A general method is proposed for predicting the asymptotic percolation threshold of networks with bottlenecks, in the limit that the subnet mesh size goes to zero. The validity of this method is tested for bond percolation on filled checkerboard and “stack-of-triangle” lattices. Thresholds for the checkerboard lattices of different mesh sizes are estimated using the gradient percolation method, while for the triangular system they are found exactly using the triangle-triangle transformation. The values of the thresholds approach the asymptotic values of 0.64222 and 0.53993, respectively, as the mesh is made finer, consistent with a direct determination based upon the predicted critical corner-connection probability.

A Brownian sieve is a spatially periodic microstructured device that combines the effects of thermal noise, local spatial asymmetry, and external forces to separate particles based on their transport properties. By treating the motion of an individual particle as a cyclical process in which the particle fluctuates away from and returns to the origin of some unit cell, I derive generalized fluctuation-dissipation and reciprocal relations for the averages (and for all moments) of the number of periodic displacements that are exact and valid for arbitrary values of the external forces. These relations hold not only for Brownian sieves, but for all molecular machines in which a nanoscale system couples two chemical, mechanical, or transport processes by a cycle in which the molecular machine itself fluctuates away from and then returns to some arbitrary reference state, in the process doing or receiving work on or from the environment.

Active Brownian motion is the complex motion of active Brownian particles. They are “active” in the sense that they can transform their internal energy into energy of motion and thus create complex motion patterns. Theories of active Brownian motion so far imposed couplings between the internal energy and the kinetic energy of the system. We investigate how this idea can be naturally taken further to include also couplings to the potential energy, which finally leads to a general theory of canonical dissipative systems. Explicit analytical and numerical studies are done for the motion of one particle in harmonic external potentials. Apart from stationary solutions, we study nonequilibrium dynamics and show the existence of various bifurcation phenomena.

We derive a simple expression for the probability of trajectories of a master equation. The expression is particularly useful when the number of states is small and permits the calculation of observables that can be defined as functionals of whole trajectories. We illustrate the method with a two-state master equation, for which we calculate the distribution of the time spent in one state and the distribution of the number of transitions, each in a given time interval. These two expressions are obtained analytically in terms of modified Bessel functions.

We apply path integrals to study nonequilibrium work theorems in the context of Brownian dynamics, deriving in particular the equations of motion governing the most typical and most dominant trajectories. For the analytically soluble cases of a moving harmonic potential and a harmonic oscillator with a time-dependent natural frequency, we find such trajectories, evaluate the work-weighted propagators, and validate Jarzynski’s equality.

We present results of the collision process of a bead onto a static granular packing. We provide, in particular, a three-dimensional (3D) extensive characterization of this process from a model experiment that allows us to propel a spherical bead onto a granular packing with a well-controlled velocity and impact angle. A collision typically produces a high-energy particle (rebound particle) and several low-energy grains (ejected particles). The collision process is recorded by means of two fast video cameras. The sequence of images from both cameras are then analyzed via image processing and the trajectories of all particles are reconstructed in 3D space. We show that the incident particle does not remain in the vertical incident plane after the rebound and that the deviation angle increases with increasing impact angle. Concerning the ejected particles, we demonstrated that the ejection angle (measured with respect to the horizontal plane) is surprisingly independent of both the impact angle and velocity of the incident particle, and is very close to 60°. The horizontal component of the ejection speed of the splashed particles is found to be weakly dependent on the incident speed and impact angle, and is relatively isotropic (no particular horizontal direction is favored). This last feature suggests that the bead packing acts as a perfect diffusive medium with respect to energy propagation.

A thin water film on a cleaved mica substrate undergoes a first-order phase transition between two values of film thickness. By inducing a finite evaporation rate of the water, the interface between the two phases develops a fingering instability similar to that observed in the Saffman-Taylor problem. We draw the connection between the two problems, and construct solutions describing the dynamics of evaporation in this system.

The self-organization of liquid crystal molecules of 4-$n$-pentyl-$4^{\prime }$-cyanobiphenyl (5CB) forming an oriented monolayer by condensation from the vapor phase onto a silicon oxynitride surface has been observed using the evanescent wave dual slab waveguide dual polarization mode interferometry (DPI) technique. Two distinct stages to the layer formation are observed: After the formation of a layer of molecules lying prone on the surface, further condensation begins to densify the layer and produces a gradual mutual alignment of the molecules until the fully condensed, fully aligned monolayer is reached. At this limit the full coverage 5CB monolayer on this surface and at a temperature of $25°\mathrm{C}$, is found to be anchored with an average molecular axis polar angle of $56\pm 1°$ and with a measured thickness of $16.6\pm 0.5\mathrm{\AA }$. These results are in reasonable agreement with the molecular dimensions provided by molecular models. The apparent precision and accuracy of these results resolves some wide disparity between earlier studies of such systems. Previous difficulties in determining optogeometrical properties of such ultrathin birefringent films using ellipsometry or in the need for complex modeling of the film layer structure using x-ray reflectivity are overcome in this instance. We provide a technique for analyzing the dual polarization data from DPI such that the bulk refractive index values, when known, can be used to determine the orientation and thickness of a layer that is on the nanometer or subnanometer scale.

Transport of a particle in a spatially periodic harmonic potential under the influence of a slowly time-dependent unbiased periodic external force is studied. The equations of motion are the same as in the problem of a slowly forced nonlinear pendulum. Using methods of the adiabatic perturbation theory we show that for a periodic external force of a general kind the system demonstrates directed (ratchet) transport in the chaotic domain on very long time intervals and obtain a formula for the average velocity of this transport. Two cases are studied: The case of the external force of small amplitude, and the case of the external force with amplitude of order one. The obtained formulas can also be used in case of a nonharmonic periodic potential.

Porous media with resurgences can be described by a double structure, namely, a continuous porous medium and capillaries with impermeable walls which relate distant points of the continuous medium. The resurgences can be either punctual or extended. The equations for flow in such media are derived; some general properties of the resulting system, which involves nonlocal aspects, are deduced. A “dilute” approximation is detailed for punctual resurgences in two-dimensional media and is illustrated by a few examples.

The Ginzburg-Landau model with two-order parameters appears in many condensed-matter problems. However, even for scalar order parameters, the most general U(1)-symmetric Landau potential with all quadratic and quartic terms contains 13 independent coefficients and cannot be minimized with straightforward algebra. Here, we develop a geometric approach that circumvents this computational difficulty and allows one to study properties of the model without knowing the exact position of the minimum. In particular, we find the number of minima of the potential, classify explicit symmetries possible in this model, establish conditions when and how these symmetries are spontaneously broken, and explicitly describe the phase diagram.

We present a systematic theory of propagation and evolution of the reaction front $A+B\leftrightarrow C$ in the reaction-diffusion system where an island of particles $A$ is surrounded by the uniform sea of particles $B$. In the first part of the work we give a systematic analysis of the crossover from the irreversible to reversible regime of front propagation in terms of the quasistatic approximation (QSA) and derive the key condition for the island death in the quasiequilibrium front regime. We show that the same as in the case of pure annihilation $A+B\to 0$ the QSA enables the description of the quasiequilibrium front propagation only to a critical point $t_c$ on approaching to which the QSA is violated. In the second part of the work under the assumption of a sufficiently large forward reaction constant $k$ we derive the perturbative expansion in powers of $1∕k$ which gives the asymptotically exact description of the quasiequilibrium front evolution up to $t\to \infty$. We demonstrate that below some critical value of the reduced backward reaction constant $g<g_c$ there appear two turning points on the front trajectory, the first of which arises at the sharp localized front stage and is due to the finite number of island particles whereas the second is a consequence of radical transformation of the front structure at the passage through the critical point (delocalization of the front). We find a remarkable property of self-similarity of the passage through the critical point, we derive scaling laws for such passage and show that in the limit $g\to 0$ these laws lead to a striking phenomenon of an abrupt delocalization of the front.

Orthogonal-unitary and symplectic-unitary crossover ensembles of random matrices are relevant in many contexts, especially in the study of time reversal symmetry breaking in quantum chaotic systems. Using skew-orthogonal polynomials we show that the same generic form of $n$-level correlation functions are obtained for the Jacobi family of crossover ensembles, including the Laguerre and Gaussian cases. For large matrices we find universal forms of unfolded correlation functions when expressed in terms of a rescaled transition parameter with arbitrary initial level density.

We study a chaotic ratchet system under the influence of a thermal environment. By direct integration of the Lindblad equation we are able to analyze its behavior for a wide range of couplings with the environment, and for different finite temperatures. We observe that the enhancement of the classical and quantum currents due to temperature depend strongly on the specific properties of the system. This makes it difficult to extract universal behaviors. We have also found that there is an analogy between the effects of the classical thermal noise and those of the finite $\hslash$ size. These results open many possibilities for their testing and implementation in kicked Bose-Einstein condensates and cold atoms experiments.

We present a molecular dynamics based method for accurately computing short-range structural forces resulting from the overlap of spatially diffuse solid-liquid interfaces at wetted grain boundaries close to the melting point. The method is based on monitoring the fluctuations of the liquid layer width at different temperatures to extract the excess interfacial free energy as a function of this width. The method is illustrated for a high-energy $\Sigma 9$ twist boundary in pure Ni. The short-range repulsion driving premelting is found to be dominant in comparison to long-range dispersion and entropic forces and consistent with previous experimental findings that nanometer-scale layer widths may be observed only very close to the melting point.

The hierarchical reference theory (HRT) and the self-consistent Ornstein-Zernike approximation (SCOZA) are two liquid state theories that both yield a largely satisfactory description of the critical region as well as the phase coexistence and equation of state in general. In two previous works, unification of these theories has been considered and general equations were established. Further it was shown that the solution of the mean spherical model and a generalized version of it can be obtained in this way. In the present work, analysis of the critical region for fluids and lattice gases is performed. A key result of our HRT-SCOZA approximation is that for the standard three-dimensional fluid, lattice gas, or Ising model, the critical index for the critical isotherm is $\delta =5$ and for the curve of coexistence it is $\beta =1∕3$ provided full scaling is assumed. More generally we find $\beta =2∕(\delta +1)$.

This article establishes a natural physical path leading from “regular” Ornstein-Uhlenbeck dynamics to “anomalous” long-memory processes and, thereafter, to fractional Brownian motion. Considering a system composed of $n$ different parts—each part conducting its own Ornstein-Uhlenbeck dynamics, and all parts being perturbed by a common external Lévy noise—we show that the collective system-dynamics, in the limit $n\to \infty$, converges to a temporal moving-average of the driving noise. The limiting moving-average process, in turn, can posses a long memory—in which case, when observed over large time scales, further yields fractional Brownian motion. The temporal correlation structure of the limiting moving-average process turns out to be determined by the structural statistical variability of the system’s composing parts. Thus, the emergence of a long memory is a consequence of the intrinsic “quenched disorder” present at the system’s formation epoch rather than the consequence of the external annealed disorder carried in continuously by the driving noise.

We report molecular dynamics simulations results for model ferrofluid films subject to an external, homogeneous magnetic field directed parallel or perpendicular to the film surfaces. The interactions between the magnetic nanoparticles are modeled via the Stockmayer potential. In a previous study [J. Jordanovic and S. H. L. Klapp, Phys. Rev. Lett. 101, 038302 (2008)] we have shown that an external field can control the number and internal structure of the layers characterizing the fluid films, in qualitative agreement with experiments. Here we explore the dependence of the layering effects on thermodynamic conditions, and we analyze the results from an energetic (microscopic and macroscopic) perspective. As a special case we investigate a monolayer to bilayer transition induced via a perpendicular field.

We consider the rupture dynamics of a homopolymer chain pulled at one end at a constant loading rate $r$. Compared to single bond breaking, the existence of the chain introduces two aspects into rupture dynamics: The non-Markovian aspect in the barrier crossing and the slow down of the force propagation to the breakable bond. The relative impact of both these processes is investigated, and the second one was found to be the most important at moderate loading rates. The most probable rupture force is found to decrease with the number of bonds as $f_{\max }\propto -{\left[\ln (\mathrm{const}N∕r)\right]}^{2∕3}$ and finally to approach a saturation value independent on $N$. All of our analytical findings are confirmed by extensive numerical simulations.