We consider the three-dimensional (3D) Bak-Tang-Wiesenfeld model in a cubic lattice. Along with analyzing the 3D problem, the geometrical structure of the two-dimensional (2D) cross section of waves is investigated. By analyzing the statistical observables defined in the cross sections, it is shown that the model in that plane (named as 2D-induced model) is in the critical state and fulfills the finite-size scaling hypothesis. The analysis of the critical loops that are interfaces of the 2D-induced model is of special importance in this paper. Most importantly, we see that their fractal dimension is $D_f=1.387\pm 0.005$, which is compatible with the fractal dimension of the external perimeter of geometrical spin clusters of 2D critical Ising model. Some hyperscaling relations between the exponents of the model are proposed and numerically confirmed. We then address the problem of conformal invariance of the mentioned domain walls using Schramm-Lowener evolution (SLE). We found that they are described by SLE with the diffusivity parameter $\kappa =2.8\pm 0.2$, nearly consistent with observed fractal dimension.

Using a method of eigenfunction expansion, a stochastic equation is developed for the generalized Schrödinger equation with random fluctuations. The wave field $\psi$ is expanded in terms of eigenfunctions: $\psi ={\sum }_n{a}_n\left(t\right){\varphi }_n\left(x\right)$, with ${\varphi }_n$ being the eigenfunction that satisfies the eigenvalue equation $H_0{\varphi }_n={\lambda }_n{\varphi }_n$, where $H_0$ is the reference “Hamiltonian” conventionally called the “unperturbed” Hamiltonian. The Langevin equation is derived for the expansion coefficient $a_n\left(t\right)$, and it is converted to the Fokker-Planck (FP) equation for a set $\left\{a_n\right\}$ under the assumption of Gaussian white noise for the fluctuation. This procedure is carried out by a functional integral, in which the functional Jacobian plays a crucial role in determining the form of the FP equation. The analyses are given for the FP equation by adopting several approximate schemes.

A question that is currently highly debated is whether the microcanonical entropy should be expressed as the logarithm of the phase volume (volume entropy, also known as the Gibbs entropy) or as the logarithm of the density of states (surface entropy, also known as the Boltzmann entropy). Rather than postulating them and investigating the consequence of each definition, as is customary, here we adopt a bottom-up approach and construct the entropy expression within the microcanonical formalism upon two fundamental thermodynamic pillars: (i) The second law of thermodynamics as formulated for quasistatic processes: $\delta Q/T$ is an exact differential, and (ii) the law of ideal gases: $PV=k_{B}NT$. The first pillar implies that entropy must be some function of the phase volume $\mathrm{\Omega }$. The second pillar singles out the logarithmic function among all possible functions. Hence the construction leads uniquely to the expression $S=k_{B}ln\mathrm{\Omega }$, that is, the volume entropy. As a consequence any entropy expression other than that of Gibbs, e.g., the Boltzmann entropy, can lead to inconsistencies with the two thermodynamic pillars. We illustrate this with the prototypical example of a macroscopic collection of noninteracting spins in a magnetic field, and show that the Boltzmann entropy severely fails to predict the magnetization, even in the thermodynamic limit. The uniqueness of the Gibbs entropy, as well as the demonstrated potential harm of the Boltzmann entropy, provide compelling reasons for discarding the latter at once.

Complex networks exhibit a wide range of collective dynamic phenomena, including synchronization, diffusion, relaxation, and coordination processes. Their asymptotic dynamics is generically characterized by the local Jacobian, graph Laplacian, or a similar linear operator. The structure of networks with regular, small-world, and random connectivities are reasonably well understood, but their collective dynamical properties remain largely unknown. Here we present a two-stage mean-field theory to derive analytic expressions for network spectra. A single formula covers the spectrum from regular via small-world to strongly randomized topologies in Watts-Strogatz networks, explaining the simultaneous dependencies on network size $N$, average degree $k$, and topological randomness $q$. We present simplified analytic predictions for the second-largest and smallest eigenvalue, and numerical checks confirm our theoretical predictions for zero, small, and moderate topological randomness $q$, including the entire small-world regime. For large $q$ of the order of one, we apply standard random matrix theory, thereby overarching the full range from regular to randomized network topologies. These results may contribute to our analytic and mechanistic understanding of collective relaxation phenomena of network dynamical systems.

A ball bouncing repeatedly on a vertically vibrating surface constitutes the famous “bouncing ball” problem, a nonlinear system used in the 1980s, and still in use nowadays, to illustrate the route to chaos by period doubling. In experiments, in order to avoid the ball escape that would be inevitable with a flat surface, a concave lens is often used to limit the horizontal motion. However, we observe experimentally that the system is not stable. The ball departs from the system axis and exhibits a pendular motion in the permanent regime. We propose theoretical arguments to account for the decrease of the growth rate and of the asymptotic-size of the trajectory when the frequency of the vibration is increased. The instability is very sensitive to the physics of the contacts, which makes it a potentially interesting way to study the collisions rules, or to test the laws used in numerical studies of granular matter.

We investigate aspects of low-magnetic-Reynolds-number flow between two parallel, perfectly insulating walls in the presence of an imposed magnetic field parallel to the bounding walls. We find a functional basis to describe the flow, well adapted to the problem of finding the attractor dimension and which is also used in subsequent direct numerical simulation of these flows. For given Reynolds and Hartmann numbers, we obtain an upper bound for the dimension of the attractor by means of known bounds on the nonlinear inertial term and this functional basis for the flow. Three distinct flow regimes emerge: a quasi-isotropic three-dimensional (3D) flow, a nonisotropic 3D flow, and a 2D flow. We find the transition curves between these regimes in the space parametrized by Hartmann number Ha and attractor dimension $d_{\text{att}}$. We find how the attractor dimension scales as a function of Reynolds and Hartmann numbers (Re and Ha) in each regime. We also investigate the thickness of the boundary layer along the bounding wall and find that in all regimes this scales as 1/Re, independently of the value of Ha, unlike Hartmann boundary layers found when the field is normal to the channel. The structure of the set of least dissipative modes is indeed quite different between these two cases but the properties of turbulence far from the walls (smallest scales and number of degrees of freedom) are found to be very similar.

We analyze the angular dynamics of a neutrally buoyant, nearly spherical particle immersed in a steady general linear flow. The hydrodynamic torque acting on the particle is obtained by means of a reciprocal theorem, a regular perturbation theory exploiting the small eccentricity of the nearly spherical particle, and by assuming that inertial effects are small but finite.

One of the major factors governing the mode of failure in disordered solids is the effective range $R$ over which the stress field is modified following a local rupture event. In a random fiber bundle model, considered as a prototype of disordered solids, we show that the failure mode is nucleation dominated in the large system size limit, as long as $R$ scales slower than $L^{\zeta }$, with $\zeta =2/3$. For a faster increase in $R$, the failure properties are dominated by the mean-field critical point, where the damages are uncorrelated in space. In that limit, the precursory avalanches of all sizes are obtained even in the large system size limit. We expect these results to be valid for systems with finite (normalizable) disorder.

Mixed spin-1/2 and spin-1 Ising ferrimagnets on a triangular lattice with sublattices A, B, and C are studied for two spin-value distributions $(S_{\mathrm{A}},S_{\mathrm{B}},S_{\mathrm{C}})=(1/2,1/2,1)$ and $(1/2,1,1)$ by Monte Carlo simulations. The nonbipartite character of the lattice induces geometrical frustration in both systems, which leads to the critical behavior rather different from their ferromagnetic counterparts. We confirm second-order phase transitions belonging to the standard Ising universality class occurring at higher temperatures, however, in both models these change at tricritical points (TCP) to first-order transitions at lower temperatures. In the model $(1/2,1/2,1)$, TCP occurs on the boundary between paramagnetic and ferrimagnetic $(\pm 1/2,\pm 1/2,\mp 1)$ phases. The boundary between two ferrimagnetic phases $(\pm 1/2,\pm 1/2,\mp 1)$ and $(\pm 1/2,\mp 1/2,0)$ at lower temperatures is always first order and it is joined by a line of second-order phase transitions between the paramagnetic and the ferrimagnetic $(\pm 1/2,\mp 1/2,0)$ phases at a critical endpoint. The tricritical behavior is also confirmed in the model $(1/2,1,1)$ on the boundary between the paramagnetic and ferrimagnetic $(0,\pm 1,\mp 1)$ phases.

In this work we study an $r$-species mixture of gases within the relativistic kinetic theory point of view. We use the relativistic covariant Boltzmann equation and incorporate the Schwarzschild metric. The method of solution of the Boltzmann equation is a combination of the Chapman-Enskog and Grad representations. The thermodynamic four-fluxes are expressed as functions of the thermodynamic forces so the generalized expressions for the Navier-Stokes, Fick, and Fourier laws are obtained. The constitutive equations for the diffusion and heat four-fluxes of the mixture are functions of thermal and diffusion generalized forces which depend on the acceleration and the gravitational potential gradient. While this dependence is of relativistic nature for the thermal force, this is not the case for the diffusion forces. We show also that the matrix of diffusion coefficients is symmetric, implying that the thermal-diffusion equals the diffusion-thermal effect, proving the Onsager reciprocity relations. The entropy four-flow of the mixture is also expressed in terms of the thermal and diffusion generalized forces, so its dependence on the acceleration and gravitational potential gradient is also determined.

We present results obtained by using nonlinear irreversible models for heat devices. In particular, we focus on the global performance characteristics, the maximum efficiency and the efficiency at maximum power regimes for heat engines, and the maximum coefficient of performance (COP) and the COP at maximum cooling power regimes for refrigerators. We analyze the key role played by the interplay between irreversibilities coming from heat leaks and internal dissipations. We also discuss the relationship between these results and those obtained by different models.

The stochastic motion in a nonhomogeneous medium with traps is studied and diffusion properties of that system are discussed. The particle is subjected to a stochastic stimulation obeying a general Lévy stable statistics and experiences long rests due to nonhomogeneously distributed traps. The memory is taken into account by subordination of that process to a random time; then the subordination equation is position dependent. The problem is approximated by a decoupling of the medium structure and memory and exactly solved for a power-law position dependence of the memory. In the case of the Gaussian statistics, the density distribution and moments are derived: depending on geometry and memory parameters, the system may reveal both the subdiffusion and enhanced diffusion. The similar analysis is performed for the Lévy flights where the finiteness of the variance follows from a variable noise intensity near a boundary. Two diffusion regimes are found: in the bulk and near the surface. The anomalous diffusion exponent as a function of the system parameters is derived.

We present generalized methods for calculating lower bounds on the ground-state entropy per site, $S_0$, or equivalently, the ground-state degeneracy per site, $W=e^{S_0/k_B}$, of the antiferromagnetic Potts model. We use these methods to derive improved lower bounds on $W$ for several lattices.

The geometrically frustrated spin-1/2 Ising-Heisenberg model on triangulated Husimi lattices is exactly solved by combining the generalized star-triangle transformation with the method of exact recursion relations. The ground-state and finite-temperature phase diagrams are rigorously calculated along with both sublattice magnetizations of the Ising and Heisenberg spins. It is evidenced that the Ising-Heisenberg model on triangulated Husimi lattices with two or three interconnected triangles-in-triangles units displays in a highly frustrated region a quantum disorder irrespective of temperature, whereas the same model on triangulated Husimi lattices with a greater connectivity of triangles-in-triangles units exhibits at low enough temperatures an outstanding quantum order due to the order-by-disorder mechanism. The quantum reduction of both sublattice magnetizations in the peculiar quantum ordered state gradually diminishes upon increasing the coordination number of the underlying Husimi lattice.

For the lattice gas models of arbitrary geometry and dimensions with absolute repulsion between particles at zero distance (a hard core identical to a single lattice site) and arbitrary repulsion or attraction at other distances, the “hole-particle” symmetry of the system potential energy has been stated and an equation of state has been derived on the basis of the classical Gibbs statistics. The equation is completely analogous to the well-known virial equation of state, except that it is more accurate at high-density states, while the virial equation has the low-density limitation. Both equations contain the common set of the so-called irreducible integrals, related to the corresponding virial coefficients, and can be used together to describe the behavior of a lattice gas in a wide range of densities.

The spreading of liquid drops on solid surfaces is a wide-spread phenomenon of both fundamental and industrial interest. In many applications, surfaces are porous and spreading patterns are very complex with respect to the case on smooth surfaces. Focusing on the inertial spreading just before the Tanner-like viscous regime, this work investigates the spreading of a low-viscosity droplet on a porous surface using lattice Boltzmann numerical simulations. The case of a flat surface is first considered, and it reveals a dependence on the solid equilibrium contact angle ${\theta }_s^{eq}$, which is in good agreement with published experimental data. We conducted numerical experiments with various surfaces perforated by a regular pattern of holes of infinite length. The results show that the global spreading dynamics is independent of the porosity morphology. Through the assumption that, for wetting, the pores can be regarded as surface patches with a contact angle of ${\theta }_{\mathrm{pore}}^{eq}={180}^{\circ }$, we deduce an effective equilibrium contact angle ${\theta }_{\mathrm{eff}}^{eq}$ on the porous surface from the Cassie-Baxter law. A spreading model is then proposed to describe both a prefactor and an exponent that are similar to a flat surface whose equilibrium contact angle is ${\theta }_{\mathrm{eff}}^{eq}$. This model compares satisfactorily with a large number of numerical experiments under varying conditions.

We study a model describing the force-extension curves of modular proteins, nucleic acids, and other biomolecules made out of several single units or modules. At a mesoscopic level of description, the configuration of the system is given by the elongations of each of the units. The system free energy includes a double-well potential for each unit and an elastic nearest-neighbor interaction between them. Minimizing the free energy yields the system equilibrium properties whereas its dynamics is given by (overdamped) Langevin equations for the elongations, in which friction and noise amplitude are related by the fluctuation-dissipation theorem. Our results, both for the equilibrium and the dynamical situations, include analytical and numerical descriptions of the system force-extension curves under force or length control and agree very well with actual experiments in biomolecules. Our conclusions also apply to other physical systems comprising a number of metastable units, such as storage systems or semiconductor superlattices.

The effect of discreteness on stochastic dynamics of chemically reacting systems is studied analytically. We apply the scheme bridging the chemical master equation and the chemical Fokker-Planck equation by a parameter representing the degree of discreteness previously proposed by the author for two concrete systems. One is an autocatalytic reaction system, and the other is a branching-annihilation reaction system. It is revealed that the change in characteristic time scales when discreteness is decreased is yielded between the two systems for different reasons. In the former system, it originates from the boundaries where one of the chemical species is zero, whereas in the latter system, it is due to modification of the most probable extinction path caused by discreteness loss.

A one-sided model of the thermal Marangoni instability owing to evaporation into an inert gas is developed. Two configurations are studied in parallel: a horizontal liquid layer and a spherical droplet. With the dynamic gas properties being admittedly negligible, one-sided approaches typically hinge upon quantifying heat and mass transfer through the gas phase by means of transfer coefficients (like in the Newton's cooling law), which in dimensionless terms eventually corresponds to using Biot numbers. Quite a typical arrangement encountered in the literature is a constant Biot number, the same for perturbations of different wavelengths and maybe even the same as for the reference state. In the present work, we underscore the relevance of accounting for its wave-number dependence, which is especially the case in the evaporative context with relatively large values of the resulting effective Biot number. We illustrate the effect in the framework of the Marangoni instability thresholds. As a concrete example, we consider HFE-7100 (a standard refrigerant) for the liquid and air for the inert gas.