On the entropy minimization problem in Statistical Mechanics

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.

Summation of power series by self-similar factor approximants

Ordered phases of elongated micelles

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.

Exact, formal, and usable solutions of the Liouville equation corresponding to the problem of independent two-level atoms in an external field are presented. Two examples are worked out: calculation of the polarization, and a model of pumping or population inversion by a $\ensuremath{\delta}$-function external field. The relation of the solutions to other problems in statistical mechanics is also discussed.

Following Richardson and using results of Kesten on First-passage percolation, we obtain an upper bound on the expected perimeter in an Eden Growth Process. Using results of the author from a problem in Statistical Mechanics, we show that the average perimeter of the lattice animals resulting from a very natural family of "growth histories" does not obey a similar bound.

Peter Mazur, one of the founders of nonequilibrium thermodynamics, died on 15 August 2001 following a short illness. Born on 11 December 1922 in Vienna, Austria, he died in Switzerland, where he had lived after his retirement. During World War II, Mazur had to go into hiding for about three years in various farms in the Dutch countryside. He then began studying chemistry at Utrecht University in the Netherlands.In 1951, Mazur obtained his doctorate under the direction of Sybren de Groot with a thesis entitled, “Thermodynamics of Transport Phenomena in Liquid Helium-2.” The results were in good agreement with experiments done at the Kamerlingh Onnes Laboratory in Leiden, the Netherlands.After many futile attempts to obtain a theoretical foundation for experimentally established reciprocal relations, Lars Onsager finally (in 1931) showed that such relations were a general consequence of the invariance of the microscopic equations of motion under time reversal. Josef Meixner in 1941 and, independently, Ilya Prigogine in 1947 set up a consistent phenomenological theory of irreversible processes, incorporating both Onsager’s reciprocity theorem and the explicit calculation for some systems of the so-called entropy source strength. Shortly thereafter, Mazur and de Groot joined this group as founding fathers of the new field of nonequilibrium thermodynamics. After a period as a postdoctoral fellow at the University of Maryland, College Park, Mazur became an associate professor at Leiden University in 1954. He and de Groot, who had also moved to Leiden, founded, in 1955, the Lorentz Institute for Theoretical Physics at Leiden University. In 1961, Mazur became a full professor, and when de Groot left in 1963, he became director of the institute. He filled this position in his own distinctive way for 25 years until he became emeritus in 1988. Under the direction of de Groot and Mazur, the institute grew substantially and eventually established the Lorentz Chair, a prestigious special professorship.In his first years at Leiden University, Mazur studied the classical and quantum molecular foundations of nonequilibrium thermodynamics. Significant results included the derivation of the Langevin equation with one of us (Oppenheim) and the classic paper on harmonic oscillator systems by George Ford, Mark Kac, and Mazur, which was published in the Journal of Mathematical Physics (in 1965). Mazur’s work in the 1950s and 1960s culminated in the publication of Nonequilibrium Thermodynamics (North-Holland and Interscience, 1962), written by de Groot and Mazur. This book, translated into several languages, became a classic in the field and was later republished as a series of classic monographs.Mazur’s work in subsequent years addressed a variety of problems in statistical mechanics. He had a good nose for problems ripe to be investigated. To describe diffusion of large particles in fluids, he introduced, together with one of us (Bedeaux), the concept of induced forces in 1974. This concept was used to derive generalizations of Faxen’s theorem and to develop a theory for the viscosity of a suspension. In 1976, Mazur, with Bedeaux and Alfonso Albano, gave the first systematic formulation of non-equilibrium thermodynamics for surfaces. This formulation opened a new field, which is still in active development. And Mazur, Wim van Saarloos, and Carlo Beenakker developed an algebraic method around 1982 to successfully describe hydrodynamic interactions between arbitrary numbers of particles using induced forces. This was a breakthrough in the field.After retiring in 1988, Mazur remained active. In 1991, he derived, with Bedeaux, the Langevin equation for a Brownian particle using only causality and time-reversal invariance. From 1994 to 2000, Mazur, together with J. Miguel Rubi, used the method of internal degrees of freedom to describe fluctuations in the context of nonequilibrium thermodynamics. In 2001, he and Bedeaux developed nonequilibrium thermodynamics for quantum systems.Many of Mazur’s colleagues and students have good memories of his flamboyant lecture style and the often heated discussions in front of the blackboard. He insisted on a crystal-clear introduction followed by a detailed derivation of the results. And he thoroughly disliked sentences such as, “It can be easily shown that….” It was great to work with him.Mazur served on the boards of the International Union of Pure and Applied Physics (1966–84) and the Dutch Foundation for Fundamental Research of Matter (1970–85). In 1987, Queen Beatrix made Mazur a Knight of the Order of the Netherlands Lion.Mazur was an excellent scientist, and superb teacher and colleague, whose knowledge and interests encompassed not only science, but almost all fields of intellectual endeavor. A great man, he was devoted to his family, friends, and colleagues. We will miss him dearly.Peter MazurC. W. J. BEENAKKERPPT|High resolution© 2002 American Institute of Physics.