Mayer Cluster Expansion Research Articles

Statistical mechanical refinement of protein structure prediction schemes. II. Mayer cluster expansion approach

We investigate the use of a Mayer cluster expansion to quantify changes to the energy landscape of an ab initio protein structure prediction Hamiltonian as side-chain excluded volume radii are increased. The expansion is generalized so that the effect of other changes in the Hamiltonian, which would alone be best treated with a cumulant expansion, can be evaluated simultaneously. The effect of changing the excluded volume radii on various thermodynamic quantities, in particular the free energy and glass transition temperature, is calculated with the expansion and compared to results from simulation. The expansion is found to be sufficiently accurate at the second-order level to be useful for calculating values of the excluded volume radii that refine the energy landscape for structure prediction. This is explicitly checked by a free energy profile analysis and examining the quality of predictions before and after the optimization. A useful method to test whether equilibrium has been attained during molecular dynamics sampling is also described.

Equation of state for hot quark-gluon plasma transitions to hadrons with full QCD potential

A practical method based on Mayer's cluster expansion to calculate critical values for a quark-gluon plasma (QGP) phase transition to hadrons is represented. It can be applied to a high-temperature QGP for clustering of quarks to mesons and baryons. The potential used is the Cornell potential, i.e., a potential containing both confining and gluon exchange terms. Debye screening effects are included. An equation of state (EOS) for hadron production is found by analytical methods, which is valid near the critical point. The example of the formation of $J/\ensuremath{\psi}$ and \ensuremath{\Upsilon} is recalculated. It is shown that in the range of temperatures available by today's accelerators, the latter particles are suppressed. This is further confirmation for heavy quarkonia suppression and, hence, for a signature of a QGP. The EOS presented here also shows that in future colliders there will be no heavy quarkonia production by the mechanism of phase transition. Hence, if there will be heavy quarkonia production, it must be based on some other mechanisms, perhaps on the basis of some recently suggested possibilities.

The equation of state of a Gaussian phantom network with defined cross-link functionality

A phantom network of Gaussian chains far from the point of gelation can be described as a gas of interacting particles represented by the cross-links. The type of particles varies with the network functionality, whereas the type of interaction depends on the properties of the connecting chains. In a mean field approximation the equation of state can be calculated using Mayer's cluster expansion. The resulting isothermal compressibility is compared for different cross-link functionalities.

Statistical theory of a solvated electron in an electrolyte

The behavior of a solvated electron in an electrolyte is investigated. The formalism of the theory is based on variational estimation of path integrals. It reduces the problem to the investigation of the self-consistent mean field produced by the ions and the electron. Mayer cluster expansions make it possible to take account of the short-range interactions and to find expressions for the effective potential of the electron and the electron-ion and electron-neutral atom correlation functions as a function of the macro-and microscopic parameters of electrolytes. In the limit of high ion densities the behavior of the electron is determined solely by the Coulomb interaction, which results in the formation of a polaron state. This state of the electron is virtually independent of the thermodynamic parameters of the electrolyte. In the opposite limit of low ion densities the electron forms a cavity state. The presence of ions results in additional localization of the electron and is manifested experimentally as a shift of the absorption band in the direction of high energies. The estimated shift for a hydrated electron agrees with the experimental data.

Quadratic fluctuation-dissipation theorem for multilayer plasmas

The author establishes the dynamical and static quadratic fluctuation-dissipation theorems (QFDTs) for multilayer classical one-component plasmas in the absence of external magnetic fields. Areal densities and spacings between layers need not be equal. The static QFDT is used to derive the lowest-order (in coupling parameter) Mayer cluster expansion for the layer-space matrix elements of the equilibrium three-point correlation function.

On “vapourliquid” phase equilibrium in a finite system

A system of a finite number of particles (atoms, molecules) is considered under the condition N. T. V.  const. A “one drop-vapour” approximation is proposed, adapting the Mayer cluster-expansion formalism to describe the heterogeneous equilibrium state of the system. The mean cluster size distribution function obtained is compared with computer simulation results by Rao, Berne and Kalos and shows quite good agreement. The difficulties of applying the Boltzmann entropy hypothesis to the heterogenous (i.e. bistable) state of the system are discussed.

Surface electronic excitations and dynamic spectral properties of adsorbates

Many-body aspects of screening and relaxation mechanisms encountered in spectroscopic studies of the electronic structure of adsorbates have recently attracted considerable attention from both experimental and theoretical physicists. Interest in these phenomena has also been augmented by the rapid improvement of experimental techniques which have enabled better resolution and analysis of various subtile components of the adsorbate spectra. Many of these spectral features have until recently been ascribed to purely chemical and initial state effects. One of the first major advances of the theoretical development in this field was to seek and attribute the origin of these structures to the many-body properties of adsorption systems and, secondly, to predict how the many-body effects would manifest themselves in surface spectroscopies. We start with a rather detailed description of the formalism of the surface electronic response and discuss the properties of the surface excitation spectrum of idealized and real metals. This formalism is then successively applied to set up a model of screening firstly in the nonbonding levels of mainly physisorbed adsorbates and later in the core and valence levels of chemisorbed species. Various modifications of the model enable a dynamic description of the final state relaxation and shake-up effects typical of spectroscopic measurements. To treat the particularly complicated problem of dynamic relaxation in the adsorbate valence levels a special perturbational approach based on Mayer's cluster expansion is developed in §5. The characteristics, and some limitations of this approach, which may also prove useful in other physical problems, are described in detail and discussed within the context of the interaction of localized adsorbate charge fluctuations with bosonic surface excitations. Experimental support for the presented theoretical framework and its applications has been very important. A qualitative comparison with the experimental findings has been given for the majority of effects predicted and some still existing controversies and interpretational problems have been pointed out.

Properties of Mayer cluster expansion

The Umbral algebra developed by Rota and his co-workers is used to show that the Mayer cluster expansion of the canonical partition function is related to the Bell polynomials. The algebra is also used to find a representation of the partition function and a rederivation of Mayer’s first theorem. Finally, it is shown that in the ’’tree approximation’’ for the cluster integrals, the summation of Mayer’s expression for the canonical-ensemble partition function for a finite number of particles could be performed using Dénes’ and Rényi’s theorems in graph theory.

Cluster theory for concentration dependence of shear viscosity for suspensions of interacting spheres. I

We present a general divergent-integral-free cluster theory for the concentration dependence, to any order in concentration, of the shear viscosity of a suspension of interacting colloidal spheres. Treating the solvent as a viscous hydrodynamic continuum, the steady state limit of vanishing shear rate is considered, so the sphere–sphere interactions enter through the Mayer cluster expansion of the various sphere equilibrium distribution functions. The spheres are taken to move such that the net hydrodynamic forces and torques acting on them identically vanish. We present general formulas for all coefficients in a virial type expansion of the suspension shear viscosity in powers of the concentration for any prescribed spherically symmetric direct interaction potential between the spheres. Previous work has been limited to the two-sphere contributions only. Every virial coefficient in our complete expansion involves integrals of the sphere equilibrium distribution functions multiplied by functions of hydrodynamic origin. These virial coefficients are perfectly convergent because of the nature of the various equilibrium distribution functions present in the integrals for the virial coefficients.

The exact fourth and fifth virial coefficients of an inverse-6 potential

Numerical values are calculated for all of the four- and five-particle diagrams in the Mayer cluster expansion of the equation of state for an inverse-6 potential. These diagrams are then summed with the appropriate weightings to give accurate values for the fourth and fifth virial coefficients, which are found to be B4 = 0.02820(B2)3 and B5 = −0.0104(B2)4, where B2 is the second virial coefficient.

Thermodynamics of a 1—d fermion system by cluster expansions

The thermodynamics of a 1—d fermion system can be perfectly mapped onto the thermodynamics of a two-component classical real gas on the surface of a cylinder. The latter system is studied by a modified Mayer cluster expansion method (Debye-Huckel theory and corrections to it). Simple reinterpretation of thermodynamic variables then leads to an exponentially activated behavior of the fermions' specific heat (spin density contribution). The exponent in Luther's recent gap formula is obtained as well as Zittartz's structure of the ground state energy. The method is applied to the case of strongly attractive spin-non-flip coupling (−1<γ∥<−3/5) when the spin-flip couplingγ⊥ is small. For largeγ⊥ there is a gap everywhere.

Correlation analysis of the space and surface distribution of galaxies

AbstractThe paper presents a number of calculations which are useful for the interpretation of catalogues of galaxies. It is suggested that a correlation function approach based on a MAYER cluster expansion for the N‐point probability function of galaxies leads to a complete description of galaxy clustering. Other statistical measures of clustering such as the distribution function for the cell population or the distribution of the next neighbour distances are shown to be reducible to a correlation approach. For the classical cell count method smoothing effect estimates are given and the ZWICKY dispersion curve technique is studied. The relation between the space distribution and the surface distribution of galaxies is discussed in cosmological models of FRIEDMANN type. Deviations from PEEBLES' scaling law occur for catalogues extending to faint magnitudes and suggest the existence of strong evolutionary effects in the distribution of galaxies. From the DODD et al. field and the Jagiellonian field one derives — comparing the data with those of the ZWICKY catalogue — that the clustering amplitude decreases like (I + z)‐s within increasing redshift, where s is of the order 6. Since the catalogues extend to small redshifts only, it is not clear if this behaviour continues to larger redshifts. Within the observationally accessible range of z, the value of s does not essentially depend neither on the galaxian luminosity function nor on the deceleration parameter q0, it depends weakly on the assumed K correction.

Degeneracy Effects in Gases in the Near-Classical Limit

The properties of Fermi gases are investigated in the near-classical limit. In this region of temperature and density the diffraction effects due to the uncertainty principle are neglected, but degeneracy effects due to the Pauli exclusion principle are included exactly. The basic approach involves the evaluation of terms in the Bloch-de Dominicis quantum-perturbation expansion of the grand canonical partition function. In the near-classical limit the perturbation terms reduce to the classical results, modified by statistics factors in the form of Fermi integrals. Using these statistics factors, a modified form of the Mayer cluster expansion that includes degeneracy is obtained. Also a new form of the Boltzmann factor including degeneracy is derived. As an illustrative example these results are applied to the calculation of the near-classical limit of the free energy for plasmas.

On the gas-liquid phase transition

A new approach to the problem of the gas-liquid phase transition, based on the Mayer cluster expansion of the partition function, is proposed. It is shown that the necessary and sufficient condition for phase transition to occur is that there exist a temperatureT= Tc > 0 such that forT ⩽Tc, all thebl (except perhaps a finite number of them) are positive, where thebl, are the cluster integrals (as defined by Mayer) in the thermodynamic limit. Explicit expressions for the isotherms for gas-saturated vapor and liquid phases are given.

Mean Field Theory and Infinite Dimensionality in Widom—Rowlinson Model

It is known for the Ising model that the mean field theory describes exactly the limit of infinite dimensionality in the one-phase region. It is shown in this article that the same statement applies also to the Widom—Rowlinson model of penetrable cubes. This model has nonpairwise-additive, many-body interactions such as one finds in the ionic solution theory. Mayer's cluster expansion scheme is used to show that the equation of state in the limit of infinite dimensionality is identical to the mean field result.

Statistics of a Two-Dimensional Gas of Long Thin Rods

A three-dimensional gas of long thin rods exhibits an order–disorder phase transition as a function of rod concentration. Theoretically, this transition has been found using the leading term in Mayer cluster expansion theory. It is shown here that if a phase transition occurs for a two-dimensional gas of long thin rods, it cannot be found within the same approximation. This is achieved by showing that the equation obeyed by the angular distribution function of the rods in this approximation has a unique constant or isotropic solution for all concentrations.

On Mayer's ionic solution theory

This theory, which is based upon McMillan and Mayer's cluster expansion of the grand partition function, is extended to include all of the components of the potentials of average force between sets of solute molecules or ions in the solvent at infinite dilution. The resulting cluster integral sum, which is a sum of integrals over an infinite number of infinite series of Mayer prototype graphs, is transformed into the sum of a single infinite series of integrals on graphs involving new types of bonds. These bonds are defined in a way first suggested by Meeron. There results a compact expression for the cluster integral sum.