We study the crossover from low-temperature to high-temperature fluctuations including Goldstone-dominated and critical fluctuations in confined isotropic and weakly anisotropic $\mathrm{O}\left(n\right)$-symmetric systems on the basis of a finite-size renormalization-group approach at fixed dimension $d$ introduced previously [V. Dohm, Phys. Rev. Lett. 110, 107207 (2013)]. Our theory is formulated within the ${\phi }^4$ lattice model in a $d$-dimensional block geometry with periodic boundary conditions. We calculate the finite-size scaling functions $F^{\text{ex}}$ and $X$ of the excess free-energy density and the thermodynamic Casimir force, respectively, for $1\le n\le \infty , 2<d<4$. Exact results are derived for $n\to \infty$. Applications are given for $L_{\parallel }^{d-1}\times L$ slab geometry with an aspect ratio $\rho =L/L_{\parallel }>0$ and for film geometry ($\rho =0$). Good overall agreement is found with Monte Carlo (MC) data for isotropic spin models with $n=1,2,3$. For $\rho =0$, the low-temperature limits of $F^{\text{ex}}$ and $X$ vanish for $n=1$, whereas they are finite for $n\ge 2$. For $\rho >0$ and $n=1$, we find a finite low-temperature limit of $F^{\text{ex}}$, which deviates from that of the Ising model. We attribute this deviation to the nonuniversal difference between the ${\phi }^4$ model with continuous variables and the Ising model with discrete variables. For $n\ge 2$ and $\rho >0$, a logarithmic divergence of $F^{\text{ex}}$ in the low-temperature limit is predicted, in excellent agreement with MC data. For $2\le n\le \infty$ and $\rho <{\rho }_0=0.8567$ the Goldstone modes generate a negative low-temperature Casimir force that vanishes for $\rho ={\rho }_0$ and becomes positive for $\rho >{\rho }_0$. For anisotropic systems a unified hypothesis of multiparameter universality is introduced for both bulk and confined systems. The dependence of their scaling functions on $d(d+1)/2-1$ microscopic anisotropy parameters implies a substantial reduction of the predictive power of the theory for anisotropic systems as compared to isotropic systems. An exact representation is derived for the nonuniversal large-distance behavior of the bulk correlation function of anisotropic systems and quantitative predictions are made. The validity of multiparameter universality is proven analytically for the $d=2,n=1$ universality class. A nonuniversal anisotropy-dependent minimum of the Casimir force scaling function $X$ is found. Both the sign and magnitude of $X$ and the shift of the film critical temperature are affected by the lattice anisotropy.

Dynamical typicality refers to the property that two pure states, which initially exhibit (almost) the same expectation value for some given observable $A$, are very likely to exhibit also very similar expectation values when evolving in time according to the pertinent Schrödinger equation. We unify and generalize a variety of previous findings of this type for sufficiently high-dimensional quantum mechanical model systems. Particular emphasis is put on the necessary and sufficient conditions, which the initial expectation value and the spectrum of $A$ have to fulfill.

The presented model for earthquakes is based on two fundamental principles: the hierarchical structure of seismic areas and the concept of self-organized criticality. The model reproduces the basic empirical properties of seismic processes: the frequency-energy scaling relation (the Gutenberg-Richter law), the generalized Omori law for temporal decay of aftershocks, the aftershock productivity law, the fractal distributions of hypocenters (epicenters) with power-law dependencies of the number of events on distances between hypocenters (epicenters), and, finally, the $\gamma$ distribution for waiting times. In the model, the threshold energies depend on the block sizes and are distributed according to the Gauss law. After strong earthquakes they are redistributed at the decreasing average values. The change of threshold energies leads to the triggering of aftershock series.

The Lieb-Robinson bound shows that the speed of propagating information in a nonrelativistic quantum lattice system is bounded by a finite velocity, which entails the clustering of correlations. In this paper, we extend the Lieb-Robinson bound to quantum systems at finite temperature by calculating the dynamical correlation function at nonzero temperature for systems whose interactions are, respectively, short range, exponentially decaying, and long range. We introduce a simple way of counting the clusters in a cluster expansion by using the combinatoric generating functions of graphs. Limitations and possible applications of the obtained bound are also discussed.

The second law of thermodynamics states that a system in contact with a heat bath can undergo a transformation if and only if its free energy decreases. However, the “if” part of this statement is only true when the effective heat bath is infinite. In this article we remove this idealization and derive corrections to the second law in the case where the bath has a finite size, or equivalently finite heat capacity. This can also be translated to processes lasting a finite time, and we show that thermodynamical reversibility is lost in this regime. We do so in full generality, without assuming any particular model for the bath; the only parameters defining the bath are its temperature and heat capacity. We find connections with second order Shannon information theory, in particular, in the case of Landauer erasure. We also consider the case of nonfluctuating work and derive finite-bath corrections to the min and max free energies employed in single-shot thermodynamics.

A theoretical investigation of the modulational instability (MI) in a composite system with a nonlocal response function is presented. A composite system of silver nanoparticles in acetone is chosen, whose nonlinearity can be delicately varied by controlling the volume fraction of the constituents, thus enabling the possibility of nonlinearity management. A pump-probe counterpropagation configuration has been assumed, and the interplay between the competing nonlinearities and the nonlocalities in the MI dynamics is systematically explored. A different class of nonlocalities have been considered, and the study reveals that the nonlocality critically depends on the kind of nonlocal function. However, the general behavior is that the strength of nonlocality suppresses the MI gain, while for a rectangular function it assists the emergence of new spectral windows. We also show that the cross coupling effects are significant in enhancing MI, especially in the defocusing nonlinearity. We also emphasize the impact of the relative strength of the nonlinearities in the MI dynamics at different settings of competing nonlinearities. Thus, we emphasize the importance of the different class of nonlocal response in the MI dynamics and explore the interplay between the higher order nonlinear effects and nonlocalities in the counterpropagating configurations.

Usage of a Hamiltonian perturbation theory for a nonconservative system is counterintuitive and, in general, a technical impossibility by definition. However, the time-independent dual Hamiltonian formalism for the nonconservative systems has opened the door for using various conservative perturbation theories for investigating the dynamics of such systems. Here we demonstrate that the Lie transform Hamiltonian perturbation theory can be adapted to find the perturbative solutions and the frequency corrections for the dissipative oscillatory systems. As a further application, we use the perturbation theory to analytically calculate the Hannay angle for the van der Pol oscillator's limit cycle trajectory when its parameters—the strength of the nonlinearity and the frequency of the linear part—evolve cyclically and adiabatically. For this van der Pol oscillator, we also numerically calculate the corresponding geometric phase and establish its equivalence with the Hannay angle.

We demonstrate that nonlinearity plays a constructive role in supporting the robustness of dynamical localization in a system which is discrete in one dimension and continuous in the orthogonal one. In the linear regime, time-periodic modulation of the gradient strength along the discrete axis leads to the usual rapid spread of an initially confined wave packet. Addition of the cubic nonlinearity makes the dynamics drastically different, inducing robust localization of moving wave packets. Similar nonlinearity-induced effects are also produced in the presence of a combination of static and oscillating linear potentials. The predicted dynamical localization in the nonlinear medium can be realized in photonic lattices and Bose-Einstein condensates.

We analyzed a generic relaxation oscillator under moderately strong forcing at a frequency much greater that the natural intrinsic frequency of the oscillator. Additionally, the forcing is of the same sign and, thus, has a nonzero average, matching neuroscience applications. We found that, first, the transition to high-frequency synchronous oscillations occurs mostly through periodic solutions with virtually no chaotic regimes present. Second, the amplitude of the high-frequency oscillations is large, suggesting an important role for these oscillations in applications. Third, the 1:1 synchronized solution may lose stability, and, contrary to other cases, this occurs at smaller, but not at higher frequency differences between intrinsic and forcing oscillations. We analytically built a map that gives an explanation of these properties. Thus, we found a way to substantially “overclock” the oscillator with only a moderately strong external force. Interestingly, in application to neuroscience, both excitatory and inhibitory inputs can force the high-frequency oscillations.

Despite the innate complexity of the cell, emergent scale-invariant behavior is observed in many biological systems. We investigate one example of this phenomenon: the dynamics of large complexes in the bacterial cytoplasm. The observed dynamics of these complexes is scale invariant in three measures of dynamics: mean-squared displacement (MSD), velocity autocorrelation function, and the step-size distribution. To investigate the physical mechanism for this emergent scale invariance, we explore minimal models in which mobility is modeled as diffusion on a rough free-energy landscape in one dimension. We discover that all three scale-invariant characteristics emerge generically in the strong disorder limit. (Strong disorder is defined by the divergence of the ensemble-averaged hop time between lattice sites.) In particular, we demonstrate how the scale invariance of the relative step-size distribution can be understood from the perspective of extreme-value theory in statistics (EVT). We show that the Gumbel scale parameter is simply related to the MSD scaling parameter. The EVT mechanism of scale invariance is expected to be generic to strongly disordered systems and therefore a powerful tool for the analysis of other systems in biology and beyond.

We develop a framework for the general interpretation of the stochastic dynamical system near a limit cycle. Such quasiperiodic dynamics are commonly found in a variety of nonequilibrium systems, including the spontaneous oscillations of hair cells of the inner ear. We demonstrate quite generally that in the presence of noise, the phase of the limit cycle oscillator will diffuse, while deviations in the directions locally orthogonal to that limit cycle will display the Lorentzian power spectrum of a damped oscillator. We identify two mechanisms by which these stochastic dynamics can acquire a complex frequency dependence and discuss the deformation of the mean limit cycle as a function of temperature. The theoretical ideas are applied to data obtained from spontaneously oscillating hair cells of the amphibian sacculus.

Often the analysis of time-dependent chemical and biophysical systems produces high-dimensional time-series data for which it can be difficult to interpret which individual features are most salient. While recent work from our group and others has demonstrated the utility of time-lagged covariate models to study such systems, linearity assumptions can limit the compression of inherently nonlinear dynamics into just a few characteristic components. Recent work in the field of deep learning has led to the development of the variational autoencoder (VAE), which is able to compress complex datasets into simpler manifolds. We present the use of a time-lagged VAE, or variational dynamics encoder (VDE), to reduce complex, nonlinear processes to a single embedding with high fidelity to the underlying dynamics. We demonstrate how the VDE is able to capture nontrivial dynamics in a variety of examples, including Brownian dynamics and atomistic protein folding. Additionally, we demonstrate a method for analyzing the VDE model, inspired by saliency mapping, to determine what features are selected by the VDE model to describe dynamics. The VDE presents an important step in applying techniques from deep learning to more accurately model and interpret complex biophysics.

We consider the reshaping of closed Janus filaments acquiring intrinsic curvature upon actuation of a driven component—a nematic elastomer elongating upon phase transition. Linear stability analysis establishes instability thresholds of circles with no imposed twist, dependent on the ratio $q$ of the intrinsic curvature to the inverse radius of the original circle. Twisted circles are proven to be absolutely unstable but the linear analysis well predicts the dependence of the looping number of the emerging configurations on the imposed twist. Modeling stable configurations by relaxing numerically the overall elastic energy detects multiple stable and metastable states with different looping numbers. The bifurcation of untwisted circles turns out to be subcritical, so that nonplanar shapes with a lower energy exist at $q$ below the critical value. The looping number of stable shapes generally increases with $q$.

We investigated the transient behavior of immiscible two-liquid interfaces initiated by a single rising gas bubble and characterized by liquid “column” and “film” morphologies. To analyze the effect of the buoyancy force, viscosity, and interfacial tension on these morphologies, the single-solution density was controlled continuously by association with the rising velocity of the bubble. It was observed that the extension of the liquid column further into the upper liquid phase owing to the wake flow under the bubble is driven by the buoyancy force, with the velocity decreasing gradually with the distance between the bubble and the liquid-liquid interface. Based on this mechanism, we determined that a strong dimensionless correlation exists between the lifetime of the column and the physical properties of the two liquid phases. On the other hand, gravitational drainage does not affect the film lifetime. However, marginal pinching is dominant, probably owing to the existence of a surface tension gradient between the film and the meniscus.

The stationary flow field in a quasi-two-dimensional hopper is investigated experimentally. The behavior of materials consisting of beads and elongated particles with different aspect ratio is compared. We show, that while the vertical velocity in the flowing region can be fitted with a Gaussian function for beads, in the case of elongated grains the flowing channel is narrower and is bordered with sharper velocity gradient. For this case, we quantify deviations from the Gaussian velocity profile. Relative velocity fluctuations are considerably larger and slower for elongated grains.

We investigate a lattice model representing a granular gas in a thin channel. We deduce the hydrodynamic description for the model from the microscopic dynamics in the large-system limit, including the lowest finite-size corrections. The main prediction from hydrodynamics, when finite-size corrections are neglected, is the existence of a steady “uniform longitudinal flow” (ULF), with the granular temperature and the velocity gradient both uniform and directly related. Extensive numerical simulations of the system show that such a state can be observed in the bulk of a finite-size system by attaching two thermostats with the same temperature at its boundaries. The relation between the ULF state and the shocks appearing in the late stage of a cooling gas of inelastic hard rods is discussed.

In very recent work the mean field theory of the jamming transition in infinite-dimensional hard sphere models was presented. Surprisingly, this theory predicts quantitatively the numerically determined characteristics of jamming in two and three dimensions. This is a rare and unusual finding. Here we argue that this agreement is nongeneric: only for hard sphere models does it happen that sufficiently close to the jamming density (at any temperature) the effective interactions are binary, in agreement with mean field theory, justifying the truncation of many-body interactions (which is the exact protocol in infinite dimensions). Any softening of the bare hard sphere interactions results in many-body effective interactions that are not mean field at any density, making the $d=\infty$ results not applicable.

The angle of directional change of tracer trajectories in rotating Rayleigh-Bénard convection is studied as a function of the time increment $\tau$ between two instants of time along the trajectories, both experimentally and with direct numerical simulations. Our aim is to explore the geometrical characterization of flow structures in turbulent convection in a wide range of timescales and how it is affected by background rotation. We find that probability density functions (PDFs) of the angle of directional change $\theta (t,\tau )$ show similar behavior as found in homogeneous isotropic turbulence, up to the timescale of the large-scale coherent flow structures. The scaling of the averaged (over particles and time) angle of directional change $\mathrm{\Theta }\left(\tau \right)=\langle \left|\theta \right(t,\tau \left)\right|\rangle$ with $\tau$ shows a transition from the ballistic regime $[\mathrm{\Theta }\left(\tau \right)\sim {\tau }^c$ with $c=1]$ for $\tau \lesssim {\tau }_{\eta }$, with ${\tau }_{\eta }$ the Kolmogorov timescale, to a scaling with smaller exponent $c$ for ${\tau }_{\eta }\lesssim \tau \lesssim T_L$, with $T_L$ the Lagrangian integral timescale. This scaling exponent is approximately constant in the weakly rotating regime (Rossby number $\text{Ro}\gtrsim 2.5$) and is decreasing for increasing rotation rates when $\text{Ro}\lesssim 2.5$. We show that this trend in the scaling exponent is related with the large-scale coherent structures in the flow; the large-scale circulation for $\text{Ro}\gtrsim 2.5$ and vertically aligned vortices emerging from the boundary layers (BLs) near the top and bottom plates and penetrating into the bulk for $\text{Ro}\lesssim 2.5$. In the viscous BLs, the PDFs of $\theta (t,\tau )$ and scaling properties of $\mathrm{\Theta }\left(\tau \right)$ are in general different from those measured in the bulk and depend on the type of boundary layer, in particular whether the BL is of Prandtl-Blasius type ($\text{Ro}\gtrsim 2.5$) or of Ekman type ($\text{Ro}\lesssim 2.5$). When it is of Ekman type, a stronger dynamic coupling exists between the BL and the bulk of the flow, resulting in similar scaling exponents in BL and bulk.

The present study focuses on the cross-stream migration of a neutrally buoyant two-dimensional drop in a Poiseuille flow in a channel under the influence of an electric field. In the absence of an electric field, the important nondimensional parameters describing this problem are the viscosity ratio $\left(\lambda \right)$ between the drop fluid and the surrounding medium, the ratio of drop diameter to channel height $\left(a^{*}\right),$ and the capillary number (Ca). The influence of all these parameters on drop migration is investigated. It is observed that a large drop moves slowly as compared to a smaller drop, but attains a steady shape at the center line of the channel. The increase in value of the capillary number enhances the cross-stream migration rate, while the increase in viscosity ratio reduces the tendency of the drops to move towards the channel center line. The presence of an electric field introduces additional interfacial stresses at the drop interface, which in turn alters the dynamics observed in the absence of an electric field. Extensive computations are carried out to analyze the combined effect of the electric field and the shear flow on the cross-stream migration of the drop. The computational results for a perfect dielectric indicate that the droplet migration enhances in the presence of an electric field. The permittivity ratio $\left(S\right)$ and the electric field strength $\left(E\right)$ play major roles in drop migration and deformation. Computations using the leaky dielectric model also show that for certain combinations of electrical properties the drop undergoes immense elongation along the direction of the electric field. The conductivity ratio $\left(R\right)$ is again a vital parameter in such a system of fluids. It is further observed that for certain conditions the leaky dielectric drops exhibit rotation together with translation.

A networked-oscillator-based analysis is performed to examine and control the transfer of kinetic energy for periodic bluff body flows. The dynamics of energy fluctuations in the flow field are described by a set of oscillators defined by conjugate pairs of spatial proper orthogonal decomposition (POD) modes. To extract the network of interactions among oscillators, impulse responses of the oscillators to amplitude and phase perturbations are tracked. Tracking small energy inputs and using linear regression, a networked-oscillator model is constructed that reveals energy exchange among the modes. The model captures the nonlinear interactions among the modal oscillators through a linear approximation. A large collection of system responses is aggregated to capture the general network structure of oscillator interactions. The present networked-oscillator model describes the modal perturbation dynamics more accurately than the empirical Galerkin reduced-order model. The linear network model for nonlinear dynamics is subsequently utilized to design a model-based feedback controller. The controller suppresses the modal amplitudes that result in wake unsteadiness leading to drag reduction. The strength of the proposed approach is demonstrated for a canonical example of two-dimensional unsteady flow over a circular cylinder. The present formulation enables the characterization of modal interactions to control fundamental energy transfers in unsteady bluff body flows.