The emergence of order in natural systems is a constant source of inspiration for both physical and biological sciences. While the spatial order characterizing for example the crystals has been the basis of many advances in contemporary physics, most complex systems in nature do not offer such high degree of order. Many of these systems form complex networks whose nodes are the elements of the system and edges represent the interactions between them. 
Traditionally complex networks have been described by the random graph theory founded in 1959 by Paul Erdohs and Alfred Renyi. One of the defining features of random graphs is that they are statistically homogeneous, and their degree distribution (characterizing the spread in the number of edges starting from a node) is a Poisson distribution. In contrast, recent empirical studies, including the work of our group, indicate that the topology of real networks is much richer than that of random graphs. In particular, the degree distribution of real networks is a power-law, indicating a heterogeneous topology in which the majority of the nodes have a small degree, but there is a significant fraction of highly connected nodes that play an important role in the connectivity of the network. 
The scale-free topology of real networks has very important consequences on their functioning. For example, we have discovered that scale-free networks are extremely resilient to the random disruption of their nodes. On the other hand, the selective removal of the nodes with highest degree induces a rapid breakdown of the network to isolated subparts that cannot communicate with each other. 
The non-trivial scaling of the degree distribution of real networks is also an indication of their assembly and evolution. Indeed, our modeling studies have shown us that there are general principles governing the evolution of networks. Most networks start from a small seed and grow by the addition of new nodes which attach to the nodes already in the system. This process obeys preferential attachment: the new nodes are more likely to connect to nodes with already high degree. We have proposed a simple model based on these two principles wich was able to reproduce the power-law degree distribution of real networks. Perhaps even more importantly, this model paved the way to a new paradigm of network modeling, trying to capture the evolution of networks, not just their static topology.

/pdf/statistical-mechanics-of-complex-networks-1zfjpvxipu.pdf

Statistical mechanics of complex networks

The exact density functional for the ground-state energy is strictly self-interaction-free (i.e., orbitals demonstrably do not self-interact), but many approximations to it, including the local-spin-density (LSD) approximation for exchange and correlation, are not. We present two related methods for the self-interaction correction (SIC) of any density functional for the energy; correction of the self-consistent one-electron potenial follows naturally from the variational principle. Both methods are sanctioned by the Hohenberg-Kohn theorem. Although the first method introduces an orbital-dependent single-particle potential, the second involves a local potential as in the Kohn-Sham scheme. We apply the first method to LSD and show that it properly conserves the number content of the exchange-correlation hole, while substantially improving the description of its shape. We apply this method to a number of physical problems, where the uncorrected LSD approach produces systematic errors. We find systematic improvements, qualitative as well as quantitative, from this simple correction. Benefits of SIC in atomic calculations include (i) improved values for the total energy and for the separate exchange and correlation pieces of it, (ii) accurate binding energies of negative ions, which are wrongly unstable in LSD, (iii) more accurate electron densities, (iv) orbital eigenvalues that closely approximate physical removal energies, including relaxation, and (v) correct longrange behavior of the potential and density. It appears that SIC can also remedy the LSD underestimate of the band gaps in insulators (as shown by numerical calculations for the rare-gas solids and CuCl), and the LSD overestimate of the cohesive energies of transition metals. The LSD spin splitting in atomic Ni and $s\ensuremath{-}d$ interconfigurational energies of transition elements are almost unchanged by SIC. We also discuss the admissibility of fractional occupation numbers, and present a parametrization of the electron-gas correlation energy at any density, based on the recent results of Ceperley and Alder.

Self-interaction correction to density-functional approximations for many-electron systems

Current studies in density functional theory and density matrix functional theory are reviewed, with special attention to the possible applications within chemistry. Topics discussed include the concept of electronegativity, the concept of an atom in a molecule, calculation of electronegativities from the Xα method, the concept of pressure, Gibbs-Duhem equation, Maxwell relations, stability conditions, and local density functional theory.

Density-functional theory of atoms and molecules

/pdf/scalable-molecular-dynamics-with-namd-3j7xyc70g4.pdf

Scalable Molecular Dynamics with NAMD

CASTEP Computer program / Density functional theory / Pseudopotentials / ab initio study / Plane-wave method / Computational crystallography Abstract. The CASTEP code for first principles electro- nic structure calculations will be described. A brief, non- technical overview will be given and some of the features and capabilities highlighted. Some features which are un- ique to CASTEP will be described and near-future devel- opment plans outlined.

/pdf/first-principles-methods-using-castep-2ke0q7ufxk.pdf

First principles methods using CASTEP

Scaled equations of state for fluids and magnets are studied near the critical point, with particular emphasis on specific-heat predictions. The important of fitting both the exponent and the critical amplitudes is emphasized. Previous proposals, such as the Missoni, Levelt Sengers, and Green (MLSG) and "linear-model" equations are examined, and the corresponding amplitude ratio $\frac{A}{{A}^{\ensuremath{'}}}$ calculated as a function of the parameters. The linear model is found to be inapplicable to Heisenberg-like systems in which the exponent $\ensuremath{\alpha}$ is negative, and $\frac{A}{{A}^{\ensuremath{'}}}g1$. Specific-heat data on Xe, C${\mathrm{O}}_{2}$, Ni, and EuO are compared to predictions based on the MLSG and linear-model equations, with parameters previously determined using pressure, volume, and temperature ($\mathrm{PVT}$) and magnetization, field, and temperature ($\mathrm{MHT}$) data. There is a small but probably significant discrepancy for the fluids, and a large deviation in the magnetic case. A "modified MLSG" equation is proposed, with an additional parameter, by means of which both $\mathrm{PVT} (\mathrm{MHT})$ and specific-heat data may be fitted. Using this equation, an estimate is made for the effect of small fields on rounding the specific-heat singularity in magnetic systems. In EuO it is found that a field as small as the earth's field has a perceptible effect on the specific heat rounding near ${T}_{c}$.

Erratum: Scaled-equation-of-state analysis of the specific heat in fluids and magnets near critical point

Publisher Summary This chapter discusses the density functional theory (DFT). It discusses two basic theorems expressing the ground state of a many-electron system in terms of the particle density distribution, and of the variational solution in terms of a single-particle Schrodinger equation. In conclusion, some challenges and directions for further progress in this field are indicated. Time-dependent density functional theory has been formally developed, but it has not yet found a practical incarnation of an ease and utility comparable to local density approximation for stationary ground and excited states and thermal ensembles. The practical implementations of DFT all take the uniform electron gas as their starting point for approximate representations of exchange-correlation effects. But in bounded systems the boundary region(s), where the density tends to zero and where the Kohn-Sham wave-functions evanesce rather than oscillate, are essentially different from a region of a uniform electron-gas, even if the density does not vary rapidly.

The Beginnings and Some Thoughts on the Future

Landau's superfluid hydrodynamics is applied to the vibration spectrum of a lattice of rectilinear vortices in both charged and neutral superfluid systems. The resulting vortex dynamics is identical with that of classical hydrodynamics: each vortex moves with the local superfluid velocity at its core. The only mode considered is one in which the vortices move without bending. This mode is unstable for all lattice structures in a neutral system (liquid helium II); in a charged system (type-II superconductors) the mode is unstable for a square lattice but stable for a triangular lattice. The corresponding long-wavelength dispersion relation is $\ensuremath{\omega}=(\frac{\mathrm{eB}}{\mathrm{mc}}){q}^{2}\ensuremath{\lambda}d{(\frac{\sqrt{3}}{32\ensuremath{\pi}})}^{\frac{1}{2}}$, where $B$ is the magnetic induction, $\ensuremath{\lambda}$ is the London penetration depth, $q$ is the wave number, and $d$ is the lattice spacing. An elasticity theory of the lattice vibrations in the charged system is shown to predict identical results. These calculations agree qualitatively with those of de Gennes and Matricon but disagree with those of Abrikosov, Kemoklidze, and Khalatnikov; the discrepancies are discussed in detail.

Stability of a Lattice of Superfluid Vortices

The relaxational models introduced in a previous treatment of critical dynamics are studied in detail using renormalization-group methods. The earlier results are justified by an analysis to all order in $\ensuremath{\epsilon}=4\ensuremath{-}d$, where $d$ is the dimensionality. The diagrammatic formalism of the full dynamic renormalization group is presented, and applied to the earlier models. A generalization of Wilson's Feynman-graph expansion method is used to calculate the exponents to second order in $\ensuremath{\epsilon}$. In model $C$, where a nonconserved order parameter is coupled to a conserved energy field, ambiguities were found in the earlier recursion-relation treatment for $2lnl4$, $d\ensuremath{\approx}4$ ($n$ is the number of components of the order parameter). These ambiguities are discussed in the present work, but are not fully resolved.

Renormalization-group methods for critical dynamics: II. Detailed analysis of the relaxational models

The presence of a free surface leads to an inhomogeneous distribution of the local magnetization and other quantities describing the magnetic state of the system. After a brief outline of a general thermodynamic theory, the mean field treatment is discussed and shown to be inconsistent with a more general scaling theory, even if one uses a temperature-dependent extrapolation length. More reliable information is obtained by numerical techniques such as high temperature series expansions and Monte Carlo calculations, which are applied to Ising and Heisenberg models. These numerical results are shown to be consistent with the scaling theory, but it is also shown that the magnetization of the surface layer m 1 depends strongly on the local conditions (e.g., exchange constant J s different from J in the bulk). Therefore the observability of the universal asymptotic exponents may be restricted to a very narrow region around the (bulk) critical temperature T cb . The case where the surface layer orders at a temperature T_{cs} > T_{cb} is also considered. The consequences of the theory for thin magnetic films and small particles are outlined and experimental applications are briefly discussed.

P. C. Hohenberg

Papers

Erratum: Scaled-equation-of-state analysis of the specific heat in fluids and magnets near critical point

The Beginnings and Some Thoughts on the Future

Stability of a Lattice of Superfluid Vortices

Renormalization-group methods for critical dynamics: II. Detailed analysis of the relaxational models

Monte Carlo and series expansion investigations of magnetic surfaces