The $k$-core decomposition of a network has thus far mainly served as a powerful tool for the empirical study of complex networks. We now propose its explicit integration in a theoretical model. We introduce a hard-core random network (HRN) model that generates maximally random networks with arbitrary degree distribution and arbitrary $k$-core structure. We then solve exactly the bond percolation problem on the HRN model and produce fast and precise analytical estimates for the corresponding real networks. Extensive comparison with real databases reveals that our approach performs better than existing models, while requiring less input information.

We focus on the front propagation of diffusive flames obtained from the downward burning of inclined thermally thin solid fuels. This process consists of a pyrolysis reaction in the solid-phase and a combustion reaction in the gas phase. The solid-phase model is based on two coupled one-dimensional equations of temperature and solid density. We reduce the system into a single one-dimensional equation from which we obtain an analytical expression for the flame front speed. This expression may be understood as an upper bound of the burning spread rate in inclined samples. The gas-phase model is based on four coupled two-dimensional equations. These are employed to derive a criterion for determining the critical inclination angle beyond which the flame behavior becomes unstable. The comparison with the experiments confirms the validity of our predictions.

We performed laboratory experiments of Rayleigh-Bénard convection with liquid gallium under various intensities of a uniform imposed horizontal magnetic field. An ultrasonic velocity profiling method was used to visualize the spatiotemporal structure of the flows with simultaneous monitoring of the temperature fluctuations in the liquid gallium layer. The explored Rayleigh numbers Ra range from the critical value for onset of convection to ${10}^5$; the Chandrasekhar number $Q$ covers values up to 1100. A regime diagram of the convection patterns was established in relation to the Ra and $Q$ values for a square vessel with aspect ratio 5. We identified five flow regimes: (I) a fluctuating large-scale pattern without rolls, (II) weakly constrained rolls with fluctuations, (III) a continuous oscillation of rolls, (IV) repeated roll number transitions with random reversals of the flow direction, and (V) steady two-dimensional (2D) rolls. These flow regimes are classified by the Ra/$Q$ values, the ratio of the buoyancy to the Lorentz force. Power spectra from the temperature time series indicate that regimes I and II have the features of developed turbulence, while the other regimes do not. The region of steady 2D rolls (Busse balloon) extends to high Ra values in the present setting by a horizontal magnetic field and regime V is located inside the Busse balloon. Concerning the instabilities of the steady 2D rolls, regime III is the traveling wave convection developed from the oscillatory instability. Regime IV can be regarded as a state of phase turbulence, which is induced by intermittent occurrences of the skewed-varicose instability.

Theory of two-population lattice Boltzmann equations for thermal flow simulations is revisited. The present approach makes use of a consistent division of the conservation laws between the two lattices, where mass and the momentum are conserved quantities on the first lattice, and the energy is conserved quantity of the second lattice. The theory of such a division is developed, and the advantage of energy conservation in the model construction is demonstrated in detail. The present fully local lattice Boltzmann theory is specified on the standard lattices for the simulation of thermal flows. Extension to the subgrid entropic lattice Boltzmann formulation is also given. The theory is validated with a set of standard two-dimensional simulations including planar Couette flow and natural convection in two dimensions.

In the framework of superstatistics it has been shown that one can calculate the entropy of nonextensive statistical mechanics. We follow a similar procedure; we assume a $\Gamma \left({\chi }^2\right)$ distribution depending on $\beta$ that also depends on a parameter $p_l$. From it we calculate the Boltzmann factor and show that it is possible to obtain the information entropy $S=k{\sum }_{l=1}^{\Omega }s\left(p_l\right)$, where $s\left(p_l\right)=1-p_l^{p_l}$. By maximizing this information measure, $p_l$ is calculated as function of $\beta E_l$ and, at this stage of the procedure, $p_l$ can be identified with the probability distribution. We show the validity of the saddle-point approximation and we also briefly discuss the generalization of one of the four Khinchin axioms. The modified axioms are then in accordance with the proposed entropy. As further possibilities, we also propose other entropies depending on $p_l$ that resemble the Kaniakadis and two possible Sharma-Mittal entropies. By expanding in series all entropies in this work we have as a first term the Shannon entropy.

The quantum ergordic theorem for a large class of quantum systems was proved by von Neumann [Z. Phys. 57, 30 (1929)] and again by Reimann [Phys. Rev. Lett. 101, 190403 (2008)] in a more practical and well-defined form. However, it is not clear whether the theorem applies to quantum chaotic systems. With a rigorous proof still elusive, we illustrate and verify this theorem for quantum chaotic systems with examples. Our numerical results show that a quantum chaotic system with an initial low-entropy state will dynamically relax to a high-entropy state and reach equilibrium. The quantum equilibrium state reached after dynamical relaxation bears a remarkable resemblance to the classical microcanonical ensemble. However, the fluctuations around equilibrium are distinct: The quantum fluctuations are exponential while the classical fluctuations are Gaussian.

A model of a single ferromagnetic particle with a finite coupling energy of the magnetic moment with the body of the particle is formulated, and regimes of its motion in a rotating magnetic field are investigated. Regimes are possible that are synchronous and asynchronous with the field. In a synchronous regime the easy axis of the particle is in the plane of the rotating magnetic field at low frequencies (a planar regime) and on the cone at high frequencies (a precession regime). The stability of these regimes is investigated, and it is shown that the precession regime is stable for field strengths below the critical value. In a particular range of field strength value, irreversible jumps of the magnetic moment take place in the asynchronous planar regime. The stability of this regime is investigated, and it is shown that it is stable for field strengths above the critical value, which depends on the frequency. The implications of these results for the energy dissipation in a rotating field are analyzed, and it is shown that the maximum of the heat production near the transition to the synchronous regime is flattened out by the transition to the precession regime.

We present a free energy model for structural transitions of the DNA double helix driven by tensile and torsional stress. Our model is coarse grained and is based on semiflexible polymer descriptions of B-DNA, underwound L-DNA, and highly overwound P-DNA. The statistical-mechanical model of plectonemic supercoiling previously developed for B-DNA is applied to semiflexible polymer models of P- and L-DNA to obtain a model of DNA structural transitions in quantitative accord with experiment. We identify two distinct plectonemic states, one “inflated” by electrostatic repulsion and thermal fluctuations and the other “collapsed,” with the two double helices inside the supercoils driven to close contact. We find that supercoiled B and L are stable only in the inflated form, while supercoiled P is always collapsed. We also predict the behavior and experimental signatures of highly underwound “Q”-DNA, the left-handed analog of P-DNA; as for P, supercoiled Q is always collapsed. Overstretched “S”-DNA and strand-separated “stress-melted” DNA are also included in our model, allowing prediction of a global phase diagram for forces up to 1000 pN and torques between $\pm$60 pN nm, or, in terms of linking number density, from $\sigma =-5$ to $+3$.

The available literature supports the hypothesis that the morphology of the inner mitochondrial membrane is regulated by different energy states, that the three-dimensional morphology of cristae is dynamic, and that both are related to biochemical function. Examination of the correlation between the inner mitochondrial membrane (IMM) structure and mitochondrial energetic function is critical to an understanding of the links between mesoscale morphology and function in progressive mitochondrial dysfunction such as aging, neurodegeneration, and disease. To investigate this relationship, we develop a model to examine the effects of three-dimensional IMM morphology on the electrochemical potential of mitochondria. The two-dimensional axisymmetric finite element method is used to simulate mitochondrial electric potential and proton concentration distribution. This simulation model demonstrates that the proton motive force ($\Delta p$) produced on the membranes of cristae can be higher than that on the inner boundary membrane. The model also shows that high proton concentration in cristae can be induced by the morphology-dependent electric potential gradient along the outer side of the IMM. Furthermore, simulation results show that a high $\Delta p$ is induced by the large surface-to-volume ratio of an individual crista, whereas a high capacity for ATP synthesis can primarily be achieved by increasing the surface area of an individual crista. The mathematical model presented here provides compelling support for the idea that morphology at the mesoscale is a significant driver of mitochondrial function.

The rotating reduced Maxwell-Bloch (RMB) equations, which describe the propagation of few-cycle optical pulses in a transparent media with two isotropic polarized electronic field components, are derived from a system of complete Maxwell-Bloch equations without using the slowly varying envelope approximations. Two hierarchies of the obtained rational solutions, including rogue waves, which are also called few-cycle optical rogue waves, of the rotating RMB equations are constructed explicitly through degenerate Darboux transformation. In addition to the above, the dynamical evolution of the first-, second-, and third-order few-cycle optical rogue waves are constructed with different patterns. For an electric field $E$ in the three lower-order rogue waves, we find that rogue waves correspond to localized large amplitude oscillations of the polarized electric fields. Further a complementary relationship of two electric field components of rogue waves is discussed in terms of analytical formulas as well as numerical figures.

The ability of semiclassical initial-value representation (IVR) methods to determine approximate energy levels for bound systems is limited due to problems associated with long classical trajectories. These difficulties become especially severe for large or classically chaotic systems. This work attempts to overcome such problems by developing an IVR expression that is classically equivalent to Bogomolny's formula for the transfer matrix [E. B. Bogomolny, Nonlinearity 5, 805 (1992); Chaos 2, 5 (1992)] and can be used to determine semiclassical energy levels. The method is adapted to levels associated with states of desired symmetries and applied to two two-dimensional quartic oscillator systems, one integrable and one mostly chaotic. For both cases, the technique is found to resolve all energy levels in the ranges investigated. The IVR method does not require a search for special trajectories obeying boundary conditions on the Poincaré surface of section and leads to more rapid convergence of Monte Carlo phase space integrations than a previously developed IVR technique. It is found that semiclassical energies can be extracted from the eigenvalues of transfer matrices of dimension close to the theoretical minimum determined by Bogomolny's theory. The results support the assertion that the present IVR theory provides a different semiclassical approximation to the transfer matrix than that of Bogomolny for $\hslash \ne 0$. For the chaotic system investigated the IVR energies are found to be generally more accurate than those predicted by Bogomolny's theory.

The physical mechanisms responsible for the electrical orientation and electrical rotation of metal nanowires suspended in an electrolyte as a function of frequency of the applied ac electric field are examined theoretically and experimentally. The alignment of a nanowire in an ac field with a fixed direction is called electro-orientation. The induced constant rotation of a nanowire in a rotating electric field is called electrorotation. In both situations, the applied electric field interacts with the induced charge in the electrical double layer at the metal-electrolyte interface, causing rotation due to the torque on the induced dipole, and also from induced-charge electro-osmotic flow around the particle. First, we describe the dipole theory that describes electro-orientation and electrorotation of perfectly polarizable metal rods. Second, based on a slender approximation, an analytical theory that describes induced-charge electro-orientation and electrorotation of metal nanowires is provided. Finally, experimental measurements of the electro-orientation and electrorotation of metal nanowires are presented and compared with theory, providing a comprehensive study of the relative importance between induced-dipole rotation and induced-charge electro-osmotic rotation.

We introduce an explicit solution for the nonequilibrium steady state (NESS) of a ring that is coupled to a thermal bath, and is driven by an external hot source with log-wide distribution of couplings. Having time scales that stretch over several decades is similar to glassy systems. Consequently there is a wide range of driving intensities where the NESS is like that of a random walker in a biased Brownian landscape. We investigate the resulting statistics of the induced current $I$. For a single ring we discuss how sign of $I$ fluctuates as the intensity of the driving is increased, while for an ensemble of rings we highlight the fingerprints of Sinai physics on the distribution of the absolute value of $I$.

Problems involving the capture of a moving entity by a trap occur in a variety of physical situations, the moving entity being an electron, an excitation, an atom, a molecule, a biological object such as a receptor cluster, a cell, or even an animal such as a mouse carrying an epidemic. Theoretical considerations have almost always assumed that the particle motion is translationally invariant. We study here the case when that assumption is relaxed, in that the particle is additionally subjected to a harmonic potential. This tethering to a center modifies the reaction-diffusion phenomenon. Using a Smoluchowski equation to describe the system, we carry out a study which is explicit in one dimension but can be easily extended for arbitrary dimensions. Interesting features emerge depending on the relative location of the trap, the attractive center, and the initial placement of the diffusing particle.

We study the diffusion of an ensemble of overdamped particles sliding over a tilted random potential (produced by the interaction of a particle with a random polymer) with long-range correlations. We found that the diffusion properties of such a system are closely related to the correlation function of the corresponding potential. We model the substrate as a symbolic trajectory of a shift space which enables us to obtain a general formula for the diffusion coefficient when normal diffusion occurs. The total time that the particle takes to travel through $n$ monomers can be seen as an ergodic sum to which we can apply the central limit theorem. The latter can be implemented if the correlations decay fast enough in order for the central limit theorem to be valid. On the other hand, we presume that when the central limit theorem breaks down the system give rise to anomalous diffusion. We give two examples exhibiting a transition from normal to anomalous diffusion due to this mechanism. We also give analytical expressions for the diffusion exponents in both cases by assuming convergence to a stable law. Finally we test our predictions by means of numerical simulations.

We propose a lattice statistical model to investigate the phase diagrams and the soft responses of nematic liquid-crystal elastomers. Using suitably scaled infinite-range interactions, we obtain exact self-consistent equations for the tensor components of the nematic order parameter in terms of temperature, the distortion and stress tensors, and the initial nematic order. These equations are amenable to simple numerical calculations, which are used to characterize the low-temperature soft regime. We find a peculiar phase diagram, in terms of temperature and the diagonal component of the distortion tensor along the stretching direction, with first- and second-order transitions to the soft phase, and the prediction of tricritical points. This behavior is not qualitatively changed if we use different values of the initial nematic order parameter.

We demonstrate the accurate calculation of entropies and free energies for a variety of liquid metals using an extension of the two-phase thermodynamic (2PT) model based on a decomposition of the velocity autocorrelation function into gas-like (hard sphere) and solid-like (harmonic) subsystems. The hard sphere model for the gas-like component is shown to give systematically high entropies for liquid metals as a direct result of the unphysical Lorentzian high-frequency tail. Using a memory function framework we derive a generally applicable velocity autocorrelation and frequency spectrum for the diffusive component which recovers the low-frequency (long-time) behavior of the hard sphere model while providing for realistic short-time coherence and high-frequency tails to the spectrum. This approach provides a significant increase in the accuracy of the calculated entropies for liquid metals and is compared to ambient pressure data for liquid sodium, aluminum, gallium, tin, and iron. The use of this method for the determination of melt boundaries is demonstrated with a calculation of the high-pressure bcc melt boundary for sodium. With the significantly improved accuracy available with the memory function treatment for softer interatomic potentials, the 2PT model for entropy calculations should find broader application in high energy density science, warm dense matter, planetary science, geophysics, and material science.

The structure and growth of molecular NaCl crystals in bulk and in a narrow, nanometer-sized apolar confinement are examined by explicit-water molecular dynamics computer simulations. It is demonstrated that fast crystallization and subsequent diffusion-controlled cluster growth in bulk is triggered by supersaturations that exceed a certain threshold value. In confinement, simulated in a pseudo grand canonical setup, salt is shown to be expelled from the narrow apolar slab region, and the effective ion concentration inside the nanoconfinement is always considerably lower than the reservoir salt concentration so that no fast crystallization takes place. For very small slab widths ($d<1.5$ nm) salt is almost entirely expelled while water remains in the slab, indicating a capillary evaporation phenomenon for the polar ions. If forced into the apolar confinement by simulating in a strictly canonical setup, we find stable crystals only if at least three crystalline planes fit into the slab, which happens above a 2-nm slab width. In this case the (100) plane of the bulk crystal is oriented parallel to the apolar surface delimited by a subnanometer thin hydration layer. This work presents molecular-level insight of salt crystallization in apolar confinements of a nanometer scale with possible implications in double-layer supercapacitor physics and geological salt weathering.

Dielectric saturation originates from the upper bound to dipole orientation, reached when all dipoles are aligned “perfectly” with respect to the electric field. For Debye-type dynamics, it is well established that the saturation effect is diminished at high frequencies relative to its steady-state value. Here, it is argued that a similar frequency dependence of this nonlinear dielectric effect is expected also for dispersive dynamics, provided that the system is dynamically homogeneous. By contrast, more realistic relaxation time dispersions based upon heterogeneous dynamics display a strongly reduced frequency dependence of dielectric saturation. Calculations demonstrate this effect in terms of both the fundamental and the third harmonic frequency susceptibilities. The relations of signatures of nonlinearity in different Fourier components of the response are briefly discussed.