We develop a method to analyze stochastic jump processes with a finite number of states $\{0,1,...,N\}$. It is known that the system can be analyzed using su(2) Lie algebra and the spin coherent state path integral. First, we analyze a linear susceptible-infected-susceptible epidemic model by using the Lie algebraic method and path integral. Then, we apply the instanton approximation to more general models where stochastic noise can induce bimodal distributions. By using density variables, we show that instanton solution can be interpreted as a solution of the equation of motion of the deterministic system corresponding to the stochastic system. It is also shown that instanton approximation correctly reproduces the noise-induced bimodality of the system.

We focus on a paradigmatic two-dimensional model of a nanoscale heat engine, the so-called Brownian gyrator, whose stochastic dynamics is described by a pair of coupled Langevin equations with different temperature noise terms. This model is known to produce a curl-carrying non-equilibrium steady-state with persistent angular rotations. We generalize the original model introducing constant forces doing work on the gyrator, for which we derive exact asymmetry relations, that are reminiscent of the standard fluctuation relations. Unlike the latter, our relations concern instantaneous and not time averaged values of the observables of interest. We investigate the full two-dimensional dynamics as well as the dynamics projected on the $x$ and $y$ axes, so that information about the state of the system can be obtained from just a part of its degrees of freedom. Such a state is characterized by effective “temperatures” that can be measured in nanoscale devices, but do not have a thermodynamic nature. Remarkably, the effective temperatures appearing in full dynamics are distinctly different from the ones emerging in its projections, confirming that they are not thermodynamic quantities, although they precisely characterize the state of the system.

We investigate the transport via the Boltzmann equation with anomalous collisions, obtaining a thermal-equilibrium velocity distribution function that deviates from the Maxwell-Boltzmann distribution. Here, we consider two types of anomalous collisions: the super-elastic-inelastic collisions and binary collisions, in which both generalized moments and the mass and momentum are conserved. Then, we investigated the transport characteristics of these two collision types, numerically, using the direct simulation Monte Carlo method. Finally, we analyzed the transport characteristics using the extended geometric Brownian motion, presenting numerical results obtained for both types of anomalous collisions.

We derive an algorithm for constructing an optimal waveform that ensures an entrainment of a spiking neuron to an external periodic stimulation with the minimum mean absolute value of the stimulating current. Such an optimization problem is relevant for the design of stimulation waveforms employed by implantable pulse generators for the treatment of brain disorders. Specifically, it is motivated by the desire to minimize tissue damage. The optimization problem is solved by involving the phase reduction theory and Pontriagyn's maximum principle. We show that the optimal waveform is of bang-off-bang type. The theoretical results are confirmed by three specific examples: an analytically solvable Stuart-Landau oscillator, a randomly generated phase response curve, and the classical Hodgkin-Huxley neuron model.

We employ Floquet analysis to study the spectral properties of a double-kicked top (DKT) system. This is a classically nonintegrable dynamical system, which also shows chaos. However, even for the underlying classically chaotic dynamics, the quantum quasienergy spectrum of this system does not follow the random matrix conjecture which was proposed for the quantum spectrum of any classically chaotic systems. Instead the quasienergy spectrum of the DKT system shows a butterfly-like self-similar fractal spectrum. Here we investigate the relation between the quasienergy spectrum and the energy spectrum of the corresponding time-independent Floquet Hamiltonian. This Hamiltonian is determined by factorizing the Floquet time-evolution operator into three terms: an initial kick and a final kick, and in between a time-independent evolution dictated by a time-independent Hermitian operator which is called the Floquet Hamiltonian. Like any other generic systems, the Floquet Hamiltonian of the DKT system is also not possible to determine exactly. We apply a recently proposed perturbation theory to obtain the approximate Floquet Hamiltonian at the high-frequency driving limit. We then study the parameter regime where the quasienergy spectrum of the Floquet time-evolution operator matches the energy spectrum of the approximate Floquet Hamiltonian. We have also done a comparative analysis of how the two butterfly spectra disappear with the variation of a system parameter. Finally, we also explore the self-similar property of the energy spectrum of the approximate Floquet Hamiltonian and find its connection with the Farey sequence. Unlike all previous studies, here we have extensively investigated the self-similar property of the whole DKT butterfly.

We study the problem of random search in finite networks with a tree topology, where it is expected that the distribution of the first-passage time $F\left(t\right)$ decays exponentially. We show that the slope $\alpha$ of the exponential tail is independent of the initial conditions of entering the tree in general, and scales exponentially or as a power law with the extent of the tree $L$, depending on the tendency $p$ to jump toward the target node. It is unfeasible to uniquely determine $L$ and $p$ from measuring $\alpha$ or the mean first-passage time (MFPT) of an ordinary diffusion along the tree. To unravel the structure, we consider lazy random walkers that take steps with probability $m$ when jumping on the nodes and return with probability $q$ from the leaves. By deriving an exact analytical expression for the MFPT of the intermittent random walk, we verify that the structural information of the tree can be uniquely extracted by measuring the MFPT for two randomly chosen types of tracer particles with distinct experimental parameters $m$ and $q$. We also address the applicability of our approach in the presence of disorder in the structure of the tree or statistical uncertainty in the experimental parameters.

DNA is a flexible molecule, but the degree of its flexibility is subject to debate. The commonly-accepted persistence length of $l_p\approx 500\AA$ is inconsistent with recent studies on short-chain DNA that show much greater flexibility but do not probe its origin. We have performed x-ray and neutron small-angle scattering on a short DNA sequence containing a strong nucleosome positioning element and analyzed the results using a modified Kratky-Porod model to determine possible conformations. Our results support a hypothesis from Crick and Klug in 1975 that some DNA sequences in solution can have sharp kinks, potentially resolving the discrepancy. Our conclusions are supported by measurements on a radiation-damaged sample, where single-strand breaks lead to increased flexibility and by an analysis of data from another sequence, which does not have kinks, but where our method can detect a locally enhanced flexibility due to an AT domain.

Trajectories and conformations of uni- and multiflagellar bacteria are studied with a coarse-grained model of a cell comprised of elastic flagella connected to a cell body. The elasticities of both the hook protein (connecting cell body and flagellum) and flagella are varied. Flexibility plays contrasting roles for uni- and multiflagellar swimmers. For a uniflagellar swimmer, hook and/or flagellar buckling occurs above a critical flexibility relative to the torque exerted by the flagellar motor. The addition of a second flagellum greatly expands the parameter regime of stable locomotion, because flexible hooks that would lead to buckling instability in the uniflagellar case provide the flexibility required for flagellar bundling in the biflagellar case. Similar observations hold for tri- and quadriflagellar swimmers. Indeed the stability regimes for uni- and quadriflagellar swimming are virtually inverted: to a first approximation what is stable in one case is unstable in the other. Swimming speed is also examined: it increases very weakly with the number of flagella and a simple theory is developed that explains this observation.

Polymer fibers, promising materials for tissue engineering and stretchable electronics, are intrinsically unstable due to their large surface-to-volume ratio. Recent experiments revealed that the addition of nanoparticles (NPs) to polymer fibers might enhance their stability. The addition of NPs, however, sometimes increases surface tension, which may facilitate the structural disruption of polymer fibers to polymer globules. It remains, therefore, elusive how NPs would affect the stability and the structural disruption of polymer fibers. In this work, we perform molecular dynamics simulations for polymer fibers with NPs of different types of intermolecular interactions. We prepare unstable polymer fibers that disrupt spontaneously into globules without NPs. We find that upon the addition of NPs, the structural disruption of unstable polymer fibers is hindered and the time (${\tau }_b$) taken for the fibers to disrupt to globules is increased. This indicates that the free energy barrier for the transition between the fiber and the globule would be increased due to NPs. The mechanism for the stability enhancement differs for NPs of different intermolecular interactions. When the interaction between polymers and NPs is quite attractive, NPs are located mostly at the center of the fiber around the axis and slow down the polymer mobility. On the other hand, when the interaction between polymers and NPs is not attractive, NPs are likely to be located at the surface of polymer fibers. When NPs are located at the fiber center, ${\tau }_b$ increases with a decrease in temperature. On the other hand, ${\tau }_b$ decreases with a decrease in temperature when NPs are located at the fiber surface.

We identify the maximally dense lattice packings of tangent-disk trimers with fixed bond angles ($\theta ={\theta }_0$) and contrast them to both their nonmaximally-dense-but-strictly-jammed lattice packings as well as the disordered jammed states they form for a range of compression protocols. While only ${\theta }_0=0,{60}^{\circ },\mathrm{and}{120}^{\circ }$ trimers can form the triangular lattice, maximally-dense maximally-symmetric packings for all ${\theta }_0$ fall into just two categories distinguished by their bond topologies: half-elongated-triangular for $0<{\theta }_0<{60}^{\circ }$ and elongated-snub-square for ${60}^{\circ }<{\theta }_0<{120}^{\circ }$. The presence of degenerate, lower-symmetry versions of these densest packings combined with several families of less-dense-but-strictly jammed lattice packings act in concert to promote jamming.

Spatial growth patterns are natural recording media (NRMs) that preserve important historical information, which can be accessed and analyzed to reconstruct past environmental conditions and events. Here, we propose the Bayesian inversion method, which can reconstruct the evolution of target parameters from purely spatial data by extending data assimilation (DA), a method for integrating numerical simulations with time-series observations. Our method converts discrete spatial observation data to time-series data with the help of a law representing the NRM's time-evolution dynamics and Gaussian process regression, enabling us to directly compare the observations with a numerical simulation based on the DA framework. The method's effectiveness is demonstrated using a synthetic inversion problem, namely reconstructing the $\mathit{pressure}–\mathit{temperature}–\mathit{time}$ ($P–T–t$) path of a metamorphic rock from chemical composition profiles of its zoned minerals. The proposed method is broadly applicable to a wide variety of NRMs.

We developed a powerful computational approach to elaborate on the onset mechanisms of deterministic chaos due to complex homoclinic bifurcations in diverse systems. Its core is the reduction of phase space dynamics to symbolic binary representations that lets one detect regions of simple and complex dynamics as well as fine organization structures of the latter in parameter space. Massively parallel simulations shorten the computational time to disclose highly detailed bifurcation diagrams to a few seconds.

In this work we address the classical statistical mechanical problem of calculating the residual entropy of ice models. The numerical work found in the literature is usually based on extrapolating to infinite-size results obtained for finite-size systems with periodic boundary conditions. In this work we investigate how boundary conditions affect the calculation of the residual entropy for square, cubic, and hexagonal lattices using periodic, antiperiodic, and open boundary conditions. We show that periodic boundary conditions lead to noticeable oscillations in the entropy as a function of lattice size, and we calculate in open finite systems the contribution to the entropy from the open boundary. For our calculations we introduce a variation on multicanonical simulation methods that directly calculate the number of states in the ground state without the need of a Hamiltonian.

Droplets can exhibit complex dynamics when vertically and sinusoidally forced by a liquid surface from which they remain separated by a thin air cushion. Here we extend previous studies to include a family of periodic forcing functions that vary smoothly from sinusoidal to square wave by changing a single parameter. Through analytical and numerical work we find that the dynamics of the droplets and transitions between regular and chaotic regimes are effectively controlled by the impulse imparted on the droplets over a half-period. We also find that having nonsinusoidal forcing lowers the threshold amplitudes for most of the dynamical regimes. This is explained on the basis of a correlation between impulse increases and subsequent energy increases.

To inquire into whether two enantiomers can be absolutely distinguished on the sole basis of their different degrees of orientational ordering when dissolved in a chiral nematic liquid crystal, several enantiomeric pairs of hard helical particles are dissolved in a cholesteric (nematic) liquid crystal made of slender hard helical particles as well as in a screwlike nematic liquid crystal made of tortuous hard helical particles. While in the former ordinary chiral nematic solvent their nematic order parameters are (almost) coincident, in the latter new chiral nematic solvent the two enantiomeric solutes not only usually have more appreciably different nematic order parameters, but also always have significantly different screwlike order parameters. If also the latter orientational order parameter could be measurable in real experiments, it would constitute an additional decisive piece of information on the way to absolutely distinguishing two enantiomers on the sole basis of their different degrees of orientational order when dissolved in a chiral nematic liquid crystal that is in the apter screwlike nematic phase. Even in that event, however, the general absence of regularity and systematicity in the trend of the orientational order parameters, already manifest in these elementary hard helical-particle binary systems, would make this coveted achievement experimentally arduous without the assistance of very accurate and precise theoretical calculations.

With today's discrete element modeling (DEM) coupled with pore-scale finite volume (PFV) scheme, the impact of an internal fluid flow on the mechanical stability of a granular material is assessed at the scale of a three-dimensional representative elementary volume. To isolate the specific impact of the fluid on the contact network, a granular sample in which rattlers have been initially removed is considered while a fluid pressure drop is applied under constant mechanical loading. Although the initial state is reputed to be unstable according to the second-order work criterion, it is shown that (i) an internal fluid flow is able to trigger the underlying instability on its own and (ii) the flow direction has no influence because of local fluid force fluctuations. In addition, the microscale mechanisms occurring during the sample collapse are investigated and it is shown that the fluid perturbation induces a decrease in kinematic constraints around force chains, which is characteristic of a local loss of stability.

Using the bilinear Markov chain approach, we study statistical properties of natural random symbolic sequences with complex correlation properties. In the limit of weak correlations, we present analytically the entropy of sequence by means of correlation functions of the second and third orders. We illustrate the applicability of the developed theory to some sequences naturally arising in biology and chaotic dynamics. We evaluate numerically the entropy of DNA nucleotide sequences and sequences obtained by dichotomization of a logistic map. Using the connection between the compressibility of random sequence and the algorithmic entropy, we compare the levels of entropies, obtained by means of a combination of analytical and numerical methods, with the algorithmic entropy calculated by the standard files archivers. We show that our method gives a much lower level of entropy as compared to the best archivers. Numerical simulations show also that an account of third-order correlation functions significantly decreases the entropy calculated in the framework of the additive Markov chain approach.