Semiclassical statistical mechanics of hard-body fluid mixtures

The thermodynamic properties of semiclassical hard-body fluid mixtures are studied. Explicit expressions are given for the free-energy, equation of state and virial coefficients of the classical hard convex-body fluid mixtures. The numerical results are discussed under different conditions. The agreement with the exact data is good in all cases. The first-order quantum corrections are also studied. The quantum effects depend on the condition, shape parameters L11* and L22*, and concentrations x1 and x2 in general and increase with an increase of packing fraction eta, in particular.     

The statistical mechanics of two-dimensional vesicles

Recent work on the statistical mechanics of model membranes and vesicles in two dimensions by Leibler, Singh and the author [l,2] is briefly summarized.     

Semiclassical approach to model quantum fluids using the statistical associating fluid theory for systems with potentials of variable range

Thermodynamic properties of quantum fluids are described using an extended version of the statistical associating fluid theory for potentials of variable range (SAFT-VR) that takes into account quantum corrections to the Helmholtz free energy A, based on the Wentzel-Kramers-Brillouin approximation. We present the theoretical background of this approach (SAFT-VRQ), considering two different cases depending on the continuous or discontinuous nature of the particles pair interaction. For the case of continuous potentials, we demonstrate that the standard Wigner-Kirkwood theory for quantum fluids can be derived from the de Broglie-Bohm formalism for quantum mechanics that can be incorporated within the Barker and Henderson perturbation theory for liquids in a straightforward way. When the particles interact via a discontinuous pair potential, the SAFT-VR method can be combined with the perturbation theory developed by Singh and Sinha [J. Chem. Phys. 67, 3645 (1977); and ibid. 68, 562 (1978)]. We present an analytical expression for the first-order quantum perturbation term for a square-well potential, and the theory is applied to model thermodynamic properties of hydrogen, deuterium, neon, and helium-4. Vapor-liquid equilibrium, liquid and vapor densities, isochoric and isobaric heat capacities, Joule-Thomson coefficients and inversion curves are predicted accurately with respect to experimental data. We find that quantum corrections are important for the global behavior of properties of these fluids and not only for the low-temperature regime. Predictions obtained for hydrogen compare very favorably with respect to cubic equations of state.     

Theory for non-equilibrium statistical mechanics

This paper reviews a new theory for non-equilibrium statistical mechanics. This gives the non-equilibrium analogue of the Boltzmann probability distribution, and the generalization of entropy to dynamic states. It is shown that this so-called second entropy is maximized in the steady state, in contrast to the rate of production of the conventional entropy, which is not an extremum. The relationships of the new theory to Onsager's regression hypothesis, Prigogine's minimal entropy production theorem, the Langevin equation, the formula of Green and Kubo, the Kawasaki distribution, and the non-equilibrium fluctuation and work theorems, are discussed. The theory is worked through in full detail for the case of steady heat flow down an imposed temperature gradient. A Monte Carlo algorithm based upon the steady state probability density is summarized, and results for the thermal conductivity of a Lennard-Jones fluid are shown to be in agreement with known values. Also discussed is the generalization to non-equilibrium mechanical work, and to non-equilibrium quantum statistical mechanics. As examples of the new theory two general applications are briefly explored: a non-equilibrium version of the second law of thermodynamics, and the origin and evolution of life.     

Eigenvalues of a statistical mechanics matrix

We consider the problem of finding the spectrum of an n × n matrix which arises in the study of a certain model of long-range interactions in a one-dimensional statistical mechanics system. Our analysis exhibits a curious resemblance of the suitably normalized distribution of eigenvalues to the Marčenko–Pastur law in the limit n → ∞.     

Modeling mercury porosimetry using statistical mechanics

We consider mercury porosimetry from the perspective of the statistical thermodynamics of penetration of a nonwetting liquid into a porous material under an external pressure. We apply density functional theory to a lattice gas model of the system and use this to compute intrusion/extrusion curves. We focus on the specific example of a Vycor glass and show that essential features of mercury porosimetry experiments can be modeled in this way. The lattice model exhibits a symmetry that provides a direct relationship between intrusion/extrusion curves for a nonwetting fluid and adsorption/desorption isotherms for a wetting fluid. This relationship clarifies the status of methods that are used for transforming mercury intrusion/extrusion curves into gas adsorption/desorption isotherms. We also use Monte Carlo simulations to investigate the nature of the intrusion and extrusion processes.     

Thermodynamics and statistical mechanics of multilayer adsorption

The multilayer adsorption models of Brunauer-Emmett-Teller and Guggenheim-Anderson-de Boer are reconsidered. The relationship between the fitting parameters and the physical parameters of the equation is discussed. The preexponential factors of the parameters are shown to be in general far different from unity, contrary to a widespread use. A thermodynamical derivation illuminates the hypothesis on which the multilayer sorption equation is dependent and frees it from too restrictive hypothesis usually taken as necessary for its validity. Equations are derived for the number fraction of sorption sites occupied by different numbers of molecules. The Guggenheim-Anderson-de Boer equation is shown to imply incomplete occupation (jamming) of the first sorption layer at saturation.     

A formulation of statistical mechanics of ordered systems

Expressions are derived for the thermodynamic functions (Gibbs free energy, Helmholtz free energy, etc.) of an ordered system in terms of the single-particle distribution function,p(x), and correlation functions. The thermodynamic functions are treated as functionals of the single-particle distribution function. By minimizing the Helmholtz free energy with respect top(x) under constraints of constantT, V andN, an integral equation is obtained from whichp(x) can be determined. The correlation function of the ordered state in the region near the coexistence surface between ordered and disordered state is expanded about the correlation function of the disordered state, and the series is truncated. Methods for calculating the thermodynamic functions and the single-particle distribution function are presented, and our result is discussed in relation to other treatments of phase coexistence in the literature.     

Size-distribution of droplets in emulsions by statistical mechanics calculation

This report is concerned with theoretical demonstration of the spontaneous emulsification which has been observed in a soft contact of nitrobenzene with water without surfactant [K. Aoki, M. Li, J. Chen, T. Nishiumi, Electrochem. Commun. 11 (2009) 239]. The demonstration is based on the model of spherical oil droplets with any size in equilibrium. The droplets are composed of the smallest droplets, the total number of which is given. An assembly of small droplets has larger surface energy than that of large ones because the surface energy is proportional to the surface area. The former has larger configurational entropy than the latter because the number of small droplets is bigger than that of the large ones. Since the free energy is determined by the competition between the surface energy and the entropy, it is not clear which assembly has lower free energy. This question was solved numerically here by statistical mechanics calculation of the size distributions, which contained only a parameter of the surface energy. The results of the computation at small number of droplets were used for deriving approximate equations for extremely large number of droplets. The size distribution was localized both to the smallest and the largest droplets. The diameter of the largest droplet was estimated from the dynamics in which coalescence by diffusion of droplets is disturbed by gravitational convection. The size then predicted was of the order of micrometer, being close to experimental values.     

Phase transitions of two-dimensional dipolar fluids in external fields

In this work, we study condensation phase transitions of two-dimensional Stockmayer fluids under additional external fields using Monte-Carlo (MC) simulations in the grand-canonical ensemble. We employ two recently developed methods to determine phase transitions in fluids, namely Wang-Landau (WL) MC simulations and successive-umbrella (SU) sampling. Considering first systems in zero field (and dipolar coupling strengths μ(2)∕εσ(3) ≤ 6), we demonstrate that the two techniques yield essentially consistent results but display pronounced differences in terms of efficiency. Indeed, comparing the computation times for these systems on a qualitative level, the SU sampling turns out to be significantly faster. In the presence of homogeneous external fields, however, the SU method becomes plagued by pronounced sampling difficulties, yielding the calculation of coexistence lines essentially impossible. Employing the WL scheme, on the other hand, we find phase coexistence even for strongly field-aligned systems. The corresponding critical temperatures are significantly shifted relative to the zero-field case.     

A new cubic simplified perturbed hard-body equation of state

A new cubic equation of state (EOS) was developed in this study for vapor-liquid equilibrium (VLE) calculations of nonpolar fluids. The repulsive term of this EOS reexpressed the results of Walsh and Gubbins (1990) from their modified thermodynamic perturbation theory of polymerization into a simple form in which a non-spherical parameter was employed to account for the different shapes of molecules. The repulsive compressibility factors calculated from this EOS agree well with the molecular simulation data for various kinds of hard bodies ranging from a single hard sphere to tangent or fused long chain molecules. A simple attractive term was then coupled with the repulsive to complete the EOS in a cubic form. Equation parameters were determined for a diversity of nonpolar real fluids. These parameters were expressed in generalized forms for engineering computations. Satisfactory results from this EOS on the saturated properties of pure nonpolar fluids were obtained. This EOS was also extended to calculate the VLE of nonpolar fluid mixtures. The results are again satisfactory over wide ranges of temperature and pressure.     

Explicit factorization of external coordinates in constrained statistical mechanics models

If a macromolecule is described by curvilinear coordinates or rigid constraints are imposed, the equilibrium probability density that must be sampled in Monte Carlo simulations includes the determinants of different mass-metric tensors. In this work, the authors explicitly write the determinant of the mass-metric tensor G and of the reduced mass-metric tensor g, for any molecule, general internal coordinates and arbitrary constraints, as a product of two functions; one depending only on the external coordinates that describe the overall translation and rotation of the system, and the other only on the internal coordinates. This work extends previous results in the literature, proving with full generality that one may integrate out the external coordinates and perform Monte Carlo simulations in the internal conformational space of macromolecules.     

Quantum statistical mechanics of closed‐ring molecular chains

A model for a closed‐ring unhindered three‐dimensional macromolecular chain, based on Quantum Mechanics, is presented. Upon starting from an exact non‐relativistic Hamiltonian operator, we integrate out all electronic degrees of freedom, in the Born‐Oppenheimer framework, giving rise to an effective vibro‐rotational Hamiltonian for the chain. Then, assuming a harmonic oscillator‐like vibrational potential between nearest‐neighbour atoms, the integration of the atomic radial degrees of freedom is carried in the limit of high frequencies. Thus, all bond lengths become fixed, including the one which makes the chain to become a closed ring. This formulation leads to a specific Hamiltonian for the remaining angular variables of the closed‐ring chain, and constitutes an alternative in comparison with standard Gaussian models, which do not. Use is made of a variational inequality by Peierls to find an approximate quantum partition function for the angular variables of the system. We then proceed to obtain approximately another representation for the angular partition function in the classical limit. Several features of the classical partition function are discussed.     

A statistical mechanics description of environmental variability in metabolic networks

Many of the chemical reactions that take place within a living cell are irreversible. Due to evolutionary pressures, the number of allowable reactions within these systems are highly constrained and thus the resulting metabolic networks display considerable asymmetry. In this paper, we explore possible evolutionary factors pertaining to the reduced symmetry observed in these networks, and demonstrate the important role environmental variability plays in shaping their structural organization. Interpreting the returnability index as an equilibrium constant for a reaction network in equilibrium with a hypothetical reference system, enables us to quantify the extent to which a metabolic network is in disequilibrium. Further, by introducing a new directed centrality measure via an extension of the subgraph centrality metric to directed networks, we are able to characterise individual metabolites by their participation within metabolic pathways. To demonstrate these ideas, we study 116 metabolic networks of bacteria. In particular, we find that the equilibrium constant for the metabolic networks decreases significantly in-line with variability in bacterial habitats, supporting the view that environmental variability promotes disequilibrium within these biochemical reaction systems.     

Semiclassical approach to electrorheological fluids. Influence of solid volume fraction on the suspension yield stress

 A semiclassical approach, previously proposed in matter and momentum transport phenomena, has been employed to deal with the electrorheological effect. The vibrational Helmholtz free energy of a solidlike medium, consisting of phononic and dielectric contributions, has been developed in terms of Debye, optical and characteristic system frequencies and constrained to a stationary condition with respect to mass-density variations. It turns out that application of an electric field results in the creation of bound energy states, which can be related to the solid fraction and the yield stress through a phenomenological equation. Application to rheological data, from polyaniline particles and chlorinated paraffin in silicon oil, silica spheres in corn oil and numerical simulations, is presented. Received: 11 December 2000 Accepted: 18 August 2001     

Thermodynamic and orientational properties of two-dimensional multicomponent hard rectangle fluids

A molecular statistical model is developed for a two-dimensional multicomponent fluid composed of hard rectangles. The configurational partition function of such a system on a square lattice is evaluated by applying the orientation dependent version of the Flory-Huggins probability method. Unlike earlier treatments of similar problems, the final result for the configurational entropy is independent of the order of placement of the rectangular plate-like particles. The present approach has been extended to enable an analysis of both facewise and edgewise modes of adsorption of rectangular molecules on a planar surface. Further, the limit of infinitesimal lattice size has been considered as the generalized continuous version of the corresponding lattice model. The results obtained within the present continuous approach are compared with those of studies of related one-component hard body fluids.     

Thermodynamic and orientational properties of two-dimensional multicomponent hard rectangle fluids

A molecular statistical model is developed for a two-dimensional multicomponent fluid composed of hard rectangles. The configurational partition function of such a system on a square lattice is evaluated by applying the orientation dependent version of the Flory-Huggins probability method. Unlike earlier treatments of similar problems, the final result for the configurational entropy is independent of the order of placement of the rectangular plate-like particles. The present approach has been extended to enable an analysis of both facewise and edgewise modes of adsorption of rectangular molecules on a planar surface. Further, the limit of infinitesimal lattice size has been considered as the generalized continuous version of the corresponding lattice model. The results obtained within the present continuous approach are compared with those of studies of related one-component hard body fluids.