Natural numbers can be divided in two nonoverlapping infinite sets, primes and composites, with composites factorizing into primes. Despite their apparent simplicity, the elucidation of the architecture of natural numbers with primes as building blocks remains elusive. Here, we propose a new approach to decoding the architecture of natural numbers based on complex networks and stochastic processes theory. We introduce a parameter-free non-Markovian dynamical model that naturally generates random primes and their relation with composite numbers with remarkable accuracy. Our model satisfies the prime number theorem as an emerging property and a refined version of Cramér's conjecture about the statistics of gaps between consecutive primes that seems closer to reality than the original Cramér's version. Regarding composites, the model helps us to derive the prime factors counting function, giving the probability of distinct prime factors for any integer. Probabilistic models like ours can help to get deeper insights about primes and the complex architecture of natural numbers.

We determine with unprecedented accuracy the lowest 900 eigenvalues of two quantum constant-width billiards from resonance spectra measured with flat, superconducting microwave resonators. While the classical dynamics of the constant-width billiards is unidirectional, a change of the direction of motion is possible in the corresponding quantum system via dynamical tunneling. This becomes manifest in a splitting of the vast majority of resonances into doublets of nearly degenerate ones. The fluctuation properties of the two respective spectra are demonstrated to coincide with those of a random-matrix model for systems with violated time-reversal invariance and a mixed dynamics. Furthermore, we investigate tunneling in terms of the splittings of the doublet partners. On the basis of the random-matrix model we derive an analytical expression for the splitting distribution which is generally applicable to systems exhibiting dynamical tunneling between two regions with (predominantly) chaotic dynamics.

A two-dimensional (2D) lattice model defined on a triangular lattice with nearest- and next-nearest-neighbor interactions based on the Taylor-Socolar monotile is known to have a limit-periodic ground state. The system reaches that state during a slow quench through an infinite sequence of phase transitions. We study the model as a function of the strength of the next-nearest-neighbor interactions and introduce closely related 3D models with only nearest-neighbor interactions that exhibit limit-periodic phases. For models with no next-nearest-neighbor interactions of the Taylor-Socolar type, there is a large degenerate class of ground states, including crystalline patterns and limit-periodic ones, but a slow quench still yields the limit-periodic state. For the Taylor-Socolar lattic model, we present calculations of the diffraction pattern for a particular decoration of the tile that permits exact expressions for the amplitudes and identify domain walls that slow the relaxation times in the ordered phases. For one of the 3D models, we show that the phase transitions are first order, with equilibrium structures that can be more complex than in the 2D case, and we include a proof of aperiodicity for a geometrically simple tile with only nearest-neighbor matching rules.

The computation of interfacial free energies between coexisting phases (e.g., saturated vapor and liquid) by computer simulation methods is still a challenging problem due to the difficulty of an atomistic identification of an interface and interfacial fluctuations on all length scales. The approach to estimate the interfacial tension from the free-energy excess of a system with interfaces relative to corresponding single-phase systems does not suffer from the first problem but still suffers from the latter. Considering $d$-dimensional systems with interfacial area $L^{d-1}$ and linear dimension $L_z$ in the direction perpendicular to the interface, it is argued that the interfacial fluctuations cause logarithmic finite-size effects of order $ln\left(L\right)/L^{d-1}$ and order $ln\left(L_z\right)/L^{d-1}$, in addition to regular corrections (with leading-order $const/L^{d-1}$). A phenomenological theory predicts that the prefactors of the logarithmic terms are universal (but depend on the applied boundary conditions and the considered statistical ensemble). The physical origin of these corrections are the translational entropy of the interface as a whole, “domain breathing” (coupling of interfacial fluctuations to the bulk order parameter fluctuations of the coexisting domains), and capillary waves. Using a new variant of the ensemble switch method, interfacial tensions are found from Monte Carlo simulations of $d=2$ and $d=3$ Ising models and a Lennard-Jones fluid. The simulation results are fully consistent with the theoretical predictions.

We propose a simple yet very predictive form, based on a Poisson's equation, for the functional dependence of the cost from the density of points in the Euclidean bipartite matching problem. This leads, for quadratic costs, to the analytic prediction of the large $N$ limit of the average cost in dimension $d=1,2$ and of the subleading correction in higher dimension. A nontrivial scaling exponent, ${\gamma }_d=\frac{d-2}{d}$, which differs from the monopartite's one, is found for the subleading correction. We argue that the same scaling holds true for a generic cost exponent in dimension $d>2$.

The question of how best to estimate a continuous probability density from finite data is an intriguing open problem at the interface of statistics and physics. Previous work has argued that this problem can be addressed in a natural way using methods from statistical field theory. Here I describe results that allow this field-theoretic approach to be rapidly and deterministically computed in low dimensions, making it practical for use in day-to-day data analysis. Importantly, this approach does not impose a privileged length scale for smoothness of the inferred probability density, but rather learns a natural length scale from the data due to the tradeoff between goodness of fit and an Occam factor. Open source software implementing this method in one and two dimensions is provided.

We consider the one-dimensional driven $ABC$ model under particle-conserving and particle-nonconserving processes. Two limiting cases are studied: (a) The rates of the nonconserving processes are vanishingly slow compared with the conserving processes in the thermodynamic limit and (b) the two rates are comparable. For case (a) we provide a detailed analysis of the phase diagram and the large deviations function of the overall density, $\mathcal{G}\left(r\right)$. The phase diagram of the nonconserving model, derived from $\mathcal{G}\left(r\right)$, is found to be different from the conserving one. This difference, which stems from the nonconvexity of $\mathcal{G}\left(r\right)$, is analogous to ensemble inequivalence found in equilibrium systems with long-range interactions. An outline of the analysis of case (a) was given in an earlier letter. For case (b) we show that, unlike the conserving model, the nonconserving model exhibits a moving density profile in the steady state with a velocity that remains finite in the thermodynamic limit. Moreover, in contrast with case (a), the critical lines of the conserving and nonconserving models do not coincide. These are new features which are present only when the rates of the conserving and nonconserving processes are comparable. In addition, we analyze $\mathcal{G}\left(r\right)$ in case (b) using macroscopic fluctuations theory. Much of the derivation presented in this paper is applicable to any driven-diffusive system coupled to an external particle bath via a slow dynamics.

We introduce and implement a Monte Carlo scheme to study the equilibrium statistics of polymers in the globular phase. It is based on a model of “interacting elastic lattice polymers” and allows a sufficiently good sampling of long and compact configurations, an essential prerequisite to study the scaling behavior of free energies. By simulating interacting self-avoiding rings at several temperatures in the collapsed phase, we estimate both the bulk and the surface free energy. Moreover from the corresponding estimate of the entropic exponent $\alpha -2$ we provide evidence that, unlike for swollen and $\Theta$-point rings, the hyperscaling relation is not satisfied for globular rings.

We present simultaneous measurement of dispersed and continuous phase flow fields for liquid-liquid systems obtained using ultrafast magnetic resonance imaging. Chemical-shift artifacts, which are otherwise highly problematic for this type of measurement, are overcome using a compressed sensing based image reconstruction algorithm that accounts for off-resonant signal components. This scheme is combined with high-temporal-resolution spiral imaging (188 frames per second), which is noted for its robustness to flow. It is demonstrated that both quantitative signal intensity and phase preconditioning are preserved throughout the image reconstruction algorithm. Measurements are acquired of oil droplets of varying viscosity rising through stagnant water. From these data it is apparent that the internal droplet flow fields are heavily influenced by the droplet shape oscillations, and that the accurate modeling of droplet shape is of critical importance in the modeling of droplet-side hydrodynamics. The application of the technique to three-component systems is also demonstrated, as is the measurement of local concentration maps of a mutually soluble species (acetone in polydimethylsiloxane-water).

The classical $n$-vector ${\varphi }^4$ model with $O\left(n\right)$ symmetrical Hamiltonian $\mathcal{H}$ is considered in a ${\infty }^2\times L$ slab geometry bounded by a pair of parallel free surface planes at separation $L$. Standard quadratic boundary terms implying Robin boundary conditions are included in $\mathcal{H}$. The temperature-dependent scaling functions of the excess free energy and the thermodynamic Casimir force are computed in the large-$n$ limit for temperatures $T$ at, above, and below the bulk critical temperature $T_{\mathrm{c}}$. Their $n=\infty$ limits can be expressed exactly in terms of the spectrum and eigenfunctions of a self-consistent one-dimensional Schrödinger equation. This equation is solved by numerical means for two distinct discretized versions of the model: in the first (“model A”), only the coordinate $z$ across the slab is discretized and the integrations over momenta conjugate to the lateral coordinates are regularized dimensionally; in the second (“model B”), a simple cubic lattice with periodic boundary conditions along the lateral directions is used. Renormalization-group ideas are invoked to show that, in addition to corrections to scaling $\propto L^{-1}$, anomalous ones $\propto L^{-1}lnL$ should occur. They can be considerably decreased by taking an appropriate $g\to \infty$ ($T_{\mathrm{c}}\to \infty$) limit of the ${\varphi }^4$ interaction constant $g$. Depending on the model A or B, they can be absorbed completely or to a large extent in an effective thickness $L_{\mathrm{eff}}=L+\delta L$. Excellent data collapses and consistent high-precision results for both models are obtained. The approach to the low-temperature Goldstone values of the scaling functions is shown to involve logarithmic anomalies. The scaling functions exhibit all qualitative features seen in experiments on the thinning of wetting layers of ${}^4$He and Monte Carlo simulations of $XY$ models, including a pronounced minimum of the Casimir force below $T_{\mathrm{c}}$. The results are in conformity with various analytically known exact properties of the scaling functions.

We consider adhesion due to London–van der Waals attraction between a thin film and a patterned surface with nanometer asperities. Depending on the surface topography and the stiffness of the film, three regimes of adhesion are identified: complete contact adhesion, partial contact adhesion, and glassy adhesion. For complete contact adhesion, the film conforms to the undulations of the surface, whereas for partial contact and glassy adhesion, the adhesive interface breaks down into microscopic areas of contact. When a film in the glassy regime is peeled off the surface, metastable states develop at which the crack front becomes arrested, analogously to the frustrated motion of the three-phase contact line across a heterogeneous surface. For this glassy regime, we use transition state theory to model the thermally activated progression of the crack front. This theoretical treatment suggests that the rate of the adhesive failure increases exponentially with the applied force.

We probe the viscous forces involved in the undulatory swimming of the model organism C. elegans. Using micropipette deflection, we attain direct measurements of lateral and propulsive forces produced in response to the motion of the worm. We observe excellent agreement of the results with resistive force theory, through which we determine the drag coefficients of this organism. The drag coefficients are in accordance with theoretical predictions. Using a simple scaling argument, we obtain a relationship between the size of the worm and the forces that we measure, which well describes our data.

The kinematics of dust particles during the early stage of mode-coupling induced melting of a two-dimensional plasma crystal is explored. It is found that the formation of the hybrid mode causes the particle vibrations to partially synchronize at the hybrid frequency. Phase- and frequency-locked hybrid particle motion in both vertical and horizontal directions (hybrid mode) is observed. The system self-organizes in a rhythmic pattern of alternating in-phase and antiphase oscillating chains of particles. The spatial orientation of the synchronization pattern correlates well with the directions of the maximal increment of the shear-free hybrid mode.

We use both lyotropic liquid crystals composed of prolate micelles and thermotropic liquid crystals made of rod-like molecules to uniformly disperse and unidirectionally align relatively large gold nanorods and other complex-shaped nanoparticles at high concentrations. We show that some of these ensuing self-assembled orientationally ordered soft matter systems exhibit polarization-dependent plasmonic properties with strongly pronounced molar extinction exceeding that previously achieved in self-assembled composites. The long-range unidirectional alignment of gold nanorods is mediated mainly by anisotropic surface anchoring interactions at the surfaces of gold nanoparticles. Polarization-sensitive absorption, scattering, and extinction are used to characterize orientations of nanorods and other nanoparticles. The experimentally measured unique optical properties of these composites, which stem from the collective plasmonic effect of the gold nanorods with long-range order in a liquid crystal matrix, are reproduced in computer simulations. A simple phenomenological model based on anisotropic surface interaction explains the alignment of gold nanorods dispersed in liquid crystals and the physical underpinnings behind our observations.

Determining the work which is supplied to a system by an external agent provides a crucial step in any experimental realization of transient fluctuation relations. This, however, poses a problem for quantum systems, where the standard procedure requires the projective measurement of energy at the beginning and the end of the protocol. Unfortunately, projective measurements, which are preferable from the point of view of theory, seem to be difficult to implement experimentally. We demonstrate that, when using a particular type of generalized energy measurements, the resulting work statistics is simply related to that of projective measurements. This relation between the two work statistics entails the existence of modified transient fluctuation relations. The modifications are exclusively determined by the errors incurred in the generalized energy measurements. They are universal in the sense that they do not depend on the force protocol. Particularly simple expressions for the modified Crooks relation and Jarzynski equality are found for Gaussian energy measurements. These can be obtained by a sequence of sufficiently many generalized measurements which need not be Gaussian. In accordance with the central limit theorem, this leads to an effective error reduction in the individual measurements and even yields a projective measurement in the limit of infinite repetitions.

A prefluidized sand bed consisting of fine particles compactifies when it is subjected to a shock. We observe that the response depends on both the shock strength and the ambient pressure, where, counterintuitively, at high ambient pressure the compaction is larger, which we connect to a decrease of the static friction inside the bed. We find that the interstitial air is trapped inside the bed during and long after compaction. We deduce this from measuring the pressure changes above and below the bed: The top pressure decreases abruptly, on the time scale of the compaction, whereas that below the bed slowly rises to a maximum. Subsequently, both gently relax to ambient values. We formulate a one-dimensional diffusion model that uses only the change in bed height and the ambient pressure as an input, and we show that it leads to a fully quantitative understanding of the measured pressure variations.

In this paper, we present and justify an effective strategy for performing three-dimensional (3D) inertial-confinement-fusion (ICF) capsule simulations. We have evaluated a frequently used strategy in which two-dimensional (2D) simulations are rotated to 3D once sufficient relevant 2D flow physics has been captured and fine resolution requirements can be restricted to relatively small regions. This addresses situations typical of ICF capsules which are otherwise prohibitively intensive computationally. We tested this approach for our previously reported fully 3D simulations of laser-driven reshock experiments where we can use the available 3D data as reference. Our studies indicate that simulations that begin as purely 2D lead to significant underprediction of mixing and turbulent kinetic energy production at later time when compared to the fully 3D simulations. If, however, additional suitable nonuniform perturbations are applied at the time of rotation to 3D, we show that one can obtain good agreement with the purely 3D simulation data, as measured by vorticity distributions as well as integrated mixing and turbulent kinetic energy measurements. Next, we present results of simulations of a simple OMEGA-type ICF capsule using the developed strategy. These simulations are in good agreement with available experimental data and suggest that the dominant mechanism for yield degradation in ICF implosions is hydrodynamic instability growth seeded by long-wavelength surface defects. This effect is compounded by drive asymmetries and amplified by repeated shock interactions with an increasingly distorted shell, which results in further yield reduction. Our simulations are performed with and without drive asymmetries in order to compare the importance of these effects to those of surface defects; our simulations indicate that long-wavelength surface defects degrade yield by approximately 60% and short-wavelength drive asymmetry degrades yield by a further 30%.

In this paper we lay out an argument that generically the psychological arrow of time should align with the thermodynamic arrow of time where that arrow is well defined. This argument applies to any physical system that can act as a memory, in the sense of preserving a record of the state of some other system. This result follows from two principles: the robustness of the thermodynamic arrow of time to small perturbations in the state, and the principle that a memory should not have to be fine-tuned to match the state of the system being recorded. This argument applies even if the memory system itself is completely reversible and nondissipative. We make the argument with a paradigmatic system, and then formulate it more broadly for any system that can be considered a memory. We illustrate these principles for a few other example systems and compare our criteria to earlier treatments of this problem.

We show evidence, based on extensive and carefully performed numerical experiments, that the system of two elastic hard-point masses in one dimension is not ergodic for a generic mass ratio and consequently does not follow the principle of energy equipartition. This system is equivalent to a right triangular billiard. Remarkably, following the time-dependent probability distribution in a suitably chosen velocity direction space, we find evidence of exponential localization of invariant measure. For nongeneric mass ratios which correspond to billiard angles which are rational, or weak irrational multiples of $\pi$, the system is ergodic, consistent with existing rigorous results.

We apply the geometric theory of swimming at low Reynolds number to the study of nearly circular swimmers in two-dimensional fluids with nonvanishing “odd,” or Hall, viscosity. The odd viscosity gives an off-diagonal contribution to the fluid stress tensor, which results in a number of striking effects. In particular, we find that a swimmer whose area is changing will experience a torque proportional to the rate of change of the area, with the constant of proportionality given by the coefficient ${\eta }^o$ of odd viscosity. After working out the general theory of swimming in fluids with odd viscosity for a class of simple swimmers, we give a number of example swimming strokes which clearly demonstrate the differences between swimming in a fluid with conventional viscosity and a fluid which also has an odd viscosity. We also include a discussion of the extension of the famous Scallop theorem of low Reynolds number swimming to the case where the fluid has a nonzero odd viscosity. A number of more technical results, including a proof of the torque-area relation for swimmers of more general shape, are explained in a set of Appendixes.