The study of freezing using perturbative classical density-functional theory is revisited, using a bridge functional approach to resum all terms beyond second order in the free energy expansion. More precisely, the first-order direct correlation function of the solid phase is written as a functional expansion about the homogeneous liquid phase, and the sum of all higher-order terms is represented as a functional of the second-order term. Information about the shape and uniqueness of this bridge functional for the case of hard spheres is obtained via an inversion procedure that employs Monte Carlo fluid-solid coexistence data from the literature. The parametric plots obtained from the inversion procedure show very little scatter in certain regions, suggesting a unique functional dependence, but large scatter in other regions. The scatter is related to the anisotropy of the solid lattice at the particle scale. Interestingly, the thermodynamic properties of the phase transition are quite insensitive to the regions where the scatter is large, and several simple closures (i.e., analytical forms of the bridge function) reproduce exactly the liquid-solid coexistence densities and Lindemann parameter from simulation. The form of these closures is significantly different from the usual closures employed in liquid-state integral equation theory.

In this paper, we investigate the behavior of the Ising model on two sparsely connected complex networks. The networks have the topology of a random graph or Barabási and Albert scale-free networks. We extend our previous analysis and show that a bistable-monostable phase transition occurs in such systems. During this transition, the magnetization undergoes a discontinuous jump. We calculate the critical temperature analytically for regular random graphs and study a more general case using an iterative map corresponding to mean-field dynamics. The calculations are confirmed by numeric simulations based on Monte Carlo approach.

We examine the influence of different forms of the free-energy functionals used in the phase-field-crystal (PFC) model, and compare them with the second-order density-functional theory (DFT) of freezing, by using bcc iron as an example case. We show that there are large differences between the PFC and the DFT and it is difficult to obtain reasonable parameters for existing PFC models directly from the DFT. Therefore, we propose a way of expanding the correlation function in terms of gradients that allows us to incorporate the bulk modulus of the liquid as an additional parameter in the theory. We show that this functional reproduces reasonable values for both bulk and surface properties of bcc iron, and therefore it should be useful in modeling bcc materials. As a further demonstration, we also calculate the grain boundary energy as a function of misorientation for a symmetric tilt boundary close to the melting transition.

We carry out a theoretical study on the isotropic-nematic phase transition and phase separation in amyloid fibril solutions. Borrowing the thermodynamic model employed in the study of cylindrical micelles, we investigate the variations in the fibril length distribution and phase behavior with respect to changes in the protein concentration, fibril’s rigidity, and binding energy. We then relate our theoretical findings to the nematic ordering experimentally observed in Hen Lysozyme fibril solution.

We define shell $\ell$ in a network as the set of nodes at distance $\ell$ with respect to a given node and define $r_{\ell }$ as the fraction of nodes outside shell $\ell$. In a transport process, information or disease usually diffuses from a random node and reach nodes shell after shell. Thus, understanding the shell structure is crucial for the study of the transport property of networks. We study the statistical properties of the shells of a randomly chosen node. For a randomly connected network with given degree distribution, we derive analytically the degree distribution and average degree of the nodes residing outside shell $\ell$ as a function of $r_{\ell }$. Further, we find that $r_{\ell }$ follows an iterative functional form $r_{\ell }=\varphi \left(r_{\ell -1}\right)$, where $\varphi$ is expressed in terms of the generating function of the original degree distribution of the network. Our results can explain the power-law distribution of the number of nodes $B_{\ell }$ found in shells with $\ell$ larger than the network diameter $d$, which is the average distance between all pairs of nodes. For real-world networks the theoretical prediction of $r_{\ell }$ deviates from the empirical $r_{\ell }$. We introduce a network correlation function $c\left(r_{\ell }\right)\equiv r_{\ell }/\varphi \left(r_{\ell -1}\right)$ to characterize the correlations in the network, where $r_{\ell }$ is the empirical value and $\varphi \left(r_{\ell -1}\right)$ is the theoretical prediction. $c\left(r_{\ell }\right)=1$ indicates perfect agreement between empirical results and theory. We apply $c\left(r_{\ell }\right)$ to several model and real-world networks. We find that the networks fall into two distinct classes: (i) a class of poorly connected networks with $c\left(r_{\ell }\right)>1$, where a larger (smaller) fraction of nodes resides outside (inside) distance $\ell$ from a given node than in randomly connected networks with the same degree distributions. Examples include the Watts-Strogatz model and networks characterizing human collaborations such as citation networks and the actor collaboration network; (ii) a class of well-connected networks with $c\left(r_{\ell }\right)<1$. Examples include the Barabási-Albert model and the autonomous system Internet network.

We study an evolutionary spatial prisoner’s dilemma game where the fitness of the players is determined by both the payoffs from the current interaction and their history. We consider the situation where the selection time scale is slower than the interaction time scale. This is done by implementing probabilistic reproduction on an individual level. We observe that both too fast and too slow reproduction rates hamper the emergence of cooperation. In other words, there exists an intermediate selection time scale that maximizes cooperation. Another factor we find to promote cooperation is a diversity of reproduction time scales.

We propose an efficient method to compute Lyapunov exponents and Lyapunov eigenvectors of long-range interacting many-particle systems, whose dynamics is described by the Vlasov equation. We show that an expansion of a distribution function using Hermite modes (in velocity variable) and Fourier modes (in configuration variable) converges fast if an appropriate scaling parameter is introduced and identified with the inverse of the temperature. As a consequence, dynamics and linear stability properties of many-particle states, both in the close-to and in the far-from equilibrium regimes, can be predicted using a small number of expansion coefficients. As an example of a long-range interacting system we investigate stability properties of stationary states, the Hamiltonian mean-field model.

We derive the sampling probability density function (pdf) of an ideal localized random electromagnetic field, its amplitude, and intensity in an electromagnetic environment that is quasistatically time-varying statistically homogeneous or static statistically inhomogeneous. The results allow for the estimation of field statistics and confidence intervals when a single spatial or temporal stochastic process produces randomization of the field. Sampling distributions are particularly significant when the number of degrees of freedom $\nu$ is relatively small (typically, $\nu <40$), e.g., in mesoscopic systems when the sample set size $N$ is relatively small by choice or by force. Results for both coherent and incoherent detection methods are derived for Cartesian, planar, and full-vectorial fields. We show that the functional form of the sampling pdf depends on whether the random variable is dimensioned (e.g., the sampled electric field proper) or is expressed in dimensionless standardized or normalized form (e.g., the sampled electric field divided by its sample standard deviation or sample mean). For dimensioned quantities, the electric field, its amplitude, and intensity exhibit different types of Bessel $K$ sampling pdfs, which differ significantly from the asymptotic Gauss normal and ${\chi }_{2p}^{\left(2\right)}$ ensemble pdfs when $\nu$ is relatively small. By contrast, for the corresponding standardized quantities, Student $t$, Fisher-Snedecor $F$, and root-$F$ sampling pdfs are obtained that exhibit heavier tails than comparable Bessel $K$ pdfs. Statistical uncertainties obtained from classical small-sample theory for dimensionless quantities are shown to be overestimated compared to dimensioned quantities. Differences in the sampling pdfs arising from denormalization versus destandardization are identified.

We present four estimators of the shared information (or interdepency) in ground states given that the coefficients appearing in the wave function are all real non-negative numbers and therefore can be interpreted as probabilities of configurations. Such ground states of Hermitian and non-Hermitian Hamiltonians can be given, for example, by superpositions of valence bond states which can describe equilibrium but also stationary states of stochastic models. We consider in detail the last case, the system being a classical not a quantum one. Using analytical and numerical methods we compare the values of the estimators in the directed polymer and the raise and peel models which have massive, conformal invariant and nonconformal invariant massless phases. We show that like in the case of the quantum problem, the estimators verify the area law with logarithmic corrections when phase transitions take place.

We show theoretically that under certain conditions colloidal particles can give rise to spontaneous emulsification of oil/water systems. The capillary penalty to create a large interface is compensated by entropic contributions connected to ionic dissociation on the colloid surfaces. The colloids themselves are absorbed on the oil/water interface. The conditions for spontaneous emulsification are: (1) oil-water interfacial tension is low (a few mN/m or lower); (2) interfacial tension between colloids and oil is smaller than between colloids and water (in the absence of charge effects); (3) density of chargeable groups on the colloids is large (order $1{\text{nm}}^{-2}$); (4) Debye length is comparable to colloid size.

The present paper considers nematogenic lattice models, involving particles of $D_{2h}$ symmetry, whose centers of mass are associated with a three-dimensional simple cubic lattice; the pair potential is isotropic in orientation space and restricted to nearest neighbors. Let two orthonormal triads define orientations of a pair of interacting particles; the simplest potential models proposed in the literature can be reduced to a linear combination involving the squares of the scalar products between corresponding unit vectors only and depending on three parameters. By now, various sets of potential parameters have been proposed and studied in the literature, some of which capable of producing biaxial orientational order at sufficiently low temperature. On the other hand, in experimental terms, mesogenic biaxial molecules mostly produce uniaxial mesophases; thus we address here two very simple cases, involving a nematic (calamitic) term as well as one (model P0M) or two (model PPM) antinematic ones, whose coefficients are set equal in magnitude; when only one antinematic coefficient is used, the third one is set to zero. The calamitic term favors the alignment of two corresponding molecular axes, whereas antinematic terms or geometric constraints tend to keep two other pairs of axes mutually orthogonal. The models were investigated by molecular-field treatments and Monte Carlo simulation and found to predict a first- or second-order transitions between uniaxial nematic and isotropic phases; the molecular-field treatments yielded results in reasonable agreement with simulation.

We present a methodology to extract the backbone of complex networks based on the weight and direction of links, as well as on nontopological properties of nodes. We show how the methodology can be applied in general to networks in which mass or energy is flowing along the links. In particular, the procedure enables us to address important questions in economics, namely, how control and wealth are structured and concentrated across national markets. We report on the first cross-country investigation of ownership networks, focusing on the stock markets of 48 countries around the world. On the one hand, our analysis confirms results expected on the basis of the literature on corporate control, namely, that in Anglo-Saxon countries control tends to be dispersed among numerous shareholders. On the other hand, it also reveals that in the same countries, control is found to be highly concentrated at the global level, namely, lying in the hands of very few important shareholders. Interestingly, the exact opposite is observed for European countries. These results have previously not been reported as they are not observable without the kind of network analysis developed here.

The dynamics of an initially localized wave packet is studied for the generalized nonlinear Schrödinger equation with a random potential, where the nonlinear term is $\beta {\left|\psi \right|}^p\psi$ and $p$ is arbitrary. Mainly short times for which the numerical calculations can be performed accurately are considered. Long time calculations are presented as well. In particular, the subdiffusive behavior where the average second moment of the wave packet is of the form $⟨m_2⟩\approx t^{\alpha }$ is computed. Contrary to former heuristic arguments, no evidence for any critical behavior as function of $p$ is found. The properties of $\alpha \left(p\right)$ for relatively short times are explored, a scaling property and a maximal value for $p\approx \frac{1}{2}$ are found.

The problem of thermally activated escape over a potential barrier is solved by means of path integrals for one-dimensional reaction dynamics with very general time dependences. For a suitably chosen but still quite simple static potential landscape, the net escape rate may be substantially reduced by temporally increasing the temperature above its unperturbed constant level.

We have shown that any pair potential function $v\left(r\right)$ possessing a Fourier transform $V\left(k\right)$ that is positive and has compact support at some finite wave number $K$ yields classical disordered ground states for a broad density range [R. D. Batten, F. H. Stillinger, and S. Torquato, J. Appl. Phys. 104, 033504 (2008)]. By tuning a constraint parameter $\chi$ (defined in the text), the ground states can traverse varying degrees of local order from fully disordered to crystalline ground states. Here, we show that in two dimensions, the “$k$-space overlap potential,” where $V\left(k\right)$ is proportional to the intersection area between two disks of diameter $K$ whose centers are separated by $k$, yields anomalous low-temperature behavior, which we attribute to the topography of the underlying energy landscape. At $T=0$, for the range of densities considered, we show that there is continuous energy degeneracy among Bravais-lattice configurations. The shear elastic constant of ground-state Bravais-lattice configurations vanishes. In the harmonic regime, a significant fraction of the normal modes for both amorphous and Bravais-lattice ground states have vanishing frequencies, indicating the lack of an internal restoring force. Using molecular-dynamics simulations, we observe negative thermal-expansion behavior at low temperatures, where upon heating at constant pressure, the system goes through a density maximum. For all temperatures, isothermal compression reduces the local structure of the system unlike typical single-component systems.

Diffusion in an external potential in a two-dimensional channel of varying cross section is considered. We show that a rigorous mapping procedure applied on the corresponding Smoluchowski equation yields a one-dimensional evolution equation of the Fick-Jacobs type corrected by an effective coefficient $D\left(x\right)$. The procedure enables us to derive this function within a recurrence scheme. We test this result on a model of stationary diffusion in a linear cone in a homogeneous potential, which is exactly solvable. Extension of the approximate formulas for $D\left(x\right)$ derived for the diffusion alone is discussed.

The free-energy distribution function of an elastic string in a quenched random potential, $P_L\left(F\right)$, is investigated with the help of the optimal fluctuation approach. The form of the far-right tail of $P_L\left(F\right)$ is found by constructing the exact solution of the nonlinear saddle-point equations describing the asymptotic form of the optimal fluctuation. The solution of the problem is obtained for two different types of boundary conditions and for an arbitrary dimension of the imbedding space $1+d$ with $d$ from the interval $0<d<2$. The results are also applicable for the description of the far-left tail of the height distribution function in the stochastic growth problem described by the $d$-dimensional Kardar-Parisi-Zhang equation.

We report noninvasive measurements of the complex field of elastic quasimodes of a silicon wafer with chaotic shape. The amplitude and phase spatial distribution of the flexural modes are directly obtained by Fourier transform of time measurements. We investigate the crossover from real mode to complex-valued quasimode, when absorption is progressively increased on one edge of the wafer. The complexness parameter, which characterizes the degree to which a resonance state is complex valued, is measured for nonoverlapping resonances, and is found to be proportional to the nonhomogeneous contribution to the line broadening of the resonance. A simple two-level model based on the effective Hamiltonian formalism supports our experimental results.

We study the evolution of a wave packet in a nonlinear Stark ladder. In the absence of nonlinearity all normal modes are spatially localized giving rise to an equidistant eigenvalue spectrum and Bloch oscillations. Nonlinearity induces frequency shifts and mode-mode interactions and destroys localization. For large strength of nonlinearity we observe single-site trapping as a transient, with subsequent explosive spreading, followed by subdiffusion. For moderate nonlinearities an immediate subdiffusion takes place. Finally, for small nonlinearities we find linear Stark localization as a transient, with subsequent subdiffusion. For single-mode excitations and weak nonlinearities, stability intervals are predicted and observed upon variation in the dc bias strength, which affects the short- and the long-time dynamics.

The pinch-off of an axisymmetric air bubble surrounded by an inviscid fluid is compared in four physical realizations: (i) cavity collapse in the wake of an impacting disk, (ii) gas bubbles injected through a small orifice, (iii) bubble rupture in a straining flow, and (iv) a bubble with an initially necked shape. Our boundary-integral simulations suggest that all systems eventually follow the universal behavior characterized by slowly varying exponents predicted by J. Eggers et al. [Phys. Rev. Lett. 98, 094502 (2007)]. However, the time scale for the onset of this final regime is found to vary by orders of magnitude depending on the system in question. While for the impacting disk it is well in the millisecond range, for the gas injection needle universal behavior sets in only a few microseconds before pinch-off. These findings reconcile the different views expressed in recent literature about the universal nature of bubble pinch-off.