Linear Closure Approximation Method for Classical 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.

A detailed account of the explicit quantum chemical solvent model QMSTAT is given. The model is presented in terms of three coupled aspects of relevance for all types of quantum chemical solvent models: the quantum chemical method, the intermolecular interactions and the statistical mechanical method. The quantum chemical method is either a compact natural orbital formulation of the standard Hartree–Fock method or a compact multiconfigurational method with a state basis. The latter method can describe excited states apart from the ground state and is for most systems an excellent approximation to the complete active space self-consistent field method. Both static and induced electrostatic interaction terms between the quantum chemical region and the solvent are included. Further, a non-electrostatic term is added to describe effects which derive from the Pauli principle. This term models both the exchange repulsion between solute and solvent and the packing effects an environment has on a molecule, in particular on diffuse states of the molecule. The statistical mechanical problem is solved with an exact Metropolis–Monte Carlo simulation that requires several similar quantum chemical problems to be solved. Since the quantum chemical problem and the statistical mechanical problem are solved as a coupled problem, the present model is especially useful for problems where electronic degrees of freedom of the solute strongly depend on the solvent distribution and vice versa. Three applications are summarized, which highlight this type of coupling present in QMSTAT and the non-electrostatic contribution. The examples are the solvation of four monatomic ions, the solvation of para-benzoquinone and the solvation of indole and the solvent shift to its absorption and fluorescence spectra

We present a procedure for the determination of the interaction potential from the knowledge of the radial pair distribution function. The method, realized inside an inverse Monte Carlo simulation scheme, is based on the application of the maximum entropy principle of information theory and the interaction potential emerges as the asymptotic expression of the transition probability. Results obtained for high density monoatomic fluids are very satisfactory and provide an accurate extraction of the potential, despite a modest computational effort.

The Mori-Zwanzig projection operator formalism is a powerful method for the derivation of mesoscopic and macroscopic theories based on known microscopic equations of motion. It has applications in a large number of areas including fluid mechanics, solid-state theory, spin relaxation theory, and particle physics. In its present form, however, the formalism cannot be directly applied to systems with time-dependent Hamiltonians. Such systems are relevant in many scenarios such as driven soft matter or nuclear magnetic resonance. In this article we derive a generalization of the present Mori-Zwanzig formalism that is able to treat also time-dependent Hamiltonians. The extended formalism can be applied to classical and quantum systems, close to and far from thermodynamic equilibrium, and even in the case of explicitly-time-dependent observables. Moreover, we develop a variety of approximation techniques that enhance the practical applicability of our formalism. Generalizations and approximations are developed for both equations of motion and correlation functions. Our formalism is demonstrated for the important case of spin relaxation in a time-dependent external magnetic field. The Bloch equations are derived together with microscopic expressions for the relaxation times.

On a transport theory for electrons in vibrating isotopically disordered crystals

On the entropy minimization problem in Statistical Mechanics

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.

These days, the demands for a new technology for producing energy without consuming fossil fuels are increasing. One of the reasons is strict regulations of emitting CO2 and NO on cars like Euro 6 in European countries. It is predicted these regulations will be much stricter in many countries in the future. Also, the demands for a new battery that is smaller and has higher energy outputs for smartphones and computers are increasing in the twenty first century. So, one of the solutions for the problems in the twenty first century is inventing new battery without emitting CO2 and NO. Above all, lithium ion battery is given attention as one of the new batteries because it has high energy density and can be stored for a long time.Lithium ion battery has some merits such as rechargeable battery, high energy density and large range of operating temperatures. With these benefits, lithium ion battery is used for electronic vehicles (EV), cellphones and computers. However, lithium ion battery has some demerits. Because its electrolyte is liquid, liquid leak and automatic firing occurs and it has low degree of design freedom. One of the solutions for these problems is all solid-state lithium ion battery whose electrolyte is solid. Now, electrolyte materials that have high Li ion transport property are required.In this study, we focus on polymer electrolyte for its high brittleness. In polymer electrolyte, it is known that Li ion transports by four diffusion mechanisms, inter-segmental hopping, intra-segmental hopping, inter-chain hopping and co-diffusion. We focus on polyethylene oxiside (PEO) and P(2EO-MO) as the electrolyte materials for this study. PEO is known for the most common polymer electrolyte material that has high Li ion transport property. P(2EO-MO) was reported in 2018 [1] that it has higher transference number than PEO but lower ionic conductivity than PEO. However, the reason why P(2EO-MO) has different Li ion transport property from PEO have not been investigated. To reveal the reason in molecular level is required because the knowledge will help to produce a new polymer that has better Li ion transport property in the future. For that, Li ion transport mechanisms in PEO and P(2EO-MO) need to be revealed.The objective of this study is to reveal the Li ion transport mechanism in PEO and P(2EO-MO). In particular, the ratio of each four types of Li ion transport mechanisms was investigated using molecular dynamics (MD) simulation because the Li ion diffusion is nano scale dynamics. The simulation systems are constructed with 30 LiTFSI molecules as Lithium salts for r = 0.01 and either 30 PEO chains or 30 P(2EO-MO) chains. The salt concentration ratio r is set 0.01, 0.04, 0.08, 1.2 and 1.6. The r is defined by the equation r = [Li]/[O]. Here, [O] means all O atoms concentration in the polymer. TraPPE-UA forcefield is used for all calculations with LAMMPS and the motion equations are integrated with timestep of 1 fs by Velocity Verlet method. Pressure and temperature are controlled by Nose-Hoover barostat and thermostat and 3-dimensional boundary conditions are applied. After equilibrating the system, NVT simulation was applied 300 ns for production. The temperature and pressure were set as 400 K and 1 bar, respectively.As the analysis of structural property, radial distribution function (RDF) was calculated. As the analysis of transport property, self-diffusion coefficient of Li ion was calculated by mean square displacement (MSD) and each Li ion displacements were also calculated. Additionally, with calculating oxygen index, Li ion transport mechanism was identified. With LiTFSI cluster analysis, the relationship between the Li or TFSI diffusion and the salt concentration ratio was explained.Reference[1] Zheng, Q. et al. Optimizing Ion Transport in Polyether-Based Electrolytes for Lithium Batteries. Macromolecules 51, 2847–2858 (2018)

Summary form only given. Correlation effects influencing the optical properties of semiconductors that cannot be described at the basis of Hartree-Fock approximation are intensively studied at the present time. Equations of motion methods, the Green's function formalism or the cumulant expansion method with fluctuation-dissipation theorem are successfully used to explain distinctive features of optical spectra of semiconductors. In spite of the fact that all these different theories deal with just the same problem, the physical mechanisms and results obtained by using these theoretical methods are various in detail drastically. Especially it concerns the coherently driven plasmons in high excited semiconductors. Our aim is to derive the equations of motion which contain the results of another theories and allow to include into considerations multi-quantum optical processes involving photon and several mixed phonon-plasmons. We get the semiconductor Bloch equations using the fluctuation-dissipation theorem to calculate four operator expectation values with account of coherent memory effects.

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).