A nonperturbative theory is developed, aiming at an exact and efficient evaluation of a general quantum system interacting with arbitrary bath environment at any temperature and in the presence of arbitrary time-dependent external fields. An exact hierarchical equations of motion formalism is constructed on the basis of a calculus-on-path-integral algorithm, via the auxiliary influence generating functionals related to the interaction bath correlation functions in a parametrization expansion form. The corresponding continued-fraction Green’s functions formalism for quantum dissipation is also presented. Proposed further is the principle of residue correction, not just for truncating the infinite hierarchy, but also for incorporating the small residue dissipation that may arise from the practical difference between the true and parametrized bath correlation functions. The final residue-corrected hierarchical equations of motion can therefore be used practically for the evaluation of arbitrary dissipative quantum systems.

We study minimal surfaces which arise in wetting and capillarity phenomena. Using conformal coordinates, we reduce the problem to a set of coupled boundary equations for the contact line of the fluid surface and then derive simple diagrammatic rules to calculate the nonlinear corrections to the Joanny–de Gennes energy. We argue that perturbation theory is quasilocal—i.e., that all geometric length scales of the fluid container decouple from the short-wavelength deformations of the contact line. This is illustrated by a calculation of the linearized interaction between contact lines on two opposite parallel walls. We present a simple algorithm to compute the minimal surface and its energy based on these ideas. We also point out the intriguing singularities that arise in the Legendre transformation from the pure Dirichlet to the mixed Dirichlet-Neumann problem.

We report experiments on a ferroelectric membrane and droplets with tunable surface properties. In smectic membranes the configuration of the $\mathbf{c}$-director field near inclusions may be rearranged drastically with temperature. The transformation of the $\mathbf{c}$-director field results from the competition between the elastic and polar properties of the membranes. We demonstrate that anchoring conditions on the inclusion boundary are not fixed but depend on the temperature. A dipolar $\mathbf{c}$-director configuration near droplets can evolve to a mixed configuration and to a quadrupolar one. These modifications of the $\mathbf{c}$-director field near the inclusions lead to a change of the interaction between the inclusions, their self-organization, and even to the destruction of structures already formed by the inclusions. Our observations open new possibilities for manipulating inclusions and controlling their self-organization.

We report on the geometry and mechanics of knotted stiff strings. We discuss both closed and open knots. Our two main results are that (i) their equilibrium energy as well as the equilibrium tension for open knots depends on the type of knot as the square of the bridge number and (ii) braid localization is found to be a general feature of stiff string entanglements, while angle and knot localizations are forbidden. Moreover, we identify a family of knots for which the equilibrium shape is a circular braid. Two other equilibrium shapes are found from Monte Carlo simulations. These three shapes are confirmed by rudimentary experiments. Our approach is also extended to the problem of the minimization of the length of a knotted string with a maximum allowed curvature.

The maximum likelihood method is a basic statistical technique for estimating parameters and variables, and is the starting point for many more sophisticated methods, like Bayesian methods. This paper shows that maximum likelihood fails to identify the true trajectory of a chaotic dynamical system, because there are trajectories that appear to be far more (infinitely more) likely than truth. This failure occurs for unbounded noise and for bounded noise when it is sufficiently large and will almost certainly have consequences for parameter estimation in such systems. The reason for the failure is rather simple; in chaotic dynamical systems there can be trajectories that are consistently closer to the observations than the true trajectory being observed, and hence their likelihood dominates truth. The residuals of these truth-dominating trajectories are not consistent with the noise distribution; they would typically have too small standard deviation and many outliers, and hence the situation may be remedied by using methods that examine the distribution of residuals and are not entirely maximum likelihood based.

In this paper, we examine how base flows affect interfacial stabilities in the presence of surfactants. A thin-film flow model, subjected to various base-flow conditions, is employed to mimic a wide class of practical interfacial flows. The base flow can be driven by an external force (e.g., pressure forcing or gravity), an interfacial stress, or their combination. For long-wave perturbations, we show that the stability is governed by a coupled set of evolution equations for the interface and surfactant concentration, so the origin of the stability can be unraveled analytically in line with simpler physical arguments. We also demonstrate that the system can exhibit a variety of stability states; it can be neutral, conditionally stable or unstable, or definitely stable or unstable, determined solely by the nature of the base flow and how it regulates surfactant transport. Two modes are found to determine the stability, the interface and surfactant modes, and characterized by the ratio of the basic interfacial shear force to the external force. The routes to the instability are also identified through the action of the base flow. The base flow plays a dual role in affecting the stability: although the imposition of an interfacial shear destabilizes the interface, an external force can cause stabilization. The competition between these two effects gives rise to stability or instability in a range of the force ratio. The underlying mechanisms are elucidated in detail. A generalized criterion for the onset of instability is also established for one- or two-fluid interfacial flows.

A short wavelength approximation of a boundary integral operator for two-dimensional isotropic and homogeneous elastic bodies is derived from first principles starting from the Navier-Cauchy equation. Trace formulas for elastodynamics are deduced connecting the eigenfrequency spectrum of an elastic body to the set of periodic rays where mode conversion enters as a dynamical feature.

We report a high-resolution x-ray scattering study of the smectic liquid crystal octylcyanobiphenyl (8CB) confined to aligned colloidal aerosil gels. The aligned gels introduce orientational fields that promote long-range nematic order while imposing positional random fields that couple to the smectic density wave and disrupt the formation of an ordered smectic phase. At low densities of aerosil, the low-temperature scattering intensity is consistent with the presence of a topologically ordered $XY$ Bragg glass phase that is predicted to form in response to such anisotropic quenched disorder. The observed features of the phase include an algebraic decay of the smectic correlations, which is truncated at large length scales due to the imperfect nematic order, and a power-law exponent that agrees closely with the universal value predicted for the $XY$ Bragg glass. At higher aerosil densities, deviations from the $XY$ Bragg glass form are apparent. At high temperature, the scattering intensity displays pretransitional dynamic fluctuations associated with the destroyed nematic to smectic-$A$ transition. The fluctuations obey quasicritical behavior over an extended range of reduced temperature. The effective critical exponents for the correlation lengths and smectic susceptibility differ systematically from those of pure 8CB, indicating that coupling of the nematic order to the gel suppresses its role in the smectic critical behavior.

We demonstrate that nonlocally coupled limit-cycle oscillators subject to spatiotemporally white Gaussian noise can exhibit a noise-induced transition to turbulent states. After illustrating noise-induced turbulent states with numerical simulations using two representative models of limit-cycle oscillators, we develop a theory that clarifies the effective dynamical instabilities leading to the turbulent behavior using a hierarchy of dynamical reduction methods. We determine the parameter region where the system can exhibit noise-induced turbulent states, which is successfully confirmed by extensive numerical simulations at each level of the reduction.

A lattice Boltzmann model is proposed for solving low Mach number thermal flows with viscous dissipation and compression work in the double-distribution-function framework. A distribution function representing the total energy is defined based on a single velocity distribution function, and its evolution equation is derived from the continuous Boltzmann equation. A lattice Boltzmann equation model with clear physics and a simple structure is then obtained from a kinetic model for the decoupled hydrodynamic and energy equations. The model is tested by simulating a thermal Poiseuille flow and natural convection in a square cavity, and it is found that the numerical results agree well with the analytical solutions and/or the data reported in previous studies.

Scaling properties of patterns formed by large contact forces are studied as a function of the applied shear stress, in two-dimensional static packings generated from the force network ensemble. An anisotropic finite-size-scaling analysis shows that the applied shear does not affect the universal scaling properties of these patterns, but simply induces different length scales in the principal directions of the macroscopic stress tensor. The ratio of these length scales quantifies the anisotropy of the force networks, and is found not to depend on the details of the underlying contact network, in contrast with other properties such as the yield stress.

A self-consistent mean-field method is used to study critical wetting transitions under nonequilibrium conditions by analyzing Kardar-Parisi-Zhang (KPZ) interfaces in the presence of a bounding substrate. In the case of positive KPZ nonlinearity a single (Gaussian) regime is found. On the contrary, interfaces corresponding to negative nonlinearities lead to three different regimes of critical behavior for the surface order parameter: (i) a trivial Gaussian regime, (ii) a weak-fluctuation regime with a trivially located critical point and nontrivial exponents, and (iii) a highly nontrivial strong-fluctuation regime, for which we provide a full solution by finding the zeros of parabolic-cylinder functions. These analytical results are also verified by solving numerically the self-consistent equation in each case. Analogies with and differences from equilibrium critical wetting as well as nonequilibrium complete wetting are also discussed.

We show that in driven systems the Gaussian nature of the fluctuating force and time reversibility are equivalent properties. This result together with the potential condition of the external force drastically restricts the form of the probability distribution function, which can be shown to satisfy time-independent relations. We have corroborated this feature by explicitly analyzing a model for the stretching of a polymer and a model for a suspension of noninteracting Brownian particles in steady flow.

We present molecular dynamics calculations of the thermal conductivity and viscosities of a model colloidal suspension with colloidal particles roughly one order of magnitude larger than the suspending liquid molecules. The results are compared with estimates based on the Enskog transport theory and effective medium theories (EMT) for thermal and viscous transport. We find, in particular, that EMT remains well applicable for predicting both the shear viscosity and thermal conductivity of such suspensions when the colloidal particles have a “typical” mass, i.e., much larger than the liquid molecules. Very light colloidal particles on the other hand yield higher thermal conductivities, in disagreement with EMT. We also discuss the consequences of these results for some proposed mechanisms for thermal conduction in nanocolloidal suspensions.

We propose a simple model for mass transport within a fungal hypha and its subsequent growth. Inspired by the role of microtubule-transported vesicles, we embody the internal dynamics of mass inside a hypha with mutually excluding particles progressing stochastically along a growing one-dimensional lattice. The connection between long-range transport of materials for growth and the resulting extension of the hyphal tip has not previously been addressed in the modeling literature to our knowledge. We derive and analyze mean-field equations for the model and present a phase diagram of its steady-state behavior, which we compare to simulations. We discuss our results in the context of the filamentous fungus Neurospora crassa.

The objective of the present work is to propose a method for nonfeedback anticontrol of chaos with perturbations of minimum power for a preset control goal. The noted Lorenz system is employed as the test model for chaotification with the target state specified by a prescribed positive value of the largest Lyapunov exponent (LLE), $\tilde{\lambda }>0$. Periodic and quasiperiodic perturbations are used as control signals, and the signals parameters are optimized using a genetic algorithm under restriction of minimum power. Performance of the optimized signals in triggering chaos at an ordered state, fixed point or periodic, as well as further enhancing chaoticity at a chaotic state is explored. The present numerical experiments reveal the following interesting physics about chaotification. In general, the power for chaotification increases with the preset value of $\tilde{\lambda }$ and quasiperiodic signals can achieve the control goal with a lower power than periodic ones. Given the same increment of LLE from that of the uncontrolled state $\left({\lambda }_{1,0}\right)$, i.e., $\mathrm{\Delta }\lambda =\tilde{\lambda }-{\lambda }_{1,0}$, the further enhancement of chaoticity in a chaotic state needs a higher control power than the triggering of chaos from an ordered state. The minimum power required for chaotification of an ordered state increases relatively slowly for lower $\tilde{\lambda }$ but increases drastically as the preset target LLE reaches a certain critical value. Most strikingly, the numerical experiments demonstrate that this critical value of $\tilde{\lambda }$ corresponds to LLE of the nearest chaotic state in the neighborhood of the uncontrolled state. Robustness of applying the present method in the presence of external noise is also demonstrated.

We determine probabilities of recurrence time into finite-sized, physically meaningful subsets of phase space. We consider three different autonomous chaotic systems: (i) scattering in a three-peaked potential, (ii) connected billiards, and (iii) Lorenz equations. We find multipeaked probability distributions, similar to the distributions found in (driven) stochastically resonant systems. In nondriven systems, such as ours, only monotonic decaying distributions (exponentials, stretched exponentials, power laws, and slight variations or combinations of these) have hitherto been reported. Discrete peaks in autonomous systems have as yet escaped attention in autonomous systems and correspond to specific trajectory subsets involving an integer number of loops.

We consider the dynamics of a driven Bose-Einstein condensate with positive scattering length. Employing an accustomed variational treatment we show that when the scattering length is time modulated as $a\{1+ϵ\sin \left[\omega \left(t\right)t\right]\}$, where $\omega \left(t\right)$ increases linearly in time, i.e., $\omega \left(t\right)=\gamma t$, the response frequency of the condensate locks to the eigenfrequency for small values of $ϵ$ and $\gamma$. A simple analytical model is presented which explains this phenomenon by mapping it to an auto-resonance, i.e., close to resonance the reduced equations describing the collective behavior of the condensate are equivalent to those of a virtual particle trapped in a finite-depth energy minimum of an effective potential.

A well-characterized set of large-scale laboratory experiments is presented, illustrating forced imbibition displacements in the presence of irreducible wetting phase saturation in a cylindrical, homogeneous Berea sandstone rock. Experiments are designed to operate in the regime of compact microscopic flows and large-scale viscous instability. The distribution of fluid phases during the flow process is visualized by high-resolution computed tomography imaging. Linear stability analysis and high-accuracy numerical simulations are employed to analyze the ability of macroscopic continuum equations to provide a consistent approximation of the displacement process. The validity of the equilibrium relative permeability functions, which form the basis for the continuum model, is fundamentally related to the stability of the displacement process. It is shown that not only is the stable flow regime modeled accurately by existing continuum models, but the onset of instability as well as the initial unstable modes are also determined with reasonable accuracy for unstable flows. However, the continuum model is found to be deficient in the case of fully developed unstable flows.