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.

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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

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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.

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First principles methods using CASTEP

It is pointed out that a rigorous inequality first proved by Bogoliubov may be used to rule out the existence of quasi-averages (or long-range order) in Bose and Fermi systems for one and two dimensions and $T\ensuremath{\ne}0$.

Existence of Long-Range Order in One and Two Dimensions

The effects of thermal fluctuations on the convective instability are considered. It is shown that the Langevin equations for hydrodynamic fluctuations are equivalent, near the instability, to a model for the crystallization of a fluid in equilibrium. Unlike the usual models, however, the free energy of the present system does not possess terms cubic in the order parameter, and therefore the system undergoes a second-order transition in mean-field theory. The effects of fluctuations on such a model were recently discussed by Brazovskii, who found a first-order transition in three dimensions. A similar argument also leads to a discontinuous transition for the convective model, which behaves two dimensionally for sufficiently large lateral dimensions. The magnitude of the jump is unobservably small, however, because of the weakness of the thermal fluctuations being considered. The relation of the present analysis to the work of Graham and Pleiner is discussed.

Hydrodynamic fluctuations at the convective instability

Shobhana Narasimhan’s article ‘A Tryst With Density’ in this issue of Resonance describes a remarkable
theorem proved by Hohenberg and Kohn in 1964, and its aftermath. These authors considered a system of
electrons in an external potential. This is an excellent idealised model of an atom, molecule or solid because
the nuclei are so much heavier than the electrons and can be treated as an external potential. The difficulty
lies in accounting for the interaction between the electrons – what one electron is doing depends on what
all the others are doing. It therefore came as a shock when the ground state energy of this very general
and nontrivial system was shown by Hohenberg and Kohn to be the result of minimising an expression
which only contained a function of three variables. This function was just the number of electrons per unit
volume as a function of position, something which chemists and crystallographers had long dealt with as a
very meaningful and measurable quantity. The theorem showed that such an expression exists, but did not
provide the form, which could contain nonlinear terms, derivatives, and integrals of the electron density.
While approximate theories of this kind had been proposed before, it was unexpected that the idea of using
the electron density could be made exact. What a marvellous turn of events that an existence theorem,
proved by contradiction in a modestly written paper would revolutionise the modeling of materials half a
century later and win a Nobel Prize in Chemistry for one of the authors!

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Inhomogeneous electron gas

The usual static scaling laws are generalized to nonequilibrium phenomena by making assumptions on the behavior of time-dependent correlation functions near the critical point of second-order phase transitions. At any temperature different from ${T}_{c}$, the correlation functions are assumed to reflect the hydrodynamic behavior of the system, for sufficiently long wavelengths and low frequencies. As the critical temperature is approached, however, the range of spatial correlations in the system diverges, and the domain of applicability of hydrodynamics is reduced to a vanishingly small region of wavelengths and frequencies. The dynamic-scaling assumptions lead to predictions for the behavior of the hydrodynamic parameters near ${T}_{c}$, as well as for the form of the correlation functions for macroscopic distances and times, outside the hydrodynamic range. In particular, singularities are predicted to occur in the temperature dependence of transport coefficients, and anomalies are expected in the frequency spectrum of certain operators, which are observable by inelastic scattering of neutrons or light. A distinction is made between the restricted dynamic-scaling hypothesis, which refers to the order parameter only, and extended dynamic scaling, which applies to other operators and involves stronger assumptions. Applications are discussed to antiferromagnets, ferromagnets, the gas-liquid critical point, and the $\ensuremath{\lambda}$ transition in superfluid helium. Specific experiments are suggested to test the scaling assumptions, and existing experimental evidence is briefly reviewed. Finally, a comparison is made with other theories of dynamical behavior near critical points.

Scaling Laws for Dynamic Critical Phenomena

http://www.lorentz.leidenuniv.nl/~saarloos/Papers/PhysicaD_56_303.pdf

P. C. Hohenberg

Papers

Existence of Long-Range Order in One and Two Dimensions

Hydrodynamic fluctuations at the convective instability

Inhomogeneous electron gas

Scaling Laws for Dynamic Critical Phenomena

Fronts, pulses, sources and sinks in generalized complex Ginzberg-Landau equations