Calculation Of Partition Function Research Articles

Notes on wormhole cancellation and factorization

In AdS/CFT, partition functions of decoupled CFTs living on separate asymptotic boundaries factorize. However, the presence of bulk wormholes connecting different boundaries tends to spoil the factorization of the bulk partition function, which leads to a disagreement between the two sides. In this paper, we present two examples where wormhole contributions cancel each other in bulk partition function calculations, thus the bulk factorization can be realized. The first example is in 2-dimensional Jackiw–Teitelboim (JT) gravity, where the proposed way of realizing the cancellation resides in the extra complex phases associated with different wormholes. The phases arise due to the degenerate vacua structure. In the example of the Sachdev–Ye–Kitaev (SYK) model, the cancellation can be achieved due to the distribution of the wormhole saddles on a complex plane. The two examples demonstrate a way of realizing bulk partition function factorization by extending the Hilbert space and dressing wormhole saddles with extra phases.

Ensemble averages of ℤ2 orbifold classes of Narain CFTs

In this work we study families of ℤ2 orbifolds of toroidal conformal field theories based on both factorizable and non-factorizable target space tori. For these classes of theories, we analyze their moduli spaces, and compute their partition functions. Building on previous work, we express the calculated partition functions in terms of suitable Siegel-Narain theta functions that allow us to determine their ensemble averages. We express the derived averaged partition functions of the studied families of conformal field theories in a manifest modular invariant finite sum of products of real analytic Eisenstein series. We speculate on a tentative holographic three-dimensional dual bulk interpretations for the considered ℤ2 orbifold classes of ensembles of conformal field theories.

On the dependence of thermodynamic variables on the relative velocity vns in a superfluid counterflowing helium

Based on the theory of the thermodynamic equilibrium in a system of quantum vortices in superfluids in the presence of a counterflow, the influence of a vortex tangle on various thermodynamic phenomena in quantum liquids is studied. Using the early calculated partition function, we study some of the properties of He II related to counterflow, such as the distribution of vortex loops in their length, the suppression of the superfluid density ρs, and the shift Tλ. The physics behind this issue is related to the fact that the partition function describing the ensemble of chaotic vortex filaments depends on the relative velocity vns. The partition function, in turn, depends on relative velocity due to the Gibbs distribution with the specific velocity-dependent Hamiltonian. Good agreement with the earlier obtained results is a fairly strong argument in favor of the point of view that a collection of chaotic quantum vortices can, indeed, be considered as a kind of gas of quasiparticles at high temperatures, especially near a phase transition. The work is closely related to nonlinear physics, which studies chaotic processes, and is currently in the stage of active development, resulting in many meaningful and expressive results. The application of the developed formalism to the theory of quantum turbulence is briefly discussed.

Differentiable partition function calculation for RNA.

Ribonucleic acid (RNA) is an essential molecule in a wide range of biological functions. In 1990, McCaskill introduced a dynamic programming algorithm for computing the partition function of an RNA sequence. McCaskill's algorithm is widely used today for understanding the thermodynamic properties of RNA. In this work, we introduce a generalization of McCaskill's algorithm that is well-defined over continuous inputs. Crucially, this enables us to implement an end-to-end differentiable partition function calculation. The derivative can be computed with respect to the input, or to any other fixed values, such as the parameters of the energy model. This builds a bridge between RNA thermodynamics and the tools of differentiable programming including deep learning as it enables the partition function to be incorporated directly into any end-to-end differentiable pipeline. To demonstrate the effectiveness of our new approach, we tackle the inverse folding problem directly using gradient optimization. We find that using the gradient to optimize the sequence directly is sufficient to arrive at sequences with a high probability of folding into the desired structure. This indicates that the gradients we compute are meaningful.

Calculation of partition function of Ising model on quantum computer

We study the partition function of the Ising model on a graph with the help of quantum computing. The Boltzmann factor is modeled on a quantum computer as a trace of some evolution operator with effective Hamiltonian over ancilla spins (qubits) corresponding to graph links. We propose two methods for this which are based on effective Hamiltonian with three-spin interaction and on two-spin interaction. The limit of small temperatures allows us to find the ground state of the system that is related to the discrete combinatorial optimization problem. The partition function of the Ising model for two-spin clusters is calculated on IBM's quantum computer. The possibility of finding ground state is also demonstrated for two-spin clusters.

MSTor 2023: A new version of the computer code for multistructural torsional anharmonicity, now with automatic torsional identification using redundant internal coordinates

The MSTor program is a computer program for calculating partition functions and thermodynamic functions of complex gas-phase molecules with multiple torsions; the multi-structural approximation with torsional anharmonicity (MS-T) can be used based on either a coupled torsional potential or an uncoupled torsional potential. The program can also carry out calculations in the multiple-structure local harmonic (MS-LH) approximation or multi-structural local quasiharmonic (MS-LQ) approximation or by the dual-level MS-T method. Furthermore, the program package includes eight utility codes that can be used as stand-alone programs to help the user to generate the input files for the code or to generate comparison results. The 2023 version of MSTor includes a new capability, namely automatic identification of torsional modes with redundant internal coordinates. New version program summaryProgram title: MSTor 2023CPC Library link to program files:https://doi.org/10.17632/t4y5wds856.1Licensing provisions: Apache 2.0Programming language: Fortran 90, C, and PerlJournal references of previous versions: Comput. Phys. Comm. 183 (2012) 1803; Comput. Phys. Comm. 184 (2013) 2032Does the new version supersede the previous version?: Yes.Nature of problem: Calculation of the partition functions and thermodynamic functions (standard-state energy, enthalpy, entropy, and free energy as functions of temperature) of molecules or transition states involving multiple torsional motions.Solution method: The program uses the multi-structural approximation with torsional anharmonicity, a coupled torsional potential, and delocalized torsions MS-T(CD) [1]. The program also provides results for the multi-structural local harmonic approximation or local quasiharmonic approximation, and it can also be used for the original multi-structural approximation with torsional anharmonicity and a coupled torsional potential MS-T(C) [2] or an uncoupled torsional potential MS-T(U) [3]. The program can also carry out dual-level [4] MS-T(CD), MS-T(C), and MS-T(U) calculations.Reasons for new version: The new version increases functionality and is more user friendly. The main enhancement is the capability for automatic identification of torsional modes with redundant internal coordinates.Summary of revisions: A newly developed torsional identification approach for complex molecules has been implemented in the MSTor program. The method, called multi-structural torsion method with a coupled torsional potential and delocalized torsions MS-T(CD), is based on Baker et al.'s delocalized internal-coordinates method and on Zheng and Truhlar's torsional projection method. By using a redundant-internal-coordinate auto-generation procedure, the MS-T(CD) method circumvents the need in the original MS-T method for the user to define nonredundant internal coordinates, and it can straightforwardly identify and separate the coupled torsions from the Cartesian Hessian. This greatly simplifies the user input, and it helps users to obtain robust and consistent results. MSTor now sets the redundant internal coordinate scheme as the default method.A new utility code, DLMSTor.exe, is provided to allow dual-level calculations. The $framechain and $framedef sections were reintroduced to the code as optional sections. If these sections are used, the D matrix is calculated using the scheme of Kilpatrick and Pitzer and one should always get correct values for the moments of inertia.Various utility codes were modified to help ensure that structures that are equivalent under rotation to other structures (and thus indistinguishable when considering overall rotation and torsions) are still properly included in the Voronoi tessellation step (which does not fully exploit overall rotational symmetry).The mcvorm.exe utility code was modified to correctly calculate uncertainties and to take advantage of rotational symmetries (at least for linear-chain molecules) and mirror-image symmetries, if present, in the calculation of the uncertainty estimates.We added a new keyword, MTUMME, that provides an option to write a density-of-states file for version 2023 (in preparation) of the TUMME [5] master equation program.Some minor bugs are fixed so the code now compiles properly with the GNU gfortran compiler (version 12.2.1), the NVidia HPC Fortran compiler pgfortran (version 21.3), and the Intel compiler ifort. The output format is cleaned, and unused variables and arrays are removed. The new version has an increased number of test runs, and the manual is improved and updated.The program authors are J. Zheng, W. Chen, S. L. Mielke, J. L. Bao, R. Meana-Pañeda, K. L. Clarkson, X. Xu, and D. G. Truhlar; the new version announcement authors are W. Chen, J. Zheng, J. L. Bao, D. G. Truhlar, and X. Xu; the primary corresponding author is X. Xu.

CRNAsp12 Web Server for the Prediction of Circular RNA Secondary Structures and Stabilities

Circular RNAs (circRNAs) are a novel class of non-coding RNA that, unlike linear RNAs, form a covalently closed loop without the 5' and 3' ends. Growing evidence shows that circular RNAs play important roles in life processes and have great potential implications in clinical and research fields. The accurate modeling of circRNAs structure and stability has far-reaching impact on our understanding of their functions and our ability to develop RNA-based therapeutics. The cRNAsp12 server offers a user-friendly web interface to predict circular RNA secondary structures and folding stabilities from the sequence. Through the helix-based landscape partitioning strategy, the server generates distinct ensembles of structures and predicts the minimal free energy structures for each ensemble with the recursive partition function calculation and backtracking algorithms. For structure predictions in the limited structural ensemble, the server also provides users with the option to set the structural constraints of forcing the base pairs and/or forcing the unpaired bases, such that only structures that meet the criteria are enumerated recursively.

Identification of Torsional Modes in Complex Molecules Using Redundant Internal Coordinates: The Multistructural Method with Torsional Anharmonicity with a Coupled Torsional Potential and Delocalized Torsions.

Identification of internal-rotation modes in the normal-mode analysis of complex molecules is important for accurately describing the thermodynamic properties and kinetics of complex molecules when it is necessary to treat the anharmonicity of torsions and the multiconformer anharmonicity caused by the internal rotations. However, identifying and distinguishing torsional modes are very challenging because they are coupled to one another. In this work, we present a new strategy to automatically identify torsional vibrations and separate them from the other vibrational modes. By combining a redundant-internal-coordinate auto-generation procedure with torsional projection techniques, we automate the procedure of identifying and separating the coupled torsions, and we show that we can obtain robust and consistent results with various reasonable definitions of redundant-internal-coordinate sets. This model has been implemented in a new development version of the MSTor program to reduce the user input needed for multistructural and torsional anharmonicity (MS-T) calculations. The new method is called multistructural and torsional anharmonicity with a coupled torsional potential and delocalized torsions ([MS-T(CD)]. As example applications, we consider MS-T(CD) calculations on three molecules (2-hexyl radical, n-propylbenzene, and 5-hydroperoxy-6-oxohexanoylperoxy radical) that have multiple rotors and that provide challenges to choosing good sets of nonredundant-internal coordinates, and we compare the performance of the new strategy to five other torsion identification methods. The new strategy is demonstrated to be efficient in separating the torsional and nontorsional elements in the Hessian matrix, as well as in providing reasonable projected nontorsional frequencies to be used for calculations of partition function and thermochemistry.

Metric-Based Assessment Method for MS-T Formalism with Small Subsets of Torsional Conformers.

The multi-structural approximation with torsional anharmonicity (MS-T) method and its variants have been widely used for calculating conformational-rovibrational partition functions of large molecules. The present work aimed to propose a systematic method to assess and explain the performance of various variants of the MS-T method. First, we proposed the simplest variant MS-T(2NN) (two nearest neighborhood torsions are coupled) and systematically validated it for large alkanes n-CnH2n+2 (n = 6-10) and their transition states of hydrogen abstraction reactions. Second, we proposed a metric-based method to explain the underlying reason for the good performance of MS-T(2NN)─it includes the torsional conformers that have dominant contributions to the partition function calculations. These conformers are closer to the lowest-energy conformer in the space of dihedral and energy metrics. Third, the same observation and explanation apply to the other two variants, MS-2DT (any two torsions are coupled) and MS-3DT (any three torsional are coupled), which contain increasingly more torsional conformers than MS-T(2NN) but are subsets of the complete set of torsional conformers considered by the MS-T method. Overall, the present method provides a mathematically rigorous and computationally effective diagnosis tool to assess various MS-T methods dealing with the torsional anharmonicity of large molecules in the partition function calculation.

Density functional thermodynamic description of spin, phonon and displacement degrees of freedom in antiferromagnetic-to-paramagnetic phase transition in YNiO3

This work demonstrates a direct density functional description of the finite-temperature thermodynamic properties of solids exhibiting phase transitions through positional and spin symmetry breaking degrees of freedom. A classic example addressed here is the rare-earth (R) nickelates RNiO3 where the ground state is characterized by crystallographic and magnetic (e.g., antiferromagnetic) long-range order (LRO), whereas the higher temperature paramagnetic phase manifests a range of local spin and positional symmetry breaking motifs with short-range order (SRO). Unlike time-dependent simulations of spin and positional degrees of freedom, in the present work, phases are described via a superposition of static configurations constructed by populating a periodic base lattice supercell allowing for the formation of energy lowing distribution of positional and spin local motifs. The thermal populations of the configurations in such a superposition phase are obtained from the energy-minimized Density Functional Theory (DFT)-calculated partition functions at different temperatures. This approach offers flexible inclusion of different physical contributions to the free energy, such as elastic, electronic and phonon free energies, all obtained from the same underlying DFT total energy calculations of periodic structures. The thermodynamic and magnetic properties of both LRO and SRO crystallographic and spin phases, including antiferromagnetic (AFM) to paramagnetic (PM) Néel phase transition in YNiO3 are studied. Including spin and phonon contributions, we find a DFT-calculated Néel temperature to be 144 K in satisfactory agreement with the experimental value of 145 K; whereas omitting the phonon contribution, one obtains a Néel temperature of 81 K. We present phonon contributions to the DFT-calculated temperature-dependent SRO, heat capacities, and the polymorphous distribution of nonzero local magnetic moments in the PM phase. This approach thus extends to finite temperatures the symmetry-broken DFT description of both the AFM and PM phases, demonstrating that a thermodynamic superposition approach based on symmetry broken configurations evaluated by a mean-field like DFT is sufficient to obtain a consistent description of the thermal physics of the AFM, PM phases and their interconversion in 3d oxides illustrated by YNiO3.

Replicated entanglement negativity for disjoint intervals in the Ising conformal field theory

We calculate the N interval replicated negativity for the Ising conformal field theory using correlation functions of branch point twist fields. For some subset P ⊂ N, this is a calculation of where n is the replica index. This can be reformulated as a calculation of partition functions over superelliptic Riemann surfaces, and for the models in question, this partition function can be expressed in terms of the period matrix of this surface. We detail how to construct the period matrices for these surfaces, giving an analytic expression for . The results are expressed such that when P = Ø, which corresponds to calculating the Rényi entropy, the formulas aligns with known results for the N interval Rényi entropy.

Rigid Body Approximation for the Anharmonic Description of Molecule-Surface Vibrations.

We present an anharmonic approach for molecule-surface vibrations that employs rigid body coordinates, based on the Rodrigues rotation formula, to describe curvilinear displacements (rotations and translations) of the molecule along normal modes. These displacements are used to calculate energy data points from which one-dimensional polynomial potentials are fitted using cubic splines. In these potentials, for each of the six rigid body modes separately, one-dimensional Schrödinger equations are solved with harmonic oscillator or Fourier functions (for pure rotations) as basis sets. The anharmonic vibrational energies obtained are used to calculate partition functions and from them enthalpies, entropies, and Gibbs free energies of adsorption. Our numerical implementation has been successfully tested for Morse and cosine potentials with known analytical solutions. The methods have been applied to adsorption of CH4 on the hydroxyl group of the proton form of the chabazite zeolite (H-CHA), as well as to adsorption of CH4 and CO on the Mg2+ ions of the metal-organic framework (MOF) Mg2(dobdc). To obtain the best estimates for thermodynamic functions, we include the coupling between molecule-surface vibrations and intrasystem vibrations at the harmonic level. The calculated Gibbs free energies show that the coupling is small for CH4/H-CHA and CO/MOF (between -0.7 and +0.1 kJ/mol) but substantial for CH4/MOF (-3.4 kJ/mol). The predicted anharmonic effect on the Gibbs free energy of adsorption for CH4/H-CHA, CH4/MOF, and CO/MOF is -4.7, 0.3 ± 0.7, and -2.4 ± 0.6 kJ/mol, respectively, which results in +4.2, +0.9 ± 0.7, and -0.4 ± 0.6 kJ/mol, respectively, for the deviation from experiment. This is well within chemical accuracy limits (±4.2 kJ/mol) for the adsorption of CH4 and CO in the MOF. The larger deviation for CH4/H-CHA, at the edge of the chemical accuracy range, is most likely due to contributions from soft zeolite modes which are neglected in our approach.

ExoMol line lists – XLVI. Empirical rovibronic spectra of silicon mononitrate (SiN) covering the six lowest electronic states and four isotopologues

ABSTRACT Silicon mononitride (28Si14N, 29Si14N, 30Si14N, 28Si15N) line lists covering infrared, visible, and ultraviolet regions are presented. The SiNful line lists produced by ExoMol include rovibronic transitions between six electronic states: $X\, {}^{2}\Sigma ^{+}$, $A\, {}^{2}\Pi$, $B\, {}^{2}\Sigma ^{+}$, $D\, {}^{2}\Delta$, $a\, {}^{4}\Sigma ^{+}$, $b\, {}^{4}\Pi$. The ab initio potential energy and coupling curves, computed at the multireference configuration interaction (MRCI/aug-cc-pVQZ) level of theory, are refined for the observed states by fitting their analytical representations to 1052 experimentally derived SiN energy levels determined from rovibronic bands belonging to the X–X, A–X, and B–X electronic systems through the MARVEL procedure. The SiNful line lists are compared to previously observed spectra, recorded and calculated lifetimes, and previously calculated partition functions. SiNful is available via the www.exomol.com database.

Elimination of Image Saturation Effects on Multifractal Statistics Using the 2D WTMM Method.

Imaging artifacts such as image saturation can restrict the computational analysis of medical images. Multifractal analyses are typically restricted to self-affine, everywhere singular, surfaces. Image saturation regions in these rough surfaces rob them of these core properties, and their exclusion decreases the statistical power of clinical analyses. By adapting the powerful 2D Wavelet Transform Modulus Maxima (WTMM) multifractal method, we developed a strategy where the image can be partitioned according to its localized response to saturated regions. By eliminating the contribution from those saturated regions to the partition function calculations, we show that the estimation of the multifractal statistics can be correctly calculated even with image saturation levels up to 20% (where 20% is the number of saturated pixels over the total number of pixels in the image).

Calculation of Gibbs partition function with imaginary time evolution on near-term quantum computers

The Gibbs partition function is an important quantity in describing statistical properties of a system in thermodynamic equilibrium. There are several proposals to calculate the partition functions on near-term quantum computers. However, the existing schemes require many copies of the Gibbs states to perform an extrapolation for the calculation of the partition function, which could be costly performed on the near-term quantum computers. We propose a scheme to calculate the Gibbs function with the imaginary time evolution. After preparing Gibbs states with different temperatures by using the imaginary time evolution, we measure the overlap between them on a quantum circuit, which allows us to calculate the Gibbs partition function. Our scheme requires only 2N qubits to calculate the Gibbs function of N qubits.

Thermodynamic Equilibrium Vortices in Counterflowing Superfluid: Calculation of Partition Function

Thermodynamic Equilibrium Vortices in Counterflowing Superfluid: Calculation of Partition Function

Grand canonical partition function of a serial metallic island system

We present a calculation of the grand canonical partition function of a serial metallic island system by the imaginary-time path integral formalism. To this purpose, all electronic excitations in the lead and island electrodes are described using Grassmann numbers. The Coulomb charging energy of the system is represented in terms of phase fields conjugate to the island charges. By the large channel approximation, the tunneling action phase dependence can also be determined explicitly. Therefore, we represent the partition function as a path integral over phase fields with a path probability given in an analytically known effective action functional. Using the result, we also propose a calculation of the average electron number of the serial island system in terms of the expectation value of winding numbers. Finally, as an example, we describe the Coulomb blockade effect in the two-island system by the average electron number and propose a method to construct the quantum stability diagram.

A Divergence-Free Wigner Transform of the Boltzmann Operator Based on an Effective Frequency Theory.

The centroid effective frequency representation of path integrals as developed by Feynman and Kleinert was originally aimed at calculating partition functions and related quantities in the canonical ensemble. In its path integral formulation, only closed paths were relevant. This formulation has been used by the present authors in order to calculate the many-body Wigner function of the Boltzmann operator, which includes also open paths. This usage of the theory outside of the original intention can lead to mathematical divergence issues for potentials with barriers, particularly at low temperature. In the present paper, we modify the effective frequency theory of Feynman and Kleinert by also including open paths in its variational equations. In this way, a divergence-free approximation to the Boltzmann operator matrix elements is derived. This generalized version of Feynman and Kleinert’s formulation is thus more robust and can be applied to all types of barriers at all temperatures. This new version is used to calculate the Wigner functions of the Boltzmann operator for a quartic oscillator and for a double well potential and both static and dynamic properties are studied at several temperatures. The new theory is found to be essentially as precise as the original one. Its advantage is that it will always deliver a well-defined, even if approximate, Wigner function, which can, for instance, be used for sampling initial conditions for molecular dynamics simulations. As will be discussed, the theory can be systematically improved by including higher-order Fourier modes into the nonquadratic part of the trial action.

Partition function and thermodynamic quantities with atomic data of Ag XLIV

In this work, the atomic parameters of Ag XLIV (Be-like Ag) are examined and evaluated by implementing the GRASP2K package with the multi-configuration Dirac–Hartree–Fock (MCDHF) method for the calculation of wave-functions. We have listed fine structure energy levels of the lowest 170 levels with radiative data for multipole moments, such as electric dipole (E1), electric quadrupole (E2), magnetic dipole (M1), and magnetic quadrupole (M2) transitions, that lie in the region of extreme ultraviolet (EUV) and soft X-ray (SXR) for Ag XLIV from the ground state within the lowest 170 levels. We have compared our GRASP2K and FAC results with theoretical results available in the literature for some levels. Additionally, we have also calculated partition function and thermodynamic quantities for temperature ranges from 104to 107K. We believe that our presented details and data may be beneficial not only in plasma modeling but also in imaging of nanostructure as well as in medicine, semiconductors, and EUV and SXR laser applications.

Dual path integral: a non-perturbative approach to strong coupling

We develop a non-perturbative method for calculating partition functions of strongly coupled quantum mechanical systems with interactions between subsystems described by a path integral of a dual system. The dual path integral is derived starting from non-interacting subsystems at zeroth order and then by introducing couplings of increasing complexity at each order of an iterative procedure. These orders of interactions play the role of a dual time and the full quantum partition function is expressed as a transition amplitude in the dual system. More precisely, it is expressed as a path integral from a deformation-operators dependent initial state at zero time/order to the inverse-temperature dependent final state at later time/order. We provide examples of strongly coupled systems with up to first-order interactions (e.g. Ising model) and arbitrary high-order interactions (e.g. 1+1hbox {D} QFT). We also discuss a possible emergence of space-time, quantum field theories and general relativity in context of the dual path integral.