Grand Partition Function Research Articles

Affine symmetries for ABJM partition function and its generalization

Partially motivated by the fact that the grand partition function of the ABJM theory or its generalization is expressed by a spectral operator enjoying symmetries of the Weyl group, it was found that the grand partition function satisfies the q-Painlevé equation, which is constructed from the affine Weyl group. In this paper we clarify the affine symmetries of the grand partition function. With the affine symmetries, we find that the grand partition function extends naturally outside the fundamental domain of duality cascades and once the Painlevé equation holds in the fundamental domain, so does it outside.

Ab initio quantification of the oxygen-hemoglobin dissociation curve

Uptake of oxygen by hemoglobin (Hb), described by the oxygen-Hb dissociation curve, is obviously important for the existence of all vertebrates. Its sigmoidal curve shape indicates that oxygen binds more tightly if sites already are occupied, commonly referred to as the cooperative effect. The effect has been challenging to understand and quantify ever since its experimental demonstration in 1904. Here, we derive an ab initio analytical expression for the dissociation curve based on the fundamental principle of uniform oxygen chemical potential and absolute activity throughout the system at equilibrium using the grand partition function. The resulting analytical dissociation expression therefore only has four molecular oxygen-Hb binding energies as free variables, which are determined by fitting the analytical expression to measured data. The corresponding resulting negative reaction enthalpies identified in increasing magnitude are, ΔH1=−41.6, ΔH2=−48.8, ΔH3=−51.2, andΔH4=−51.8kJ/mol, in the range observed experimentally. The difference between ΔH1 and ΔH4 is ∼10kJ/mol, smaller than the maximum enthalpy difference measured experimentally, ∼16.7kJ/mol. Hence, the cooperative effect can therefore be explained, from an energy point of view, as caused by the reaction enthalpy difference between ΔH1 and the three subsequent enthalpy values. No impact of Hb’s spatial and structural properties is assumed. The finding highlights the importance of identifying the ligand-receptor molecular binding energies, and thereby the reaction enthalpies, under different conditions as a way for calculating not only the oxygen-Hb but ligand-receptor dissociation curves in general under various conditions, a priori, since the procedure for determining these curves ab initio has been established.

Effect of minimal length on Landau diamagnetism and de Haas–van Alphen effect

We study Landau diamagnetism in the framework of generalised uncertainty principle(GUP). We calculate the correction to magnetisation and susceptibility by constructing the grand partition function of diamagnetic material in this framework. We explicitly show that Curie’s law gets a temperature-independent correction which vanishes when quantum gravity effects are neglected. We further consider the low-temperature limit to find how GUP affects the de Haas–van Alphen effect.

On the Thermodynamics of Particles Obeying Monotone Statistics

The aim of the present paper is to provide a preliminary investigation of the thermodynamics of particles obeying monotone statistics. To render the potential physical applications realistic, we propose a modified scheme called block-monotone, based on a partial order arising from the natural one on the spectrum of a positive Hamiltonian with compact resolvent. The block-monotone scheme is never comparable with the weak monotone one and is reduced to the usual monotone scheme whenever all the eigenvalues of the involved Hamiltonian are non-degenerate. Through a detailed analysis of a model based on the quantum harmonic oscillator, we can see that: (a) the computation of the grand-partition function does not require the Gibbs correction factor n! (connected with the indistinguishability of particles) in the various terms of its expansion with respect to the activity; and (b) the decimation of terms contributing to the grand-partition function leads to a kind of "exclusion principle" analogous to the Pauli exclusion principle enjoined by Fermi particles, which is more relevant in the high-density regime and becomes negligible in the low-density regime, as expected.

Structure ordering and glass transition in size-asymmetric ternary mixtures of hard spheres: Variation from fragile to strong glasses.

We investigate the structure and activated dynamics of a binary mixture of colloidal particles dispersed in a solvent of much smaller-sized particles. The solvent degrees of freedom are traced out from the grand partition function of the colloid-solvent mixture which reduces the system from ternary to effective binary mixture of colloidal particles. In the effective binary mixture colloidal particles interact via effective potential that consists of bare potential plus the solvent-induced interaction. Expressions for the effective potentials and pair correlation functions are derived. We used the result of pair correlation functions to determine the number of particles in a cooperatively reorganizing cluster (CRC) in which localized particles form "long-lived" nonchemical bonds with the central particle. For an event of relaxation to take place these bonds have to reorganize irreversibly, the energy involved in the processes is the effective activation energy of relaxation. Results are reported for hard sphere colloidal particles dispersed in a solvent of hard sphere particles. Our results show that the concentration of solvent can be used as a control parameter to fine-tune the microscopic structural ordering and the size of CRC that governs the glassy dynamics. We show that a small variation in the concentration of solvent creates a bigger change in the kinetic fragility which highlights a wide variation in behavior, ranging from fragile to strong glasses. We conclude that the CRC which is determined from the static pair correlation function and the fluctuations embedded in the system is probably the sole player in the physics of glass transition.

Canonical partition function and center symmetry breaking in finite density lattice gauge theories

We study the nature of the phase transition of lattice gauge theories at high temperature and high density by focusing on the probability distribution function, which represents the probability that a certain density will be realized in a heat bath. The probability distribution function is obtained by creating a canonical partition function fixing the number of particles from the grand partition function. However, if the Z_3 center symmetry, which is important for understanding the finite temperature phase transition of SU(3) lattice gauge theory, is maintained on a finite lattice, the probability distribution function is always zero, except when the number of particles is a multiple of 3. For U(1) lattice gauge theory, this problem is more serious. The probability distribution becomes zero when the particle number is nonzero. This problem is essentially the same as the problem that the expectation value of the Polyakov loop is always zero when calculating with finite volume. In this study, we propose a solution to this problem. We also propose a method to avoid the sign problem, which is an important problem at finite density, using the center symmetry. In the case of U(1) lattice gauge theory with heavy fermions, numerical simulations are actually performed, and we demonstrate that the probability distribution function at a finite density can be calculated by the method proposed in this study. Furthermore, the application of this method to QCD is discussed.

Lee-Yang zeros in the Rydberg atoms

Lee-Yang (LY) zeros play a fundamental role in the formulation of statistical physics in terms of (grand) partition functions, and assume theoretical significance for the phenomenon of phase transitions. In this paper, motivated by recent progress in cold Rydberg atom experiments, we explore the LY zeros in classical Rydberg blockade models. We find that the distribution of zeros of partition functions for these models in one dimension (1d) can be obtained analytically. We prove that all the LY zeros are real and negative for such models with arbitrary blockade radii. Therefore, no phase transitions happen in 1d classical Rydberg chains. We investigate how the zeros redistribute as one interpolates between different blockade radii. We also discuss possible experimental measurements of these zeros.

Corrections to the thermodynamic quantities of Bose system by the generalized uncertainty principle

This paper investigated the Bose system in a spherical shell close to the black hole horizon. Several thermodynamic quantities of the Bose system are derived, which are different from those in the flat spacetime, by introducing the generalized uncertainty principle (GUP) into the grand partition function of statistical mechanics. The internal energy and the pressure of the Bose system appear to have a correction term of T 6, while the entropy has a T 5 correction term where both the coefficients are functions of the spacetime component g 00 and the brick wall model parameter ϵ. Taking the Schwarzschild black hole as an example, the physical quantities of the shell such as temperature, pressure and entropy are calculated for the final stage of black hole radiation.

Coarse-grained Hamiltonian and effective one component theory of colloidal suspensions

We develop a theory to trace out the solvent degrees of freedom from the grand partition function of colloid-solvent mixtures. Our approach to coarse-graining is based on density functional formalism of density profile and the grand thermodynamic potential of solvent. The solvent-induced interaction which is many-body in character is expressed in terms of two functionals; one that couples the solvent to the colloidal density distribution and the second represents the density–density correlation function of the solvent. The nature, strength, and range of the potential depend on these functionals and therefore on the thermodynamic state of the solvent. The solvent-induced contribution to free energy functional is also derived. A self-consistent procedure is developed to calculate the effective potential between colloidal particles, colloid-solvent, and colloid-colloid correlation functions. The theory is used to investigate both additive and nonadditive binary hard-sphere mixtures. Results are reported for the two systems for several values of packing fractions ηb and ηs and particles diameter ratio q=σsσb where symbols b and s refer to colloid and solvent, respectively. Several interesting features are found: The short-range attractive part of the potential shows non-monotonic dependence on ηb; when ηb is increased from zero, initially the potential becomes more attractive but beyond a certain value of ηb that depends on q, the attraction starts weakening. The repulsive peaks formed at R∼(1+12nq) where R is a distance between centers of colloidal particles expressed in units of σb and n is an integer, become stronger on increasing ηb. These results show that many-body contribution to the effective potential depends in a subtle way on packing fractions ηb,ηs, size ratio q, and on nature of the interaction model and makes a non-negligible contribution to the coarse-grained Hamiltonian.

Exact results for interacting hard rigid rotors on a d-dimensional lattice

We study the entropy of a set of identical hard objects, of general shape, with each object pivoted on the vertices of a d-dimensional regular lattice of lattice spacing a, but can have arbitrary orientations. When the pivoting point is situated asymmetrically on the object, we show that there is a range of lattice spacings a, where in any orientation, a particle can overlap with at most one of its neighbors. In this range, the entropy of the system of particles can be expressed exactly in terms of the grand partition function of coverings of the base lattice by dimers at a finite negative activity. The exact entropy in this range is fully determined by the second virial coefficient. Calculation of the partition function is also shown to be reducible to that of the same model with discretized orientations. We determine the exact functional form of the probability distribution function of orientations at a site. This depends on the density of dimers for the given activity in the dimer problem, which we determine by summing the corresponding Mayer series numerically. These results are verified by numerical simulations.

Phase behavior of a cell model with Curie-Weiss interaction

The object of this study is a cell model with Curie-Weiss interaction potential, which is a microscopic counterpart of the mean-field approximation for a continuous system of interacting particles. We have already proved (Kozitsky et al., 2018, 2020) the possibility of a mathematically rigorous transition from a continuous system of interacting particles to such a model and made an accurate calculation of its grand partition function. In the present research, we derive an explicit accurate equation of state of the cell model with Curie-Weiss potential. It turned out that this model has a sequence of first-order phase transitions at temperatures below the critical Tc. We analyzed the mechanism of these transitions based on the behavior of the chemical potential as a function of density. Thus we obtained the points of phase coexistence without going beyond the microscopic approach. We also proposed a mathematically strict definition of the critical temperature as a function of the microscopic parameters of the model. We determined the parameters of the critical points and plotted phase diagrams which cover the area of the first three phase transitions in the sequence.

Modified Yang–Lee theory for nonlocal gravitational potential and their phase transition

In this paper, we describe the extension to study the thermodynamics of the structure formation in the large-scale Universe in the nonlocal gravity formalism using standard statistical mechanics. From the derivation of the grand partition function in a modified version of the Yang–Lee theory, we obtained the corresponding thermodynamics properties that can be in consistency with a Bose–Einstein dark matter framework and derive its gravitational phase transition.

On the Thermodynamics of the q-Particles.

Since the grand partition function for the so-called q-particles (i.e., quons), , cannot be computed by using the standard 2nd quantisation technique involving the full Fock space construction for , and its q-deformations for the remaining cases, we determine such grand partition functions in order to obtain the natural generalisation of the Plank distribution to . We also note the (non) surprising fact that the right grand partition function concerning the Boltzmann case (i.e., ) can be easily obtained by using the full Fock space 2nd quantisation, by considering the appropriate correction by the Gibbs factor in the n term of the power series expansion with respect to the fugacity z. As an application, we briefly discuss the equations of the state for a gas of free quons or the condensation phenomenon into the ground state, also occurring for the Bose-like quons .

Algebraic area enumeration of random walks on the honeycomb lattice.

We study the enumeration of closed walks of given length and algebraic area on the honeycomb lattice. Using an irreducible operator realization of honeycomb lattice moves, we map the problem to a Hofstadter-like Hamiltonian and show that the generating function of closed walks maps to the grand partition function of a system of particles with exclusion statistics of order g=2 and an appropriate spectrum, along the lines of a connection previously established by two of the authors. Reinterpreting the results in terms of the standard Hofstadter spectrum calls for a mixture of g=1 (fermion) and g=2 exclusion particles whose properties merit further studies. In this context we also obtain some unexpected Fibonacci sequences within the weights of the combinatorial factors appearing in the counting of walks.

Thermodynamics of the System of Massive Dirac Fermions in a Uniform Magnetic FieldSupported by the National Natural Science Foundation of China (Grant Nos. 11890713, 11735007, 11890711, and 11947228), and the Chinese Postdoctoral Science Foundation (Grant No. 2019M662316).

We construct the grand partition function of the system of massive Dirac fermions in a uniform magnetic field from Landau levels, through which all thermodynamic quantities can be obtained. Making use of the Abel–Plana formula, these thermodynamic quantities can be expanded as power series with respect to the dimensionless variable b = 2eB/T 2. The zero-field magnetic susceptibility is expanded at zero mass, and the leading order term is logarithmic. We also calculate scalar, vector current, axial vector current and energy-momentum tensor of the system through ensemble average approach. Mass correction to chiral separation effect is discussed. For massless chiral fermions, our results recover the chiral magnetic effect for right- and left-handed fermions, as well as chiral separation effect.

Study of Structural Behavior of Cadmium Based Alloys at Molten State

The simple statistical model or simple theory of mixing has been used to study the structural behavior of cadmium based alloys at their molten state at a temperature of 800 K by computing thermodynamic functions and structural functions. The thermodynamic functions include free energy of mixing (GM), activity (a), the heat of mixing (HM), and the entropy of mixing (SM). The structural functions include concentration fluctuation in the long-wavelength limit (SGG(0)) and Warren-Cowley short-range order parameter (α1). Interchange energy or interaction energy or ordering energy (ω) was calculated for the respective alloys system and found to be positive and temperature-dependent. Based on interchange energy (ω) and coordination number (Z), theoretical values of all the functions are calculated by applying the grand partition function. All the computed values for the mentioned functions are in good agreement with experimental values. For the cadmium based alloys, viz., Cd-Zn & Cd-In, both show the segregating in nature at temperature 800 K for the concentration of range 0.1 to 0.9, however, Cd-Zn is more segregating than Cd-In.

Hamiltonian and exclusion statistics approach to discrete forward-moving paths.

We use a Hamiltonian (transition matrix) description of height-restricted Dyck paths on the plane in which generating functions for the paths arise as matrix elements of the propagator to evaluate the length and area generating function for paths with arbitrary starting and ending points, expressing it as a rational combination of determinants. Exploiting a connection between random walks and quantum exclusion statistics that we previously established, we express this generating function in terms of grand partition functions for exclusion particles in a finite harmonic spectrum and present an alternative, simpler form for its logarithm that makes its polynomial structure explicit.

Second quantization and the spectral action

We consider both the bosonic and fermionic second quantization of spectral triples in the presence of a chemical potential. We show that the von Neumann entropy and the average energy of the Gibbs state defined by the bosonic and fermionic grand partition function can be expressed as spectral actions. It turns out that all spectral action coefficients can be given in terms of the modified Bessel functions. In the fermionic case, we show that the spectral coefficients for the von Neumann entropy, in the limit when the chemical potential μ approaches 0, can be expressed in terms of the Riemann zeta function. This recovers a result of Chamseddine-Connes-van Suijlekom.

SU(N) q-Toda equations from mass deformed ABJM theory

It is known that the partition functions of the U(N)k × U(N + M)−k ABJM theory satisfy a set of bilinear relations, which, written in the grand partition function, was recently found to be the q-Painlevé III3 equation. In this paper we have suggested that a similar bilinear relation holds for the ABJM theory with mathcal{N} = 6 preserving mass deformation for an arbitrary complex value of mass parameter, to which we have provided several non-trivial checks by using the exact values of the partition function for various N, k, M and the mass parameter. For particular choices of the mass parameters labeled by integers ν, a as m1 = m2 = −πi(ν − 2a)/ν, the bilinear relation corresponds to the q-deformation of the affine SU(ν) Toda equation in τ-form.

α-boson gas model based on α-additive entropy

In this paper, we consider the grand partition function for an α-boson gas. First, we revisit the thermodynamic laws and functions in this formalism. Then, we explore α-boson gas condensation and show that the critical temperature at which α-boson condensation begins depends on the shape of the container. In particular, we compute the critical temperature for a cubic container. Finally, we discuss the α-deformed virial expansion for α-boson gas.