Ordered phases of elongated micelles

Nijenrode Castle in Breukelen, the Netherlands, situated between Utrecht and Amsterdam, was home for seventeen days during August of last year to more than sixty physicists, chemists, and advanced students from sixteen countries. These people gathered to attend an intensive course on Fundamental Problems in Statistical Mechanics, held under the auspices of NUFFIC (the Netherlands Universities Foundation for International Cooperation) with support from NATO. Statistical mechanics seemed most appropriately chosen as the subject of the first summer school in theoretical physics in the Netherlands in view of the long tradition, begun by Van der Waals and continued by Ehrenfest. of Dutch research in this field. The lecturers at Nijenrode representing that tradition as it is active in Holland today included N. G. van Kampen, B. R. A. Nijboer, L. van Hove, N. M. Hugenholtz. E. G. D. Cohen, and P. Mazur. Visiting lecturers from abroad were K. Huang of the Massachusetts Institute of Technology. H. Wergeland of the Norwegian Technical High School in Trondheim, and G. E. Uhlenbeck of the Rockefeller Institute in New York.

This paper provides an overview of the static behavior of systems with partial orientational and positional ordering of the species. The understanding of the static behavior provides the foundation needed to understand the dynamic properties of such systems, which are described in more detail in other papers in this issue. More specifically, this paper provides an overview of the fact that species of different dimensions exhibit similar types of partial ordering of the species. This paper further provides an overview of liquid crystalline (LC) phases and molecules, including discussions of types of chemical structures (both rigid and semiflexible) in LC molecules (both nonpolymeric and polymeric); thermotropic and lyotropic LCs; nematic and multiple smectic LC phases; reentrant nematic LC phases; cholesteric LC phases and blue phases; smectic crystal phases; columnar phases; micelles and lamellar LC structures; LC phase transitions; LC glasses; viscosities of LC phases; and dynamic textures and patterns in LC phases.

On the entropy minimization problem in Statistical Mechanics

Biocompatible systems formulated for use in the food, cosmetic, and pharmaceutical fields are characterized. Ternary phase diagrams of mixtures of natural lipids (glycerol trioleate, glycerol monooleate, diglycerol monooleate, and lecithin) and water were investigated by means of optical microscopy in polarized light and by multinuclear NMR spectroscopy. All systems showed a microemulsion region at high oil content and a large area of coexistence of two liquid crystalline (hexagonal and lamellar) phases. 1H and 13C NMR self-diffusion measurements were used to characterize microstructural features of the microemulsions. On water dilution, the two-phase liquid crystalline region transforms into a creamy emulsion area where the droplets of water are stabilized by both the lamellar and the hexagonal phases, as indicated by 2H NMR measurements. Due to the very effective dispersing action of the two liquid crystalline phases, these emulsions show a high stability toward phase separation.

Calcium carbonate was synthesized according to a generic three-phase synthesis route in the presence of self-assembling liquid crystalline phases, and a comparison was made between the products formed in hexagonal and reverse hexagonal liquid crystalline phases. The surfactants used to form the liquid crystalline phases were the three different triblock copolymers Pluronic P105 (hexagonal) and P104 or P123 (reverse hexagonal), consisting of poly (propylene glycol) as the middle block and poly (ethylene glycol) as the end blocks. An aqueous solution of CaCl 2 was mixed with the Pluronics and organic solvents to make up the liquid crystalline phases, which subsequently were exposed to CO 2 (g), resulting in the formation of CaCO 3 . The specific surface areas of the CaCO 3 products obtained varied from 5 to 226 m 2 / g, depending on the initial concentration of CaCl 2 . Small-angle X-ray scattering showed that the reactant salt and the CaCO 3 crystals were situated in the water domains of the liquid crystalline phases and that these remained throughout the course of the experiment. Transmission electron microscopy studies in combination with powder X-ray diffraction revealed that CaCO 3 with high specific surface areas consisted of fibrils with the vaterite structure and formed at low CaCl 2 concentrations. Samples of lower surface areas consisted of flakes with the calcite structure and formed predominantly at higher CaCl 2 concentrations. Nitrogen adsorption showed a narrow pore size distribution for some of the materials prepared in the reverse hexagonal phases. Poly(acrylic acid) added to a hexagonal phase increased the specific surface area of the CaCO 3 substantially, whereas when added to a reverse hexagonal phase no increase in surface area was found. Powder X-ray diffraction showed that higher specific surface areas obtained with poly(acrylic acid) were associated with the formation of a larger fraction of amorphous CaCO 3 .

Summation of power series by self-similar factor approximants

For quantum spin systems it is known that for a suitable space of potentials the equilibrium states areW*-dense in the set of all translation invariant states. The problem discussed in this paper is how to recognize such equilibrium states and how to find the corresponding potential. A necessary and sufficient condition for a state to be an equilibrium state for some potential is given in Sect. 3.

Subject Materials Science Mathematical and Statistical Physics Classical Mechanics Collection: Oxford Scholarship Online

An interesting and important problem in statistical mechanics is the phospholipid main phase transition and related phenomena. A phospholipid molecule may be thought of as being composed of a polar head group molecular subunit to which are attached two alkyl chain “tails.”1 In aqueous solution, under conditions of excess water, these molecules are organized into bilayer structures. The bilayer may be viewed as a planar assembly with thickness approximately twice the length of the chain tails (~ 60 A). The polar head groups are confined to the two planes which define the surface of the bilayer and are in contact with the water on the outside of the bilayer. The hydrophobic alkyl chains are confined to the interior of the bilayer and extend from the polar head group to which they are attached to the plane which is the mid plane of the bilayer. Water is almost totally excluded from the interior of the bilayer.

The independent set polynomial of a graph has one variable for each vertex and one monomial for each independent set, comprising the product of the corresponding variables. Given a graph G on n vertices and a vector p ∈ [0,1)ⁿ, a central problem in statistical mechanics is determining whether the independent set polynomial of G is non-vanishing in the polydisk of p, i.e., whether |Z_G(x)| > 0 for every x ∈ ℂⁿ such that |x_i| ≤ p_i. Remarkably, when this holds, Z_G(-p) is a lower bound for the avoidance probability when G is a dependency graph for n events whose probabilities form vector p. A local sufficient condition for |Z_G| > 0 in the polydisk of p is the Lovász Local Lemma (LLL). 
\nIn this work we derive several new results on the efficient evaluation and bounding of Z_G. Our starting point is a monotone mapping from subgraphs of G to truncations of the tree of self-avoiding walks of G. Using this mapping our first result is a local upper bound for Z(-p), similar in spirit to the local lower bound for Z(-p) provided by the LLL. Next, using this mapping, we show that when G is chordal, Z_G can be computed exactly and in linear time on the entire complex plane, implying perfect sampling for the hard-core model on chordal graphs. We also revisit the task of bounding Z(-p) from below, i.e., the LLL setting, and derive four new lower bounds of increasing sophistication. Already our simplest (and weakest) bound yields a strict improvement of the famous asymmetric LLL, i.e., a strict relaxation of the inequalities of the asymmetric LLL without any further assumptions. This new asymmetric local lemma is sharp enough to recover Shearer’s optimal bound in terms of the maximum degree Δ(G). We also apply our more sophisticated bounds to estimate the zero-free region of the hard-core model on the triangular lattice (hard hexagons model).

A radically new approach to facility layout optimization involving nonconvex quadratic assignment problems is presented. The approach uses a simulated annealing technique originally developed to solve problems in statistical mechanics by Metropolis et al, and recently applied to VLSI chip design problems. The Metropolis algorithm is relatively simple to apply and a microcomputer model called TOPMET has been developed. TOPMET is shown to produce superior solutions to some of the more popular computer-planning techniques and hand-generated methods. The algorithm also lends itself readily to user interaction and colour graphics display, and its application is illustrated by a practical building problem. Extensions into artificial intelligence are discussed.

A description is given of a general closure principle involving the minimization of the mean square error. The procedure based upon this principle can be applied to the truncation of the BBGKY hierarchy at various stages and to the approximation of unwanted terms arising in the equation of motion method by linear combinations of the observables to be retained. On a general level the significance of the closure principle is described in terms of the geometry of function space, and several useful general properties of the principle are derived. A discussion is devoted to the relation between the closure error (i.e., the least mean square error) and the error in the end result (e.g., the free energy, the radial distribution function, etc.); however, the results, while providing some insight, are not sufficiently refined to provide upper bounds to errors in all problems of statistical mechanics where the method is applicable. On the level of specific application it is shown that the principle yields results identical to the random phase approximation and to the linearized version of the Kirkwood superposition approximation in two special cases. Later sections of the paper describe in greater than usual generality, the formalism connecting thermodynamic properties and other equilibrium properties with the microscopic equations of motion in which closure approximations have been introduced. Two illustrative examples of the application of the over-all method were made to the case of a classical system of electrons in a uniform background of compensating charge, one leading to the well-known results of Debye and the other to a more accurate and elaborate theory developed in quantitative detail elsewhere.

This dissertation deals with two classical problems in statistical mechanics: the first passage percolation on Euclidean spaces, FPP for short, in both directed and undirected settings.

The method of moving planes is used to establish a weak set of conditions under which the nonlinear equation −Δu(x)=V(|x|)e u(x),x∈ℝ2 admits only rotationally symmetric solutions. Additional structural invariance properties of the equation then yield another set of conditions which are not originally covered by the moving plane technique. For instance, nonmonotonicV can be accommodated. Results for −Δu(y)=V(y)e u(y) −c, withy∈S 2, are obtained as well. A nontrivial example of broken symmetry is also constructed. These equations arise in the context of extremization problems, but no extremization arguments are employed. This is of some interest in cases where the extremizing problem is neither manifestly convex nor monotone under symmetric decreasing rearrangements. The results answer in part some conjectures raised in the literature. Applications to logarithmically interacting particle systems and geometry are emphasized.

One of the central problems in statistical mechanics is that of finding the density of states of a system. Knowledge of the density of states of a system is equivalent to knowledge of its fundamental equation, from which all thermodynamic quantities can be obtained. Over the past several years molecular simulations have made considerable strides in their ability to determine the density of states of complex fluids and materials. In this review we discuss some of the more promising approaches proposed in the recent literature along with their advantages and limitations.