Partition Function Research Articles

Bending elasticity of the reversible freely jointed chain.

The freely jointed chain model with reversible hinges (rFJC) is the simplest theoretical model, which captures reversible transitions of the local bending stiffness along the polymer chain backbone (e.g., helix-coil-type of local conformational changes or changes due to the binding/unbinding of ligands). In this work, we analyze the bending fluctuations and the bending response of a grafted rFJC in the Gibbs (fixed-force) ensemble. We obtain a recursion relation for the partition function of the grafted rFJC under a bending force, which allows, in principle, an exact-numerical calculation of the behavior of an rFJC of arbitrary size. In contrast to stretching, we show that under sufficiently stiff conditions, the differential bending compliance and the mean fraction of closed hinges are non-monotonic functions of the force. We also obtain the persistence length Lp of the rFJC and the moments ⟨R2⟩ (mean-square end-to-end distance) and ⟨z2⟩ (mean-square transverse deflection) for the discrete chain and take the continuum limit. The tangent vector auto-correlation decays exponentially, as in the wormlike chain model (WLC). Remarkably, the expression of ⟨R2⟩ as a function of the contour length L becomes the same as that in the WLC. In the thermodynamic limit, we have calculated the exact bending response analytically. As expected, for L ≫ Lp, the boundary conditions do not matter, and the bending becomes equivalent to stretching. In contrast, for Lp ≫ L, we have shown the non-monotonicity of the bending response (the compliance and mean fraction of closed hinges).

Tau Functions from Joyce Structures

We argued in [Proc. Sympos. Pure Math., Vol. 103, American Mathematical Society, Providence, RI, 2021, 1-66, arXiv:1912.06504] that, when a certain sub-exponential growth property holds, the Donaldson-Thomas invariants of a 3-Calabi-Yau triangulated category should give rise to a geometric structure on its space of stability conditions called a Joyce structure. In this paper, we show how to use a Joyce structure to define a generating function which we call the $\tau$-function. When applied to the derived category of the resolved conifold, this reproduces the non-perturbative topological string partition function of [J. Differential Geom. 115 (2020), 395-435, arXiv:1703.02776]. In the case of the derived category of the Ginzburg algebra of the A$_2$ quiver, we obtain the Painlevé I $\tau$-function.

Holographic thermal correlators and quasinormal modes from semiclassical Virasoro blocks

Motivated by its relevance for thermal correlators in strongly coupled holographic CFTs, we refine and further develop a recent exact analytic approach to black hole perturbation problem, based on the semiclassical Virasoro blocks, or equivalently via AGT relation, the Nekrasov partition functions in the Nekrasov-Shatashvili limit. Focusing on asymptotically AdS5 black hole backgrounds, we derive new universal exact expressions for holographic thermal two-point functions, both for scalar operators and conserved currents. Relatedly, we also obtain exact quantization conditions of the associated quasinormal modes (QNMs). Our expressions for the holographic CFT4 closely resemble the well-known results for 2d thermal CFTs on ℝ1,1. This structural similarity stems from the locality of fusion transformation for Virasoro blocks. We provide numerical checks of our quantization conditions for QNMs. Additionally, we discuss the application of our results to understand specific physical properties of QNMs, including their near-extremal and asymptotic limits. The latter is related to a certain large-momentum regime of semiclassical Virasoro blocks dual to Seiberg-Witten prepotentials.

Complexity is not enough for randomness

We study the dynamical generation of randomness in Brownian systems as a function of the degree of locality of the Hamiltonian. We first express the trace distance to a unitary design for these systems in terms of an effective equilibrium thermal partition function, and provide a set of conditions that guarantee a linear time to design. We relate the trace distance to design to spectral properties of the time-evolution operator. We apply these considerations to the Brownian pp-SYK model as a function of the degree of locality pp. We show that the time to design is linear, with a slope proportional to 1/p1/p. We corroborate that when pp is of order the system size this reproduces the behavior of a completely non-local Brownian model of random matrices. For the random matrix model, we reinterpret these results from the point of view of classical Brownian motion in the unitary manifold. Therefore, we find that the generation of randomness typically persists for exponentially long times in the system size, even for systems governed by highly non-local time-dependent Hamiltonians. We conjecture this to be a general property: there is no efficient way to generate approximate Haar random unitaries dynamically, unless a large degree of fine-tuning is present in the ensemble of time-dependent Hamiltonians. We contrast the slow generation of randomness to the growth of quantum complexity of the time-evolution operator. Using known bounds on circuit complexity for unitary designs, we obtain a lower bound determining that complexity grows at least linearly in time for Brownian systems. We argue that these bounds on circuit complexity are far from tight and that complexity grows at a much faster rate, at least for non-local systems.

Tetrahedron equation and Schur functions

Abstract The tetrahedron equation introduced by Zamolodchikov is a three-dimensional generalization of the Yang–Baxter equation. Several types of solutions to the tetrahedron equation that have connections to quantum groups can be viewed as q-oscillator valued vertex models with matrix elements of the L-operators given by generators of the q-oscillator algebra acting on the Fock space. Using one of the q = 0-oscillator valued vertex models introduced by Bazhanov–Sergeev, we introduce a family of partition functions that admits an explicit algebraic presentation using Schur functions. Our construction is based on the three-dimensional realization of the Zamolodchikov–Faddeev algebra provided by Kuniba–Maruyama–Okado. Furthermore, we investigate an inhomogeneous generalization of the three-dimensional lattice model. We show that the inhomogeneous analog of (a certain subclass of) partition functions can be expressed as loop elementary symmetric functions.

Spin-refined partition functions and CRT black holes

We investigate spin-refined partition functions in AdS/CFT using Euclidean gravitational path integrals. We construct phase diagrams for ZX = Tr(e−βHX) in various dimensions and for different choices of discrete isometry X, discovering rich structures at finite temperature. When X is a reflection, ZX counts the difference between the number of even- and odd-spin microstates. The high-temperature regime is universally dominated by CRT-twisted black holes in any dimension, and in odd spacetime dimensions we examine whether complex rotating black hole solutions can contribute to spin-refined observables or potentially dominate at finite temperature. We also analyze the microcanonical ensemble. There the leading contribution almost always comes from rotating black holes, showing that the two ensembles are not necessarily equivalent.

Invariants of the quartic binary form and proofs of Chen’s conjectures on inequalities for the partition function and the Andrews’ spt function

Invariants of the quartic binary form and proofs of Chen’s conjectures on inequalities for the partition function and the Andrews’ spt function

Modular invariance as completeness

We review the physical meaning of modular invariance for unitary conformal quantum field theories in d=2. For quantum field theory models, while T invariance is necessary for locality, S invariance is not mandatory. S invariance is a form of completeness of the theory that has a precise meaning as Haag duality for arbitrary multi-interval regions. We present a mathematical proof as well as derive this result from a physical standpoint using Renyi entropies and the replica trick. For rational conformal field theories (CFTs), the failure of modular invariance or Haag duality can be measured by an index, related to the quantum dimensions of the model. We show how to compute this index from the modular transformation matrices. The index also appears in a limit of the Renyi mutual information. Cases of infinite index are briefly discussed. Part of the argument can be extended to higher dimensions, where the lack of completeness can also be diagnosed using the CFT data through the thermal partition function and measured by an index. Published by the American Physical Society 2024

Waves and non-propagating mode in stratified and rotating magnetohydrodynamic turbulence

In this study, we consider a freely decaying, stably stratified, and rotating homogeneous magneto-hydrodynamic (MHD) turbulent plasma with a vertical background magnetic field (B0=B0ẑ), aligned with the density gradient (with a constant Brunt–Váisálá frequency N) viewed in a frame rotating uniformly around the vertical axis (Ω0=Ω0ẑ). Quasi-linear theory is used to analyze the flow dynamics for an inviscid and non-diffusive Boussinesq fluid. We perform a normal mode decomposition emphasizing three types of motions: a non-propagating (NP) mode, which is no longer a vortex mode, and slow and fast magneto-inertia-gravity waves. The total energy as well as the L2 norm, say Γ, of the magnetic induction potential scalar (MIPS), which remains similar to the potential enstrophy for non-magnetized rotating and stratified flows, are inviscid invariants. In contrast with the potential vorticity for non-magnetized rotating and stratified flows, the MIPS is not affected by system rotation in the quasi-linear limit, and this is the effect of rotation which presumes an inverse cascade of energy in the equilibrium statistical mechanics. We characterized the system setting up our investigation from the point of view of equilibrium statistical mechanics in the limit of small Froude number and small Alfvén–Mach number. In this limit, the non-propagating quantity Γ can be approximated by its quadratic part that explicitly depends only on the vertical component of the fluctuating magnetic field and the density fluctuations. We demonstrate that the partition function restricted to the non-propagating manifold does not indicate an inverse cascade of energy.

Comment on “Relativistic spinless energies and thermodynamic properties of sodium dimer molecule”

In this Comment we point out two incorrect results in a paper published recently in this journal. In the first place, the fact that the maximum number of vibrational energy levels is not a variable parameter of the model because it is determined by the potential-energy function and, in particular, by the dissociation energy. In the second place, we argue that the vibrational thermodynamic functions for an excited electronic state of a diatomic molecule are of no physical utility because any physical application requires the more relevant contribution of the lower electronic states to the canonical partition function. To illustrate this point we show the calculation of the equilibrium constant for the dimerization of sodium using only spectroscopic information about the ground electronic state. The theoretical expression agrees remarkably well with the available experimental data.

Partitions in real quadratic fields

AbstractWe study partitions of totally positive integers in real quadratic fields. We develop an algorithm for computing the number of partitions, prove a result about the parity of the partition function, and characterize the quadratic fields such that there exists an element with exactly 1–5, 7, and 11 partitions.

Angular fractals in thermal QFT

We show that thermal effective field theory controls the long-distance expansion of the partition function of a d-dimensional QFT, with an insertion of any finite-order spatial isometry. Consequently, the thermal partition function on a sphere displays a fractal-like structure as a function of angular twist, reminiscent of the behavior of a modular form near the real line. As an example application, we find that for CFTs, the effective free energy of even-spin minus odd-spin operators at high temperature is smaller than the usual free energy by a factor of 1/2d. Near certain rational angles, the partition function receives subleading contributions from “Kaluza-Klein vortex defects” in the thermal EFT, which we classify. We illustrate our results with examples in free and holographic theories, and also discuss nonperturbative corrections from worldline instantons.

Vieta’s Formula, the Fabius Function and the Partition Function

We use an idea of Pólya and Szegö to give a common basis to Vieta’s formula, Fabius function and the partition function. Moreover our construction leads also to a function considered by Hallström, Bowen and Macintyre, which has, as a particular value, the Kepler-Bouwkamp constant, and to a function considered by Zondadari, that vanishes only at prime numbers.

Scaling limit of domino tilings on a pentagonal domain.

We consider the six-vertex model at its free-fermion point with domain wall boundary conditions, which is equivalent to random domino tilings of the Aztec diamond. We compute the scaling limit of a particular nonlocal correlation function, essentially equivalent to the partition function for the domino tilings of a pentagon-shaped domain, obtained by cutting away a triangular region from a corner of the initial Aztec diamond. We observe a third-order phase transition when the geometric parameters of the obtained pentagonal domain are tuned to have the fifth side exactly tangent to the arctic ellipse of the corresponding initial model.

DFT-SCRF Calculations of Nitrogen Isotopic Reduced Partition Function Ratios for Amino Acids in Aqueous Solution at pH 7.4.

Using the self-consistent reaction field (SCRF) method, the nitrogen isotopic reduction partition function ratio (RPFR), a fundamental physical quantity for theoretical consideration of equilibrium isotope effects, was evaluated by density functional theory (DFT) for 20 proteinogenic and two related amino acids under physiological conditions at pH 7.4. At this pH value, all amino acids were of the zwitterion-type form with α-NH3+ and α-COO- groups, regardless of the net charge of -1 (monoanion), 0 (zwitterion), or +1 (monocation). The largest RPFR value by far was for proline (Pro). This is due to the fact that the nitrogen atom of Pro has two C-N bonds and two N-H bonds, whereas the α-nitrogen atoms of other amino acids form one C-N bond and three N-H bonds. There was a clear correlation between the RPFR value of α-nitrogen and the distance of the N-H covalent bond of the hydrogen atom involved in the formation of the intramolecular hydrogen bond (HB); the stronger the intramolecular HB, the lower the RPFR value of α-nitrogen. It was suggested that equilibrium isotope effects were partially responsible for the nitrogen isotope fractionation in the enzymatic glutamic acid/aspartic acid transamination and in ammonia assimilation during the formation of glutamine from glutamic acid.

Understanding the Negative Apparent Activation Energy for Cu2O and CoO Oxidation Kinetics at High Temperature near Equilibrium

The pairs of Cu2O/CuO and CoO/Co3O4 as the carriers of transferring oxygen and storing heat are essential for the recently emerged high-temperature thermochemical energy storage (TCES) system. Reported research results of Cu2O and CoO oxidation kinetics show that the reaction rate near equilibrium decreases with the temperature, which leads to the negative activation energy obtained using the Arrhenius equation and apparent kinetics models. This study develops a first-principle-based theoretical model to analyze the Cu2O and CoO oxidation kinetics. In this model, the density functional theory (DFT) is adopted to determine the reaction pathways and to obtain the energy barriers of elementary reactions; then, the DFT results are introduced into the transition state theory (TST) to calculate the reaction rate constants; finally, a rate equation is developed to describe both the surface elemental reactions and the lattice oxygen concentration in a grain. The reaction mechanism obtained from DFT and kinetic rate constants obtained from TST are directly implemented into the rate equation to predict the oxidation kinetics of Cu2O without fitting experimental data. The accuracy of the developed theory is validated by experimental data obtained from the thermogravimetric analyzer (TGA). Comparing the developed theory with the traditional apparent models, the reasons why the latter cannot appropriately predict the true oxidation characteristics are explained. The reaction rate is jointly controlled by thermodynamics (reaction driving force) and kinetics (reaction rate constant). Without considering the effect of the reaction driving force, the negative apparent activation energy of Cu2O oxidation is obtained. However, for CoO oxidation, the negative apparent activation energy is still obtained although the effect of the reaction driving force is considered. According to the DFT results, the activation energy of the overall CoO oxidation reaction is negative, but the energy barriers of the elementary reactions are positive. Moreover, according to the first-principle-based rate equation theory, the pre-exponential factor in the kinetic model is dependent on the partition function ratio and decreases with the temperature for the Cu2O and CoO oxidation near equilibrium, which results in the apparent activation energy being slightly lower than the actual value.

Factorization of rational six vertex model partition functions

We show factorization formulas for a class of partition functions of rational six vertex model. First we show factorization formulas for partition functions under triangular boundary. Further, by combining the factorization formulas with the explicit forms of the generalized domain wall boundary partition functions by Belliard-Pimenta-Slavnov, we derive factorization formulas for partition functions under trapezoid boundary which can be viewed as a generalization of triangular boundary. We also discuss an application to emptiness formation probabilities under trapezoid boundary which admit determinant representations.

Correlation analysis of multifractal stock price fluctuations based on partition function

Studying the correlation analysis of stock price fluctuations helps to understand market dynamics better and improve the scientific nature of investment decisions and risk management capabilities. Most existing methods use multifractals to explore the correlation between different economic entities. However, the study of multifractals fails to fully consider the weight of each entity’s impact on the market, resulting in an inaccurate description of the overall market dynamics. To address this problem, this paper creatively proposes a weighted multifractal analysis method (WMA). The correlation analysis of government regulation, market supply and demand, and stock price index is performed using the data of A-share listed companies in Shenzhen and Shanghai as samples. First, we consider the amplitude fluctuation information the signal carries and weigh the partition function according to the proportion of variance in the segment for different amplitude changes. Secondly, we derive the theoretical analytical form of the classical multifractal model (SMA) of the scaling indicator under WMA. Finally, through numerical simulation experiments, it is confirmed that WMA is equally effective as SMA. In addition, the re-fractal correlation analysis of real financial time series also confirms that WMA can effectively utilize the amplitude fluctuation information in the series and outperforms the classical SMA method in distinguishing different signals.

Euclidean wormholes in holographic RG flows

We describe a one-parameter family of Euclidean wormhole solutions with the topology of a compact hyperbolic space times an interval in Einstein gravity minimally coupled to a massless scalar field in AdSd+1 commonly referred to as Einstein-dilaton gravity. These solutions are locally described by the same metric and dilaton profile as the single-boundary Janus domain wall solutions in the same theory which are usually studied in the context of holographic RG flows. The wormholes compute the averaged product of partition functions of CFTs on either boundary deformed by different marginal couplings to the scalar operator dual to the dilaton. We observe that the renormalised volumes of these wormholes increase monotonically with the difference in the marginal couplings on the boundary thereby showing that the pair of CFTs on the boundaries get increasingly decorrelated as the difference in the marginal couplings increases. We use the partition functions of the three-dimensional wormhole solutions to determine the variance of the OPE data of local operators between the marginally deformed 2d CFTs and quantify how the variance decays with the difference in marginal couplings. In addition, a family of wormholes sourced by a thin shell of dust determine how the variance of the matrix elements of the dual line defect decays with the difference in marginal couplings. Applying the GKPW dictionary to wormholes, we compute averages of integrated dilaton correlators treating the wormhole amplitude as a functional of the dilaton sources. We observe that the crossed two-point correlators with a dilaton insertion on either boundary decay monotonically with the difference in marginal couplings consistent with the observation that the CFTs increasingly decorrelate as the difference in marginal couplings grows.

Prediction of Molar Entropy of Gaseous Molecules for a New Pὃschl‐Teller Potential Model

ABSTRACTThe Pὃschl–Teller potential is a molecular potential energy function that has only been reported for bound state. This Pὃschl–Teller potential is a good representation of many molecules and has not been examined for any thermodynamic property irrespective of its fitness for molecular study. In this study, the molar entropy of four molecules (Pbr, BBr, CsCl, and CsO molecules) is calculated via the molar partition function. The predicted results are compared with the experimental data recorded in the National Institute of Standards and Technology (NIST) database. It is noted that the predicted values for the studied molecules perfectly agree with the experimental results with the following average absolute percentage deviation, PBr is 0.0158%, BBr is 0.0053%, CsCl is 0.0020%, and CsO is 0.0052%. The present model reproduces better results for CsCl and CsO molecules compared to the shifted Tietz–Wei potential and improved Tietz‐oscillator previously reported whose average absolute percentage deviation are 0.361% and 0.284% for CsCl and 0.272% and 0.228% for CsO, respectively.