Partition Function Of Ensemble Research Articles

Grothendieck’s dessins d’enfants in a web of dualities. III

We give a new proof of the equivalence between the dessin partition function and the partition function of the Laguerre unitary ensemble (LUE), originally found by Ambjørn and Chekhov. We also introduce a correction factor for the dessin/LUE partition function, and show that the corrected dessin/LUE partition function is a Dubrovin–Zhang type tau-function of the Toda lattice hierarchy. As an application, we use the approach of Dubrovin and Zhang for the computation of the dessin correlators. In physicists’ terminology, we establish dualities among dessin counting, generalized Penner model, and -topological sigma model.

An almost-entirely precise empirical estimation for various chemical potentials

The transverse momentum spectra of the well-identified produced particles, π+, π−, K+, K−, p, p̄, Ks0, Λ, Λ̄, Ξ−, and Ξ+ are analyzed in a statistical approach. From the partition function of grand-canonical ensemble, we propose a generic expression for the dependence of the generic chemical potential μ on the rapidity y. Then, by fitting this expression to the experimental results on the most central p⊥ and d2N/2πp⊥dp⊥dy, at energies ranging from 7.7 to 200GeV, we have introduced a generic expression for the rapidity dependence of μ, at different energies, namely μ=a+by2, where a and b are constants. We find that the resulting energy dependence sNN=c[(μ−a)/b]d/2 agrees well with the statistical thermal models, where c and d are constants. We also present precise estimations of various types of chemical potentials, μB, μS, and μQ, the baryon, the strangeness, and the charge chemical potential, respectively.

An Almost-Entirely Precise Empirical Estimation for Various Chemical Potentials

The transverse momentum spectra of the well-identified produced particles, $\pi^+$, $\pi^-$, $K^+$, $K^-$, $p$, $\bar{p}$, $K_s^0$, $\Lambda$, $\bar{\Lambda}$, $\Xi^-$, and $\Xi^+$ are analyzed in a statistical approach. From the partition function of grand-canonical ensemble, we propose a generic expression for the dependence of the generic chemical potential $\mu$ on the rapidity $y$. Then, by fitting this expression to the experimental results on the most central $p_{\perp}$ and $d^2 N/2 \pi p_{\bot} dp_{\bot} dy$, at energies ranging from $7.7$ to $200~$GeV, we have introduced a generic expression for the rapidity dependence of $\mu$, at different energies, namely $\mu=a+b y^2$, where $a$ and $b$ is constants. We find that the resulting energy dependence $\sqrt{s_{\mathtt{NN}}}=c[(\mu-a)/b]^{d/2}$ agrees well with the statistical thermal models. We also present precise estimations of various types of chemical potentials, $\mu_B$, $\mu_S$, and $\mu_Q$, the baryon, the strangeness, and the charge chemical potential, respectively.

Symmetric matrix ensemble and integrable hydrodynamic chains

The partition function of the Symmetric Matrix Ensemble is identified with the tau -function of a particular solution of the Pfaff Lattice. We show that, in the case of even power interactions, in the thermodynamic limit, the tau -function corresponds to the solution of an integrable chain of hydrodynamic type. We prove that the hydrodynamic chain so obtained is diagonalisable and admits hydrodynamic reductions in Riemann invariants in an arbitrary number of components.

Diagrammatic approaches to RNA structures with trinucleotide repeats

Trinucleotide repeat expansion disorders are associated with the overexpansion of (CNG) repeats on the genome. Messenger RNA transcripts of sequences with greater than 60–100 (CNG) tandem units have been implicated in trinucleotide repeat expansion disorder pathogenesis. In this work, we develop a diagrammatic theory to study the structural diversity of these (CNG)n RNA sequences. Representing structural elements on the chain’s conformation by a set of graphs and employing elementary diagrammatic methods, we have formulated a renormalization procedure to re-sum these graphs and arrive at a closed-form expression for the ensemble partition function. With a simple approximation for the renormalization and applied to extended (CNG)n sequences, this theory can comprehensively capture an infinite set of conformations with any number and any combination of duplexes, hairpins, multiway junctions, and quadruplexes. To quantify the diversity of different (CNG)n ensembles, the analytical equations derived from the diagrammatic theory were solved numerically to derive equilibrium estimates for the secondary structural contents of the chains. The results suggest that the structural ensembles of (CNG)n repeat sequence with n ∼60 are surprisingly diverse, and the distribution is sensitive to the ability of the N nucleotide to make noncanonical pairs and whether the (CNG)n sequence can sustain stable quadruplexes. The results show how perturbations in the form of biases on the stabilities of the various structural motifs, duplexes, junctions, helices, and quadruplexes could affect the secondary structures of the chains and how these structures may switch when they are perturbed.

Matrix integral solutions to the discrete and coupled Leznov lattice equations

Matrix integrals used in random matrix theory for the study of eigenvalues of matrix ensembles have been shown to provide τ-functions for several hierarchies of integrable equations. In this paper, we construct the matrix integral solutions to the Leznov lattice equation, semi-discrete and fully-discrete version and the Pfaffianized Leznov lattice systems, respectively. We demonstrate that the partition function of the Jacobi type unitary ensemble is a solution to the semi-discrete Leznov lattice and the partition function of the Jacobi type orthogonal/symplectic ensemble solves the Pfaffianized Leznov lattice.

Probability density correlation for PDM-Hamiltonians and superstatistical PDM-partition functions

Schrodinger equation with position-dependent mass (PDM) allows the identification of quantum wave functions in a complex environment. Following the progress of this investigation field, in this work, we consider the non-Hermitian kinetic operators associated with the PDM Schrodinger equation. We provide a simplified picture for PDM quantum systems that admit exact solutions in confining potentials. First, we investigate the solutions for a sinusoidal and an exponential PDM distributions in an infinite potential well. Next, we consider the solutions for a PDM harmonic oscillator potential associated with a power-law PDM distribution. The results presented in this work offer a way to approach new classes of solutions for PDM quantum systems in confining potential (bound states). Complementarily, we interpret the quantum partition function of the canonical ensemble of a PDM system in the context of the superstatistics, which, in turn, allows us to express the inhomogeneity of the PDM in terms of beta distribution $$f(\beta )$$ , Dirac delta distributions for $$f(\beta )$$ , and effective temperatures. Our results are, hereby, reported for the sinusoidal and the exponential PDM distributions.

Almost Entirely Empirical Estimation for Chemical Potential

Based on statistical thermal approaches, the transverse momentum distribution of the well-identified produced particles, π+, π–, K+, K–, p, and $$\bar {p}$$, is studied. We aim at introducing a novel almost entirely empirical estimation for the inclusive chemical potential μ. From the partition function of a grand-canonical ensemble, we propose a generic expression for the dependence of μ on the rapidity y. Then, by fitting this expression with the experimental results of the most central p⊥ and d2N/2πp⊥dp⊥dy, at 7.7, 11.5, 19.6, 27, 39, 130, 200 GeV, we introduce a generic expression for the rapidity dependence of for all particle yields, at different energies, μ = a + by2. The resulting energy dependence reads $$\sqrt {{{s}_{{{\text{NN}}}}}} $$ = c[(μ – a)/b]d/2. As a validation test, we show that the proposed approach reproduces excellently the rapidity spectra of various particle yields measured, at different energies.

ПОЧТИ ПОЛНОСТЬЮ ЭМПИРИЧЕСКОЕ ОПРЕДЕЛЕНИЕ ХИМИЧЕСКОГО ПОТЕНЦИАЛА

Based on statistical thermal approaches, the transverse momentum distribution of the well-identified produced particles, $\pi^+$, $\pi^-$, $K^+$, $K^-$, $p$, $\bar{p}$, is studied. From the partition function of grand-canonical ensemble, we propose a generic expression for the dependence of the full chemical potential $\mu$ on rapidity $y$. Then, by fitting this expression with the experimental results of most central $p_{\perp}$ and $d^2 N/2 \pi p_{\bot} dp_{\bot} dy$, at $7.7$, $11.5$, $19.6$, $27$, $39$, $130$, $200~$GeV, we introduce a generic expression for the rapidity dependence of $\mu$, at different energies and particle types, $\mu=a+b y^2$. The resulting energy dependence reads $\sqrt{s_{\mathtt{NN}}}=c[(\mu-a)/b]^{d/2}$. As a validation check, the proposed approach reproduces, excellently, the rapidity spectra measured at different energies.

Resolving the partition function’s paradox of the hydrogen atom

Our protagonist here is the electronic contribution to the canonical ensemble partition function of the hydrogen atom. It is well known that it poses an intriguing paradox, (Blinder, 1995). We attempt to solve it by appeal to a mathematical zeta function’s regularization technique.

Partial-Skew-Orthogonal Polynomials and Related Integrable Lattices with Pfaffian Tau-Functions

Skew-orthogonal polynomials (SOPs) arise in the study of the n-point distribution function for orthogonal and symplectic random matrix ensembles. Motivated by the average of characteristic polynomials of the Bures random matrix ensemble studied in [22], we propose the concept of partial-skew-orthogonal polynomials (PSOPs) as a modification of the SOPs, and then the PSOPs with a variety of special skew-symmetric kernels and weight functions are addressed. By considering appropriate deformations of the weight functions, we derive nine integrable lattices in different dimensions. As a consequence, the tau-functions for these systems are shown to be expressed in terms of Pfaffians and the wave vectors PSOPs. In fact, the tau-functions also admit the representations of multiple integrals. Among these integrable lattices, some of them are known, while the others are novel to the best of our knowledge. In particular, one integrable lattice is related to the partition function of the Bures random matrix ensemble. Besides, we derive a discrete integrable lattice, which can be used to compute certain vector Pad\'e approximants. This yields the first example regarding the connection between integrable lattices and vector Pad\'e approximants, for which Hietarinta, Joshi and Nijhoff pointed out that "This field remains largely to be explored." in the recent monograph [27, Section 4.4] .

Thermodynamics of small systems with conformational transitions: The case of two-state freely jointed chains with extensible units.

Several experimental methods are usually applied for stretching single molecules and provide valuable insights about the static and dynamic responses induced by externally applied forces. This analysis is even more important for macromolecules exhibiting conformational transitions, thereby corresponding to folding/unfolding processes. With the aim of introducing the statistical mechanics of such phenomena, we apply here the spin variables approach based on a set of discrete quantities able to identify the folded or unfolded state of the chain units. First, we obtain the macroscopic thermodynamics of the chain from its microscopic description. For small systems, far from the thermodynamic limit, this result depends on the applied boundary condition (e.g., isometric or isotensional), which corresponds to the considered statistical ensemble. Then, we develop the theory for the two-state extensible freely jointed chain, where the elastic constant of the units, a property often neglected, plays a central role in defining the force-extension curve. For this system, the partition function of the isometric ensemble can be written in closed form in terms of the natural generalization of the Hermite polynomials, obtained by considering negative indices. These results are relevant for the interpretation of stretching experiments, operated from the entropic regime up to the unfolding processes.

The partition function of the Bures ensemble as the τ-function of BKP and DKP hierarchies: continuous and discrete

The relationship between matrix integrals and integrable systems was revealed more than 20 years ago. As is known, matrix integrals over a Gaussian ensemble used in random matrix theory could act as the τ-function of several hierarchies of integrable systems. In this article, we will show that the time-dependent partition function of the Bures ensemble, whose measure has many interesting geometric properties, could act as the τ-function of BKP and DKP hierarchies. In addition, if discrete time variables are introduced, then this partition function could act as the τ-function of discrete BKP and DKP hierarchies. In particular, there are some links between the partition function of the Bures ensemble and Toda-type equations.

Statistical properties and condensate fluctuation of attractive Bose gas with finite number of particles

‘We study the condensate fluctuation and several statistics of weakly interacting attractive Bose gas of 7Li atoms in harmonic trap. Using exact recursion relation we calculate canonical ensemble partition function and study the thermal evolution of the condensate. As 7Li condensate is associated with collapse, the number of condensate atom is truly finite and it facilitates to study the condensate in mesoscopic region. Being highly correlated, we utilize the two-body correlated basis function to get the many-body effective potential which is further used to calculate the energy levels. Taking van der Waals interaction as interatomic interaction we calculate several quantities like condensate fraction <n0>N, root-mean-square fluctuation δn0 and different orders of central moments. We observe the effect of finite size on the calculation of condensate fluctuations and the effect of attractive interaction over the noninteracting limit. We observe the depletion of the condensate with increase in temperature. The calculated moments nicely exhibit the mesoscopic effect. The sharp fall in the root-mean-square fluctuation near the critical point signifies the possibility of phase transition.

Particle entropy and depairing in hot nuclei

The nuclear level densities and single particle entropies are predicted for nuclei in the mass region [Formula: see text] within a framework of statistical theory of hot nuclei method. In this method, particle-number and energy conservation as well as nuclear pairing correlations are included in the partition function of grand canonical ensemble. The suppression of pairing correlations is distinctly noticed in temperature dependence of entropies between the critical temperatures [Formula: see text] MeV and [Formula: see text] MeV for [Formula: see text], [Formula: see text] and [Formula: see text] isotopes of the elements. These structural thermodynamic entropies are interpreted as a remarkable signature of the superfluid to normal phase transition connected to the vanishing of pairing gap. The calculated level densities are compared with recent experimental values. In addition, the single particle entropy of intermediate-mass nuclei is depicted as half of the entropy of mid-shell nuclei in the rare-earth region. As a consequence, the [Formula: see text] shell closure of [Formula: see text]V carries low entropy at low excitation energy presents an interesting analogy to the [Formula: see text] shell closure of [Formula: see text]Ni. Merely, in the case of odd–even [Formula: see text] has higher entropy than the even–even [Formula: see text] nucleus.

Condensate fluctuation and thermodynamics of mesoscopic Bose-Einstein condensates: A correlated many-body approach

We present a correlated many-body approach to calculate the distribution function and fluctuations for a Bose-Einstein condensate with $N$ interacting atoms in the harmonic confinement. The present formulation uses the recursion relation for the canonical ensemble partition function $(Z)$. $Z$ is calculated from the energy spectrum of the many-body effective potential, which keeps all possible two-body correlations and uses the realistic interatomic interaction. The condensate statistics are in very good agreement with earlier results of an ideal gas for which exact statistical moments for all temperature are known. We also present the numerical results of condensate statistics for real experimental situations. The calculated moments nicely exhibit the mesoscopic effect for a few hundred atoms, whereas the sharp fall in the variance for the large condensate near the critical temperature shows the possibility of phase transition. We also calculate the critical temperature for the mesoscopic regime. Our present calculation mimics the JILA experiment with $^{87}\mathrm{Rb}$ atoms.

KECSA-Movable Type Implicit Solvation Model (KMTISM).

Computation of the solvation free energy for chemical and biological processes has long been of significant interest. The key challenges to effective solvation modeling center on the choice of potential function and configurational sampling. Herein, an energy sampling approach termed the “Movable Type” (MT) method, and a statistical energy function for solvation modeling, “Knowledge-based and Empirical Combined Scoring Algorithm” (KECSA) are developed and utilized to create an implicit solvation model: KECSA-Movable Type Implicit Solvation Model (KMTISM) suitable for the study of chemical and biological systems. KMTISM is an implicit solvation model, but the MT method performs energy sampling at the atom pairwise level. For a specific molecular system, the MT method collects energies from prebuilt databases for the requisite atom pairs at all relevant distance ranges, which by its very construction encodes all possible molecular configurations simultaneously. Unlike traditional statistical energy functions, KECSA converts structural statistical information into categorized atom pairwise interaction energies as a function of the radial distance instead of a mean force energy function. Within the implicit solvent model approximation, aqueous solvation free energies are then obtained from the NVT ensemble partition function generated by the MT method. Validation is performed against several subsets selected from the Minnesota Solvation Database v2012. Results are compared with several solvation free energy calculation methods, including a one-to-one comparison against two commonly used classical implicit solvation models: MM-GBSA and MM-PBSA. Comparison against a quantum mechanics based polarizable continuum model is also discussed (Cramer and Truhlar’s Solvation Model 12).

Van der Waals type equation of state for Lennard-Jones fluid and the fluctuation of the potential energy by molecular dynamics simulations

Molecular dynamics (MD) simulations of a Lennard-Jones system are performed to obtain the pVT and UVT relations. An extended van der Waals equation of state (EOS) is derived by statistical mechanics on the perturbation approximation. A hard sphere system is used as the reference system. The Ree–Hoover EOS is assumed for the hard sphere system. The attraction energy term in the canonical ensemble partition function is extended by a cluster expansion. The new EOS includes three parameters, two of which are the interaction parameters in the Lennard-Jones interaction. The last parameter is the effective volume of the hard sphere system as the reference. The extended van der Waals EOS reproduces the pVT and UVT relations, at least qualitatively, whereas the original van der Waals EOS can explain only the pVT relation. The present EOS can also explain the fluctuations of the potential energy as calculated by MD simulations with 1000 molecules in the unit cell. In this sense, the present EOS gives better heat capacities at constant volume C v, even at lower temperatures than the original van der Waals EOS, which includes the attraction part of the equation, with no temperature dependence in the internal energy U.

Replica Exchange of Self-Guided Langevin Dynamics Simulation

Based on the self-guided Langevine dynamics (SGLD) simulation method, this work presents a guiding effect based replica exchange method (RXSGLD). The key characteristic of this method is that it can be performed at constant temperature to achieve high replica exchange efficiency. By avoiding temperature elevation, this method has a higher replica exchange efficiency for large systems and uses relatively fewer replicas compared with temperature based replica exchange methods. This method is made possible through recent progress in the understanding of the conformational distribution in SGLD simulations. The partition function of the SGLD ensemble provides a quantitative relation for the calculation of exchange probabilities between replicas. RXSGLD can be utilized as an alternative to the force-momentum based self-guided Langevin dynamics (SGLDfp) to directly sample canonical ensemble without reweighting. This method can be combined with the temperature-based replica exchange method to further improve conformational search efficiency. Using skewed double well systems, we demonstrate that RXSGLD produces correct ensemble distributions while accelerating statistical convergence. Through β-hairpin folding simulations with implicit and explicit solvent, we demonstrate that RXSGLD has excellent conformational search efficiency and can be applied to large systems, while preserving the ensemble distribution.

Three hard spheres in a spherical cavity

This work is devoted to furthering the understanding of few- and many-body inhomogeneous systems in the framework of the statistical mechanics of fluids. The three-body system consisting in three hard spheres (HS) confined in a spherical cavity at constant temperature is studied. Its canonical ensemble partition function and thermodynamic properties (such as the free energy, pressures, and fluid-substrate surface tension) are analytically obtained as a function of the cavity radius. This is the first time that a three-body fluid-like system is exactly solved. Symmetry relations between this system and its dual system composed of three HS surrounding a hard spherical object are analyzed. They allow to analytically obtain the canonical partition function of the dual system and its thermodynamic properties. Finally, the behavior of the many-body system of HS in contact with a hard spherical wall in the low density limit, is studied, focusing on the curvature dependence of the fluid-substrate surface tension and finding exact expressions for the Tolman's length and the second order term in curvature.