Canonical Partition Function Research Articles

In silico concurrent multisite pH titration in proteins.

The concurrent proton binding at multiple sites in macromolecules such as proteins and nucleic acids is an important yet challenging problem in biochemistry. We develop an efficient generalized Hamiltonian approach to attack this issue. Based on the previously developed generalized-ensemble methods, an effective potential energy is constructed which combines the contributions of all (relevant) protonation states of the molecule. The effective potential preserves important phase regions of all states and, thus, allows efficient sampling of these regions in one simulation. The need for intermediate states in alchemical free energy simulations is greatly reduced. Free energy differences between different protonation states can be determined accurately and enable one to construct the grand canonical partition function. Therefore, the complicated concurrent multisite proton titration process of protein molecules can be satisfactorily simulated. Application of this method to the simulation of the pKa of Glu49, Asp50, and C-terminus of bovine pancreatic trypsin inhibitor shows reasonably good agreement with published experimental work. This method provides an unprecedented vivid picture of how different protonation states change their relative population upon pH titration. We believe that the method will be very useful in deciphering the molecular mechanism of pH-dependent biomolecular processes in terms of a detailed atomistic description.

Field-theoretic description of charge regulation interaction.

In order to find the exact form of the electrostatic interaction between two proteins with dissociable charge groups in aqueous solution, we have studied a model system composed of two macroscopic surfaces with charge dissociation sites immersed in a counterion-only ionic solution. Field-theoretic representation of the grand canonical partition function is derived and evaluated within the mean-field approximation, giving the Poisson-Boltzmann theory with the Ninham-Parsegian boundary condition. Gaussian fluctuations around the mean field are then analyzed in the lowest-order correction that we calculate analytically and exactly, using the path integral representation for the partition function of a harmonic oscillator with time-dependent frequency. The first-order (one loop) free-energy correction gives the interaction free energy that reduces to the zero-frequency van der Waals form in the appropriate limit but in general gives rise to a monopolar fluctuation term due to charge fluctuation at the dissociation sites. Our formulation opens up the possibility to investigate the Kirkwood-Shumaker interaction in more general contexts where their original derivation fails.

S-matrix approach to the equation of state of dilute nuclear matter

Based on the general analysis of the grand canonical partition function in the S-matrix framework, a method is presented to calculate the equation of state of dilute warm nuclear matter. The result is a model-independent virial series for the pressure and density that systematically includes contributions from all the ground and excited states of all the stable nuclear species and their scattering channels. The multiplicity distribution of these species to keep the matter in statistical equilibrium is found out and then the pressure, incompressibility and the symmetry energy of the system are evaluated. The calculated symmetry energy coefficients are found to be in fair agreement with the recent experimental data.

Review of Feynman’s Path Integral in Quantum Statistics: from the Molecular Schrödinger Equation to Kleinert’s Variational Perturbation Theory

AbstractFeynman’s path integral reformulates the quantum Schrödinger differential equation to be an integral equation. It has been being widely used to compute internuclear quantum-statistical effects on many-body molecular systems. In this Review, the molecular Schrödinger equation will first be introduced, together with the Born-Oppenheimer approximation that decouples electronic and internuclear motions. Some effective semiclassical potentials, e.g., centroid potential, which are all formulated in terms of Feynman’s path integral, will be discussed and compared. These semiclassical potentials can be used to directly calculate the quantum canonical partition function without individual Schrödinger’s energy eigenvalues. As a result, path integrations are conventionally performed with Monte Carlo and molecular dynamics sampling techniques. To complement these techniques, we will examine how Kleinert’s variational perturbation (KP) theory can provide a complete theoretical foundation for developing non-sampling/non-stochastic methods to systematically calculate centroid potential. To enable the powerful KP theory to be practical for many-body molecular systems, we have proposed a new path-integral method: automated integration-free path-integral (AIF-PI) method. Due to the integration-free and computationally inexpensive characteristics of our AIF-PI method, we have used it to performab initiopath-integral calculations of kinetic isotope effects on proton-transfer and RNA-related phosphoryl-transfer chemical reactions. The computational procedure of using our AIF-PI method, along with the features of our new centroid path-integral theory at the minimum of the absolute-zero energy (AMAZE), are also highlighted in this review.

Adsorption and diffusion of the antiparkinsonian drug amantadine in carbon nanotubes

Using molecular simulations, we study the adsorption and diffusion of an antiparkinsonian drug known as amantadine (or 1-aminoadamantane) in different single-walled carbon nanotubes (SWCNs) with diameters ranging from 10.9 to 27.1 Å. First, we compute the adsorption isotherm for each SWCN at using the recently developed expanded Wang–Landau (EWL) simulations. This method allows for a direct computation of the grand canonical partition function of the system studied which, in turn, gives access to thermodynamics properties important for the adsorption process. The EWL approach is especially well suited to study the adsorption of large molecules in nanoporous materials since it combines the advantages of the expanded ensemble, in which the insertion and deletion of molecules is divided into a large number of steps, and of the Wang–Landau sampling scheme, which ensures uniform sampling of the number of molecules adsorbed in the porous material. Second, to characterise transport in these systems, we use molecular dynamics simulations to determine how self-diffusivity varies with the loading in amantadine. This allows us to shed light on the interplay between structure and transport in the nanoconfined fluid. In particular, we identify a sharp change in the rate at which the self-diffusivity decreases with the loading and relate it to a transition of the confined fluid from a single cylindrical layer to a bilayer, when additional adamantine molecules are adsorbed in the central region of the pore.

Notes on Mayer expansions and matrix models

Mayer cluster expansion is an important tool in statistical physics to evaluate grand canonical partition functions. It has recently been applied to the Nekrasov instanton partition function of N=2 4d gauge theories. The associated canonical model involves coupled integrations that take the form of a generalized matrix model. It can be studied with the standard techniques of matrix models, in particular collective field theory and loop equations. In the first part of these notes, we explain how the results of collective field theory can be derived from the cluster expansion. The equalities between free energies at first orders is explained by the discrete Laplace transform relating canonical and grand canonical models. In a second part, we study the canonical loop equations and associate them with similar relations on the grand canonical side. It leads to relate the multi-point densities, fundamental objects of the matrix model, to the generating functions of multi-rooted clusters. Finally, a method is proposed to derive loop equations directly on the grand canonical model.

Leading Fisher Partition Function Zeros as Indicators of Structural Transitions in Macromolecules

We show that leading zeros, i.e., isolated complex zeros of the canonical partition function near the positive real axis, are robust indicators of conformational transitions in finite macromolecular systems. For a flexible polymer with 250 monomers, we calcu- late the complex Fisher partition function zeros from a precise estimate of the density of states that was obtained via advanced multicanonical Monte Carlo simulations. Transition temperatures estimated from the leading zeros correspond very well to regions of enhanced thermal activity as signaled by canonical thermodynamic quantities such as the specific heat and fluctuations of the radius of gyration.

Adsorption of hydrogen in covalent organic frameworks using expanded Wang–Landau simulations

Using expanded Wang–Landau simulations, we study the adsorption of hydrogen in a series of covalent organic frameworks (COF-102, COF-105 and COF-108). In the present implementation, this new method, based on the Wang–Landau sampling, allows for a uniform sampling of the number of molecules adsorbed in the COFs. We show that this method leads to a highly accurate estimate of the grand canonical partition function for the system and gives excellent results for the adsorption isotherms of , both at and , when compared to prior simulation work and experiments. Moreover, various thermodynamic properties of adsorption, including entropy, Gibbs free energy and Helmohltz free energy, as well as the desorption properties are calculated, providing insight into the adsorption phenomenon. We finally propose a ranking for the performance of the three COFs for adsorption, based on the isothermal work required to regenerate the adsorbent , and find the following order for applications at ambient temperature.

A derivation of the Grand Canonical Partition Function for systems with a finite number of binding sites using a Markov chain model for the dynamics of single molecules

We use a Markov chain to model the ligand binding dynamics of a single molecule and show that its stationary distribution coincides with the laws of the Grand Canonical Ensemble. This way of deriving the equilibrium laws has the following advantages: Firstly, the derivation is short and does not require the knowledge of the Microcanonical, Canonical or Grand Canonical Ensemble. Secondly, it provides a descriptive interpretation of the factors that contribute to the probability of a microstate. In this regard, it also shows that the chemical activity, which cannot be regarded as a probability (since it is not necessarily bounded by one), can be interpreted as the ratio of two probabilities. Thirdly, our approach allows modeling how the system reaches equilibrium. This can be a useful tool for the study of non-equilibrium states.

Classical limit of the canonical partition function

We analyse the so-called classical limit of the quantum-mechanical canonical partition function. In order to do that, we accurately define the density matrix for symmetrized and antisymmetrized wave functions only (Bose–Einstein and Fermi–Dirac), and find an exact relation between them and the density matrix for non-symmetrized functions (Maxwell–Boltzmann). Our results differ from the generally assumed in a numerical factor N!, for which we suggest a physical interpretation. We derive as well the reverse (and also exact) relation, to find the canonical partition function for non-symmetrized wave functions in terms of the corresponding function for fermions and bosons.

Partition Function of the Model of Perfect Gas of Clusters for Interacting Fluids

The last paper of A. Fronczak presents a new, combinatorial approach to the model of perfect gas of clusters of interacting fluids. In the paper the enumerative properties and combinatorial meaning of Bell polynomials have been used to reveal some properties of systems described by grand canonical ensemble . In this paper, an exact proof of one of the important assertions from the work mentioned above is given. The assertion states that for the system in which one can distinguish k non interacting clusters the canonical partition function is given by Bell polynomial of the derivatives of the grand-thermodynamic potential with respect to the fugacity. Some possible applications of the considered theorem are presented.

Partition function zeros and phase transitions for a square-well polymer chain

The zeros of the canonical partition functions for flexible square-well polymer chains have been approximately computed for chains up to length 256 for a range of square-well diameters. We have previously shown that such chain molecules can undergo a coil-globule and globule-crystal transition as well as a direct coil-crystal transition. Here we show that each of these transitions has a well-defined signature in the complex-plane map of the partition function zeros. The freezing transitions are characterized by nearly circular rings of uniformly spaced roots, indicative of a discontinuous transition. The collapse transition is signaled by the appearance of an elliptical horseshoe segment of roots that pinches down towards the positive real axis and defines a boundary to a root-free region of the complex plane. With increasing chain length, the root density on the circular ring and in the space adjacent to the elliptical boundary increases and the leading roots move towards the positive real axis. For finite-length chains, transition temperatures can be obtained by locating the intersection of the ellipse and/or circle of roots with the positive real axis. A finite-size scaling analysis is used to obtain transition temperatures in the long-chain (thermodynamic) limit. The collapse transition is characterized by crossover and specific-heat exponents of φ≈0.76(2) and α≈0.66(2), respectively, consistent with a second-order phase transition.

Thermodynamics of quantum isolated horizons with model Hamiltonians

Following a recent proposal, we consider the most general structure possible for the Hamiltonian operator associated with the Quantum Isolated Horizon(QIH) with explanations of the underlying physical motivations. An extensive thermodynamic analysis with this model Hamiltonian is presented. Considering {\it fixed} number of punctures, the canonical partition function can now be written in the usual energy ensemble. Arguing that the known classical results must follow in the correspondence limit (viz. the equilibrium temperature as observed by a local observer) the model is fixed, yielding the energy spectrum of the QIH.

Lee yang theory for cosmological many body problem

We study the phase transition in a gravitating system by analyzing grand canonical partition function as a function of complex fugacity. We extend the Yang-Lee theory to study phase transitions in the gravitational galaxy clustering of galaxies having a variety of masses. This generalizes our previous work based on the same theory for the single-component system to a multicomponent system. We find that galaxy clustering is sensitive to masses and number densities of individual galaxies at early stages while at later stages collective behavior of the particles is more pronounced. This validates our earlier work obtained from different considerations.

Subsets of configurations and canonical partition functions

We explain the physical nature of the subset solution to the sign problem in chiral random matrix theory: The subset sum is shown to project out the canonical determinant with zero quark charge from a given configuration. As the grand canonical chiral random matrix partition function is independent of the chemical potential, the zero quark charge sector provides the full result.

Stability of quantum isolated horizon: a local observer's view

It is shown that a quantum isolated horizon (QIH), as observed by a local observer, is locally unstable as a thermodynamic system. The result is derived in two different ways. Firstly, the specific heat of the QIH is shown to be negative definite through a quantum statistical analysis. Then, it is shown, in the thermal holographic approach, that the canonical partition function of the QIH diverges under Gaussian thermal fluctuations of such energy spectrum, implying local instability of such a QIH as a thermodynamic system.

Simple method for exact calculation of thermodynamic properties of the 1D Hubbard model with infinite repulsion

It is shown that the canonical partition function in the 1D Hubbard model with U = ∞ in the nearest neighbor approximation is determined by the product of canonical partition functions of spinons and holons. In this approximation, the concentration and temperature dependences of the free and internal energies, as well as of the chemical potential, entropy, and heat capacity, are calculated for electron concentrations of 0 ≤ ne < 1.

Stability of quantum isolated horizon: a local observer's view

It is shown that a quantum isolated horizon (QIH), as observed by a local observer, is locally unstable as a thermodynamic system. The result is derived in two different ways. Firstly, the specific heat of the QIH is shown to be negative definite through a quantum statistical analysis. Then, it is shown, in the thermal holographic approach, that the canonical partition function of the QIH diverges under Gaussian thermal fluctuations of such an energy spectrum, implying local instability of such a QIH as a thermodynamic system.

Implications of Poincaré symmetry for thermal field theories in finite-volume

The analytic continuation to an imaginary velocity $i\xi$ of the canonical partition function of a thermal system expressed in a moving frame has a natural implementation in the Euclidean path-integral formulation in terms of shifted boundary conditions. Writing the Boltzmann factor as $\exp[-L_0(H-i\xi.P)]$, the Poincare invariance underlying a relativistic theory implies a dependence of the free-energy on $L_0$ and the shift $\xi$ only through the combination $\beta= L_0 \sqrt{1+\xi^2}$. This in turn implies a set of Ward identities, some of which were previously derived by us, among the correlators of the energy-momentum tensor. In the infinite-volume limit they lead to relations among the cumulants of the total energy distribution and those of the momentum, i.e. they connect the energy and the momentum distributions in the canonical ensemble. In finite volume the Poincare symmetry translates into exact relations among partition functions and correlation functions defined with different sets of (generalized) periodic boundary conditions. They have interesting applications in lattice field theory. In particular, they offer Ward identities to renormalize non-perturbatively the energy-momentum tensor and novel ways to compute thermodynamic potentials. At fixed bare parameters they also provide a simple method to vary the temperature in much smaller steps than with the standard procedure.

Inequivalence of Canonical and Grand Canonical Ensembles for Bosonic Systems

For many-particle quantum systems, calculating thermodynamic quantities in the canonical ensemble is a very hard task, while this is tractable in the grand canonical ensemble. The second ensemble is then used. The results are supposed to be the same, at least in the thermodynamic limit. Is this actually the case? In this work, we consider a system of N noninteracting bosons distributed among few energy levels. We can calculate the canonical partition function in this case and deduce the canonical mean energy. We compare it to the mean energy deduced from the grand canonical ensemble for the same number of particles. We consider the case of a large number of particles.