic potentials and analyzed the thermodynamics of the resultant grand canonical ensemble. We found ordered or semiordered phases at x = 0.5 and 1.0 in Li x MnO2, verified by configurational entropy changes and simulated X-ray diffraction patterns of partially lithiated R-MnO2. The radial distribution functions show the preference of lithium for homogeneous distributions across the one-dimensional 2 × 1 ramsdellite channels accompanied by alternating reduced/oxidized manganese ions. The occupation of the interstitial sites in the channels is correlated with the calculated voltage profile, showing a sharp voltage drop at x = 0.5, which is explained by the energy penalty of shifting lithium ions from stable tetrahedral to unstable octahedral sites. To facilitate this work, our in-house software, Knowledge Led Master Code (KLMC) was extended to support massive parallelism on high-performance computers.

Plasma-Plasma Third Order Phase Transition from Type IIB Supergravity.

We show that the planar, charged black hole in asymptotically anti-de Sitter spacetime, dual to the strongly coupled quark-gluon plasma thermal state of large N, SU(N), N=4 super Yang-Mills at finite chemical potential undergoes a third-order phase transition in the grand canonical ensemble to a hairy black hole of type IIB supergravity. The hairy phase is another strongly coupled fluid with a conformal equation of state and can be interpreted as another kind of quark-gluon plasma. This new quark-gluon plasma has less entropy and, therefore, seems to characterize some form of smooth hadronization. The locus of the transition in terms of the critical values of the "baryon" chemical potential μ_{c} and the temperature T_{c} is μ_{c}=2πT_{c}.

Phase Diagrams of a Relativistic Self-Interacting Boson System

Within the Canonical Ensemble, we investigate a system of interacting relativistic bosons at finite temperatures and finite isospin densities in a mean-field approach. The mean field contains both attractive and repulsive terms. Temperature and isospin density dependences of thermodynamic quantities are obtained. It is shown that, in the case of attraction between particles in a bosonic system, a liquid-gas phase transition develops against the background of the Bose–Einstein condensate. The corresponding phase diagrams are given. We explain the reasons for why the presence of a Bose condensate significantly increases the critical temperature of the liquid-gas phase transition compared to that obtained for the same system within the framework of Boltzmann statistics. Our results may have implications for the interpretation of experimental data, in particular, how sensitive the critical point of the mixed phase is to the presence of the Bose–Einstein condensate.

RASPA3: A Monte Carlo code for computing adsorption and diffusion in nanoporous materials and thermodynamics properties of fluids.

We present RASPA3, a molecular simulation code for computing adsorption and diffusion in nanoporous materials and thermodynamic and transport properties of fluids. It implements force field based classical Monte Carlo/molecular dynamics in various ensembles. In this article, we introduce the new additions and changes compared to RASPA2. RASPA3 is rewritten from the ground up in C++23 with speed and code readability in mind. Transition-matrix Monte Carlo is added to compute the density of states and free energies. The Monte Carlo code for rigid molecules is based on quaternions, and the atomic positions needed in the energy evaluation are recreated from the center of mass position and quaternion orientation. The expanded ensemble methodology for fractional molecules, with a scaling parameter λ between 0 and 1, now also keeps track of analytic expressions of dU/dλ, allowing independent verification of the chemical potential using thermodynamic integration. The source code is freely available under the MIT license on GitHub. Using this code, we compare four Monte Carlo (MC) insertion/deletion techniques: unbiased Metropolis MC, Configurational-Bias Monte Carlo (CBMC), Continuous Fractional Component MC (CFCMC), and CB/CFCMC. We compare particle distribution shapes, acceptance ratios, accuracy and speed of isotherm computation, enthalpies of adsorption, and chemical potentials, over a wide range of loadings and systems, for the grand canonical ensemble and for the Gibbs ensemble.

Generalized hydrodynamics for the volterra lattice: ballistic and non-ballistic behavior of correlation functions

Abstract In recent years, a lot of effort has been put in describing the hydrodynamic behavior of in- tegrable systems. In this paper, we describe such picture for the Volterra lattice. Specifically, we are able to explicitly compute the susceptibility matrix and the current-field correlation matrix in terms of the density of states of the Volterra lattice endowed with a Generalized Gibbs ensemble. Furthermore, we apply the theory of linear Generalized Hydrodynamics to describe the Euler scale behavior of the correlation functions. We anticipate that the solution to the Generalized Hydrodynamics equations develops shocks at ξ0=x/t ; so this linear approximation does not fully describe the behavior of correlation functions. Intrigued but this fact, we performed several numerical investigations which show that, exactly when the solution to the hydrodynamic equations develops shock, the correlation functions show an highly oscillatory behavior. In view of this empirical observation, we believe that at this point ξ0 the diffusive contribution are not sub-leading corrections to the ballistic transport, but they are of the same order.

A thermophysical mechanism exploration of the brain: Motor cortex modeling with canonical ensemble theory

The brain, recognized as one of the most intricate systems globally, has been a focal point for scientific exploration. Researchers have made efforts to construct models of the brain based on neural dynamics and complex networks to gain insights into its workings. It is crucial to investigate the brain's working principles from various perspectives. This study presents a novel thermophysical model of the motor cortex and examines its potential thermodynamic properties. Utilizing canonical ensemble theory, we constructed the thermophysical model using spike and local field potential (LFP) signals obtained from intracortical brain-machine-interfaces (iBMIs) in two monkeys. The parameters derived from this model—namely internal energy, free energy, and entropy—were employed to assess the thermodynamic properties and observe alterations in these properties during reaching and grasping movements. Furthermore, this proposed model was applied to movement pattern decoding, highlighting its potential in neural decoding tasks. In both LFP- and spike-based thermodynamic models, there was an increase in internal energy and free energy, coupled with a decrease in entropy when the motor cortex was activated across various movement tasks. This suggests that the neural system adheres to the principles of a thermophysical system. Notably, the thermodynamic features demonstrated superior performance in decoding movement intentions compared to traditional LFP and spike features. This study represents the first construction of a comprehensive thermodynamic model of the motor cortex based on LFP and spike signals. The model exhibits remarkable stationarity and holds promise for long-term and stable evaluations of motor cortex functions.