R J Baxter 1982 London: Academic xii + 486 pp price £43.60 Over the past few years there has been a growing belief that all the twodimensional lattice statistical models will eventually be solved and that it will be Professor Baxter who solves them. Baxter has inherited the mantle of Onsager who started the process by solving exactly the two-dimensional Ising model in 1944.

Exactly Solved Models in Statistical Mechanics

The stability of two-dimensional (2D) layers and membranes is subject of a long standing theoretical debate. According to the so called Mermin-Wagner theorem, long wavelength fluctuations destroy the long-range order for 2D crystals. Similarly, 2D membranes embedded in a 3D space have a tendency to be crumpled. These dangerous fluctuations can, however, be suppressed by anharmonic coupling between bending and stretching modes making that a two-dimensional membrane can exist but should present strong height fluctuations. The discovery of graphene, the first truly 2D crystal and the recent experimental observation of ripples in freely hanging graphene makes these issues especially important. Beside the academic interest, understanding the mechanisms of stability of graphene is crucial for understanding electronic transport in this material that is attracting so much interest for its unusual Dirac spectrum and electronic properties. Here we address the nature of these height fluctuations by means of straightforward atomistic Monte Carlo simulations based on a very accurate many-body interatomic potential for carbon. We find that ripples spontaneously appear due to thermal fluctuations with a size distribution peaked around 70 \AA which is compatible with experimental findings (50-100 \AA) but not with the current understanding of stability of flexible membranes. This unexpected result seems to be due to the multiplicity of chemical bonding in carbon.

Intrinsic ripples in graphene

Journal of Physics Condensed Matter: Preface

The WIEN2k program is based on the augmented plane wave plus local orbitals (APW+lo) method to solve the Kohn-Sham equations of density functional theory. The APW+lo method, which considers all electrons (core and valence) self-consistently in a full-potential treatment, is implemented very efficiently in WIEN2k, since various types of parallelization are available and many optimized numerical libraries can be used. Many properties can be calculated, ranging from the basic ones, such as the electronic band structure or the optimized atomic structure, to more specialized ones such as the nuclear magnetic resonance shielding tensor or the electric polarization. After a brief presentation of the APW+lo method, we review the usage, capabilities, and features of WIEN2k (version 19) in detail. The various options, properties, and available approximations for the exchange-correlation functional, as well as the external libraries or programs that can be used with WIEN2k, are mentioned. References to relevant applications and some examples are also given.

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WIEN2k: An APW+lo program for calculating the properties of solids

Foreword Acknowledgements Definitions of symbols used 1. Some fundamentals 2. The harmonic approximation and lattice dynamics of very simple systems 3. Dynamics of diatomic crystals: general principles 4. How far do the atoms move? 5. Lattice dynamics and thermodynamics 6. Formal description 7. Acoustic modes and macroscopic elasticity 8. Anharmonic effects and phase transitions 9. Neutron scattering 10. Infrared and Raman spectroscopy 11. Formal quantum mechanical description of lattice vibrations 12. Molecular dynamics simulations Appendices Problems Bibliography Index.

Introduction to Lattice Dynamics

As a rule, mean-field theories applied to a fluid that can undergo a transition from saturated vapor at density ρυ to a liquid at density ρl yield a van der Waals loop. For example, isotherms of the chemical potential μ(T,ρ) as a function of the density ρ at a fixed temperature T less than the critical temperature Tc exhibit a maximum and a minimum. Metastable and unstable parts of the van der Waals loop can be eliminated by the Maxwell construction. Van der Waals loops and the corresponding double minimum potentials are mean-field artifacts. Simulations at fixed μ=μcoex for ρυ<ρ<ρl yield a loop, but for sufficiently large systems this loop does not resemble the van der Waals loop and reflects interfacial effects on phase coexistence due to finite size effects. In contrast to the van der Waals loop, all parts of the loop found in simulations are thermodynamically stable. The successive umbrella sampling algorithm is described as a convenient tool for seeing these effects. It is shown that the maximum of t...

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Beyond the Van Der Waals loop: What can be learned from simulating Lennard-Jones fluids inside the region of phase coexistence

A recently proposed method to obtain the surface free energy σ(R) of spherical droplets and bubbles of fluids, using a thermodynamic analysis of two-phase coexistence in finite boxes at fixed total density, is reconsidered and extended Building on a comprehensive review of the basic thermodynamic theory, it is shown that from this analysis one can extract both the equimolar radius Re as well as the radius Rs of the surface of tension Hence the free energy barrier that needs to be overcome in nucleation events where critical droplets and bubbles are formed can be reliably estimated for the range of radii that is of physical interest It is found that the conventional theory of nucleation, where the interface tension of planar liquid–vapor interfaces is used to predict nucleation barriers, leads to a significant overestimation, and this failure is particularly large for bubbles Furthermore, different routes to estimate the effective radius-dependent Tolman length δ(Rs) from simulations in the canonical e

http://smt.tuwien.ac.at/extra/publications/papers/2012/JChemPhys_136_064709.pdf

Numerical approaches to determine the interface tension of curved interfaces from free energy calculations.

We report a self-adapting version of the Wang-Landau algorithm that is ideally suited for application to systems with a complicated structure of the density of states. Applications include determination of two-dimensional densities of states and high-precision numerical integration of sharply peaked functions on multidimensional integration domains.

Wang-Landau sampling with self-adaptive range

Using the Wang-Landau algorithm we derive the full thermal order parameter probability distribution of the ${\ensuremath{\phi}}^{4}$ model for various displacive degrees and temperatures and calculate the resulting free energies. We obtain high-precision data on the shape of the free-energy barrier separating states of opposite order parameter values. For order-disorder-like systems, i.e., at low displacive degree we observe phase separation below the transition temperature. A model taking into account the surface free energy related to different domain shapes, which fits the simulation data extremely well at low temperatures, is constructed. The interpretation of the results in the context of Landau or Landau-Ginzburg theory is discussed and an improved setup for simulating Landau potentials is proposed.

https://homepage.univie.ac.at/Wilfried.Schranz/papers/papermitandi.pdf

Free energies of the ϕ 4 model from Wang-Landau simulations

We present a new method to determine the curvature dependence of the interface tension between coexisting phases in a finite volume from free energies obtained by Monte Carlo simulations. For the example of a lattice gas on a 3D fcc lattice with nearest neighbor three-body interactions, we demonstrate how to calculate the equimolar radius ${R}_{e}$ as well as the radius ${R}_{s}$ of the surface of tension and thus the Tolman length $\ensuremath{\delta}({R}_{s})={R}_{e}\ensuremath{-}{R}_{s}$. Within the physically relevant range of radii, $\ensuremath{\delta}({R}_{s})$ shows a pronounced ${R}_{s}$ dependence, such that the simple Tolman parametrization for the interface tension is refutable. For the present model, extrapolation of $\ensuremath{\delta}({R}_{s})$ to ${R}_{s}\ensuremath{\rightarrow}\ensuremath{\infty}$ by various methods clearly indicates a positive limiting value.

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Andreas Tröster

Papers

Beyond the Van Der Waals loop: What can be learned from simulating Lennard-Jones fluids inside the region of phase coexistence

Numerical approaches to determine the interface tension of curved interfaces from free energy calculations.

Wang-Landau sampling with self-adaptive range

Free energies of the ϕ 4 model from Wang-Landau simulations

Positive Tolman Length in a Lattice Gas with Three-Body Interactions