The persistence of a stochastic variable is the probability that it does not cross a given level during a fixed time interval. Although persistence is a simple concept to understand, it is in general hard to calculate. Here we consider zero mean Gaussian stationary processes in discrete time $n$. Few results are known for the persistence $P_0\left(n\right)$ in discrete time, except the large time behavior which is characterized by the nontrivial constant $\theta$ through $P_0\left(n\right)\sim {\theta }^n$. Using a modified version of the independent interval approximation (IIA) that we developed before, we are able to calculate $P_0\left(n\right)$ analytically in $z$-transform space in terms of the autocorrelation function $A\left(n\right)$. If $A\left(n\right)\to 0$ as $n\to \infty$, we extract $\theta$ numerically, while if $A\left(n\right)=0$, for finite $n>N$, we find $\theta$ exactly (within the IIA). We apply our results to three special cases: the nearest-neighbor-correlated “first order moving average process”, where $A\left(n\right)=0$ for $n>1$, the double exponential-correlated “second order autoregressive process”, where $A\left(n\right)=c_1{\lambda }_1^n+c_2{\lambda }_2^n$, and power-law-correlated variables, where $A\left(n\right)\sim n^{-\mu }$. Apart from the power-law case when $\mu <5$, we find excellent agreement with simulations.

This study proposes a model of a totally asymmetric simple exclusion process on a single-channel lane with functions of site assignments along the pit lane. The system model attempts to insert a new particle to the leftmost site at a certain probability by randomly selecting one of the empty sites in the pit lane, and reserving it for the particle. Thereafter, the particle is directed to stop at the site only once during its travel. Recently, the system was determined to show a self-deflection effect, in which the site usage distribution biases spontaneously toward the leftmost site, and the throughput becomes maximum when the site usage distribution is slightly biased to the rightmost site. Our exact analysis describes this deflection effect and show a good agreement with simulations.

We show that the decay of the density of active particles in the reaction $A+B\to 0$ in one dimension, with exclusion interaction, results in logarithmic corrections to the expected power law decay, when the starting initial condition (i.c.) is periodic. It is well known that the late-time density of surviving particles goes as $t^{-1/4}$ with random initial conditions, and as $t^{-1/2}$ with alternating initial conditions ($ABABAB\cdots$). We show that the decay for periodic i.c.'s made of longer blocks ($A^n{B}^n{A}^n{B}^n\cdots$) do not show a pure power-law decay when $n$ is even. By means of first-passage Monte Carlo simulations, and a mapping to a $q$-state coarsening model which can be solved in the independent interval approximation (IIA), we show that the late-time decay of the density of surviving particles goes as $t^{-1/2}{\left[\ln \left(t\right)\right]}^{-1}$ for $n$ even, but as $t^{-1/2}$ when $n$ is odd. We relate this kinetic symmetry breaking in the Glauber Ising model. We also see a very slow crossover from a $t^{-1/2}{\left[\ln \left(t\right)\right]}^{-1}$ regime to eventual $t^{-1/2}$ behavior for i.c.'s made of mixtures of odd- and even-length blocks.

We present a path integral formulation of Darcy's equation in one dimension with random permeability described by a correlated multivariate lognormal distribution. This path integral is evaluated with the Markov chain Monte Carlo method to obtain pressure distributions, which are shown to agree with the solutions of the corresponding stochastic differential equation for Dirichlet and Neumann boundary conditions. The extension of our approach to flow through random media in two and three dimensions is discussed.

A reduction in high-dimensional phenotypic states to a few degrees of freedom is essential to understand biological systems. Here, we show evolutionary robustness causes such reduction which restricts possible phenotypic changes in response to a variety of environmental conditions. First, global protein expression changes in Escherichia coli after various environmental perturbations were shown to be proportional across components, across different types of environmental conditions. To examine if such dimension reduction is a result of evolution, we analyzed a cell model—with a huge number of components, that reproduces itself via a catalytic reaction network—and confirmed that common proportionality in the concentrations of all components is shaped through evolutionary processes. We found that the changes in concentration across all components in response to environmental and evolutionary changes are constrained to the changes along a one-dimensional major axis, within a huge-dimensional state space. On the basis of these observations, we propose a theory in which such constraints in phenotypic changes are achieved both by evolutionary robustness and plasticity and formulate this proposition in terms of dynamical systems. Accordingly, broad experimental and numerical results on phenotypic changes caused by evolution and adaptation are coherently explained.

We explore minimal navigation strategies for active particles in complex, dynamical, external fields, introducing a class of autonomous, self-propelled particles which we call Markovian robots (MR). These machines are equipped with a navigation control system (NCS) that triggers random changes in the direction of self-propulsion of the robots. The internal state of the NCS is described by a Boolean variable that adopts two values. The temporal dynamics of this Boolean variable is dictated by a closed Markov chain—ensuring the absence of fixed points in the dynamics—with transition rates that may depend exclusively on the instantaneous, local value of the external field. Importantly, the NCS does not store past measurements of this value in continuous, internal variables. We show that despite the strong constraints, it is possible to conceive closed Markov chain motifs that lead to nontrivial motility behaviors of the MR in one, two, and three dimensions. By analytically reducing the complexity of the NCS dynamics, we obtain an effective description of the long-time motility behavior of the MR that allows us to identify the minimum requirements in the design of NCS motifs and transition rates to perform complex navigation tasks such as adaptive gradient following, detection of minima or maxima, or selection of a desired value in a dynamical, external field. We put these ideas in practice by assembling a robot that operates by the proposed minimalistic NCS to evaluate the robustness of MR, providing a proof of concept that is possible to navigate through complex information landscapes with such a simple NCS whose internal state can be stored in one bit. These ideas may prove useful for the engineering of miniaturized robots.

Linear stability analysis is conducted to study the onset of density-driven convection involved in solubility trapping of $\mathrm{C}{\mathrm{O}}_2$ in fractured aquifers. The effect of physical properties of a fracture network on the stability of a diffusive boundary layer in a saturated fractured porous media is investigated using the dual porosity concept. Linear stability analysis results show that both fracture interporosity flow and fracture storativity play an important role in the stability behavior of the system. It is shown that a diffusive boundary layer under the gravity field in fractured porous media with lower fracture storativity and/or higher fracture interporosity flow coefficient is more stable. We present scaling relations for the onset of convective instability in fractured aquifers with single and variable matrix block size distribution. These findings improve our understanding of density-driven flow in fractured aquifers and are important in the estimation of potential storage capacity, risk assessment, and storage site characterization and screening.

Three-dimensional plasma crystals are often described as Yukawa systems for which a phase transition between the crystal structures fcc and bcc has been predicted. However, experimental investigations of this transition are missing. We use a fast scanning video camera to record the crystallization process of 70 000 microparticles and investigate the existence of the fcc-bcc phase transition at neutral gas pressures of 30, 40, and 50 Pa. To analyze the crystal, robust phase diagrams with the help of a machine learning algorithm are calculated. This work shows that the phase transition can be investigated experimentally and makes a comparison with numerical results of Yukawa systems. The phase transition is analyzed in dependence on the screening parameter and structural order. We suggest that the transition is an effect of gravitational compression of the plasma crystal. Experimental investigations of the fcc-bcc phase transition will provide an opportunity to estimate the coupling strength $\mathrm{\Gamma }$ by comparison with numerical results of Yukawa systems.

This paper presents a simple tool for characterizing the time scale for continuum diffusion processes through layered heterogeneous media. This mathematical problem is motivated by several practical applications such as heat transport in composite materials, flow in layered aquifers, and drug diffusion through the layers of the skin. In such processes, the physical properties of the medium vary across layers and internal boundary conditions apply at the interfaces between adjacent layers. To characterize the time scale, we use the concept of mean action time, which provides the mean time scale at each position in the medium by utilizing the fact that the transition of the transient solution of the underlying partial differential equation model, from initial state to steady state, can be represented as a cumulative distribution function of time. Using this concept, we define the characteristic time scale for a multilayer diffusion process as the maximum value of the mean action time across the layered medium. For given initial conditions and internal and external boundary conditions, this approach leads to simple algebraic expressions for characterizing the time scale that depend on the physical and geometrical properties of the medium, such as the diffusivities and lengths of the layers. Numerical examples demonstrate that these expressions provide useful insight into explaining how the parameters in the model affect the time it takes for a multilayer diffusion process to reach steady state.

We investigate the long-time behavior of a passive particle evolving in a one-dimensional diffusive random environment, with diffusion constant $D$. We consider two cases: (a) The particle is pulled forward by a small external constant force and (b) there is no systematic bias. Theoretical arguments and numerical simulations provide evidence that the particle is eventually trapped by the environment. This is diagnosed in two ways: The asymptotic speed of the particle scales quadratically with the external force as it goes to zero, and the fluctuations scale diffusively in the unbiased environment, up to possible logarithmic corrections in both cases. Moreover, in the large $D$ limit (homogenized regime), we find an important transient region giving rise to other, finite-size scalings, and we describe the crossover to the true asymptotic behavior.

Using mesoscopic numerical simulations and analytical theory, we investigate the coarsening of the solvent structure around a colloidal particle emerging after a temperature quench of the colloid surface. Qualitative differences in the coarsening mechanisms are found, depending on the composition of the binary liquid mixture forming the solvent and on the adsorption preferences of the colloid. For an adsorptionwise neutral colloid, the phase next to its surface alternates as a function of time. This behavior sets in on the scale of the relaxation time of the solvent and is absent for colloids with strong adsorption preferences. A Janus colloid, with a small temperature difference between its two hemispheres, reveals an asymmetric structure formation and surface enrichment around it, even if the solvent is within its one-phase region and if the temperature of the colloid is above the critical demixing temperature $T_c$ of the solvent. Our phenomenological model turns out to capture recent experimental findings according to which, upon laser illumination of a Janus colloid and due to the ensuing temperature gradient between its two hemispheres, the surrounding binary liquid mixture develops a concentration gradient.

The behavior of a vertically vibrated granular bed is reminiscent of a liquid in that it exhibits many phenomena such as convection and Faraday-like surface waves. However, when the lateral dimensions of the bed are confined such that a quasi-one-dimensional geometry is formed, the only phenomena that remain are bouncing bed and the granular Leidenfrost effect. This permits the observation of the granular Leidenfrost state for a wide range of energy injection parameters and more specifically allows for a thorough characterization of the low-frequency oscillation (LFO) that is present in this state. In both experiments and particle simulations we determine the LFO frequency from the power spectral density of the center-of-mass signal of the grains, varying the amplitude and frequency of the driving, the particle diameter, and the number of layers in the system. We thus find that the LFO frequency (i) is inversely proportional to the fast inertial timescale and (ii) decorrelates with a typical decay time proportional to the slow dissipative timescale in the system. The latter is consistent with the view that the LFO is driven by the inherent noise that is present in the granular Leidenfrost state with a low number of particles.

Numerical simulations and finite-size scaling analysis have been carried out to study standard and inverse bond percolation of straight rigid rods on square lattices. In the case of standard percolation, the lattice is initially empty. Then, linear bond $k$-mers (sets of $k$ linear nearest-neighbor bonds) are randomly and sequentially deposited on the lattice. Jamming coverage $p_{j,k}$ and percolation threshold $p_{c,k}$ are determined for a wide range of $k$ ($1\le k\le 120$). $p_{j,k}$ and $p_{c,k}$ exhibit a decreasing behavior with increasing $k$, $p_{j,k\to \infty }=0.7476\left(1\right)$ and $p_{c,k\to \infty }=0.0033\left(9\right)$ being the limit values for large $k$-mer sizes. $p_{j,k}$ is always greater than $p_{c,k}$, and consequently, the percolation phase transition occurs for all values of $k$. In the case of inverse percolation, the process starts with an initial configuration where all lattice bonds are occupied and, given that periodic boundary conditions are used, the opposite sides of the lattice are connected by nearest-neighbor occupied bonds. Then, the system is diluted by randomly removing linear bond $k$-mers from the lattice. The central idea here is based on finding the maximum concentration of occupied bonds (minimum concentration of empty bonds) for which connectivity disappears. This particular value of concentration is called the inverse percolation threshold $p_{c,k}^i$, and determines a geometrical phase transition in the system. On the other hand, the inverse jamming coverage $p_{j,k}^i$ is the coverage of the limit state, in which no more objects can be removed from the lattice due to the absence of linear clusters of nearest-neighbor bonds of appropriate size. It is easy to understand that $p_{j,k}^i=1-p_{j,k}$. The obtained results for $p_{c,k}^i$ show that the inverse percolation threshold is a decreasing function of $k$ in the range $1\le k\le 18$. For $k>18$, all jammed configurations are percolating states, and consequently, there is no nonpercolating phase. In other words, the lattice remains connected even when the highest allowed concentration of removed bonds $p_{j,k}^i$ is reached. In terms of network attacks, this striking behavior indicates that random attacks on single nodes ($k=1$) are much more effective than correlated attacks on groups of close nodes (large $k$'s). Finally, the accurate determination of critical exponents reveals that standard and inverse bond percolation models on square lattices belong to the same universality class as the random percolation, regardless of the size $k$ considered.

We introduce two integral transforms of the quantum mechanical transition kernel that represent physical information about the path integral. These transforms can be interpreted as probability distributions on particle trajectories measuring respectively the relative contribution to the path integral from paths crossing a given spatial point (the hit function) and the likelihood of values of the line integral of the potential along a path in the ensemble (the path-averaged potential).

In this comparative study, six causality detection methods were compared, namely, the Granger vector autoregressive test, the extended Granger test, the kernel version of the Granger test, the conditional mutual information (transfer entropy), the evaluation of cross mappings between state spaces, and an assessment of predictability improvement due to the use of mixed predictions. Seven test data sets were analyzed: linear coupling of autoregressive models, a unidirectional connection of two Hénon systems, a unidirectional connection of chaotic systems of Rössler and Lorenz type and of two different Rössler systems, an example of bidirectionally connected two-species systems, a fishery model as an example of two correlated observables without a causal relationship, and an example of mediated causality. We tested not only $20000$ points long clean time series but also noisy and short variants of the data. The standard and the extended Granger tests worked only for the autoregressive models. The remaining methods were more successful with the more complex test examples, although they differed considerably in their capability to reveal the presence and the direction of coupling and to distinguish causality from mere correlation.

Multiple scales in metapopulations can give rise to paradoxical behavior: in a conceptual model for a public goods game, the species associated with a fitness cost due to the public good production can be stabilized in the well-mixed limit due to the mere existence of these scales. The scales in this model involve a length scale corresponding to separate patches, coupled by mobility, and separate time scales for reproduction and interaction with a local environment. Contrary to the well-mixed high mobility limit, we find that for low mobilities, the interaction rate progressively stabilizes this species due to stochastic effects, and that the formation of spatial patterns is not crucial for this stabilization.

In this work, computer modeling has been used to show that longer ligands allow biological cells (e.g., blood platelets) to withstand stronger flows after their adhesion to solid walls. A mechanistic model of polymer-mediated ligand-receptor adhesion between a microparticle (cell) and a flat wall has been developed. The theoretical threshold between adherent and non-adherent regimes has been derived analytically and confirmed by simulations. These results lead to a deeper understanding of numerous biophysical processes, e.g., arterial thrombosis, and to the design of new biomimetic colloid-polymer systems.

In a neuron with hyperpolarization activated current ($I_h$), the correct input frequency leads to an enhancement of the output response. This behavior is known as resonance and is well described by the neuronal impedance. In a simple neuron model we derive equations for the neuron's resonance and we link its frequency and existence with the biophysical properties of $I_h$. For a small voltage change, the component of the ratio of current change to voltage change ($dI/dV$) due to the voltage-dependent conductance change ($dg/dV$) is known as derivative conductance ($G_h^{\text{Der}}$). We show that both $G_h^{\text{Der}}$ and the current activation kinetics (characterized by the activation time constant ${\tau }_h$) are mainly responsible for controlling the frequency and existence of resonance. The increment of both factors ($G_h^{\text{Der}}$ and ${\tau }_h$) greatly contributes to the appearance of resonance. We also demonstrate that resonance is voltage dependent due to the voltage dependence of $G_h^{\text{Der}}$. Our results have important implications and can be used to predict and explain resonance properties of neurons with the $I_h$ current.