The stochastic analysis of complex reaction networks is a difficult problem because the number of microscopic states in such systems increases exponentially with the number of reactive species. Direct integration of the master equation is thus infeasible and is most often replaced by Monte Carlo simulations. While Monte Carlo simulations are a highly effective tool, equation-based formulations are more amenable to analytical treatment and may provide deeper insight into the dynamics of the network. Here, we present a highly efficient equation-based method for the analysis of stochastic reaction networks. The method is based on the recently introduced binomial moment equations [Barzel and Biham, Phys. Rev. Lett. 106, 150602 (2011)]. The binomial moments are linear combinations of the ordinary moments of the probability distribution function of the population sizes of the interacting species. They capture the essential combinatorics of the reaction processes reflecting their stoichiometric structure. This leads to a simple and transparent form of the equations, and allows a highly efficient and surprisingly simple truncation scheme. Unlike ordinary moment equations, in which the inclusion of high order moments is prohibitively complicated, the binomial moment equations can be easily constructed up to any desired order. The result is a set of equations that enables the stochastic analysis of complex reaction networks under a broad range of conditions. The number of equations is dramatically reduced from the exponential proliferation of the master equation to a polynomial (and often quadratic) dependence on the number of reactive species in the binomial moment equations. The aim of this paper is twofold: to present a complete derivation of the binomial moment equations; to demonstrate the applicability of the moment equations for a representative set of example networks, in which stochastic effects play an important role.

We present a generalization of Jarzynski's equality, applicable to quantum systems, that is related to discretized mechanical work and free-energy changes. The theory is based on a stepwise pulling protocol. We find that work distribution functions can be constructed from fluctuations of a reaction coordinate along a reaction pathway in the stepwise pulling protocol. We also propose two sets of equations to determine the two possible optimal pathways that provide the most significant contributions to free-energy changes. We find that the transitions along these most optimal pathways, satisfying both sets of equations, follow the principle of detailed balance. We then test the theory by explicitly computing the free-energy changes for a one-dimensional quantum harmonic oscillator. This approach suggests a feasible way of measuring the fluctuations to experimentally test Jarzynski's equality in many-body systems, such as Bose-Einstein condensates.

The indoor microclimate is an issue in modern society, where people spend about 90$\%$ of their time indoors. Temperature and relative humidity are commonly used for its evaluation. In this context, the two parameters are usually considered as behaving in the same manner, just inversely correlated. This opinion comes from observation of the deterministic components of temperature and humidity time series. We focus on the dynamics and the dependency structure of the time series of these parameters, without deterministic components. Here we apply the mean square displacement, the autoregressive integrated moving average (ARIMA), and the methodology for studying anomalous diffusion. The analyzed data originated from five monitoring locations inside a modern office building, covering a period of nearly one week. It was found that the temperature data exhibited a transition between diffusive and subdiffusive behavior, when the building occupancy pattern changed from the weekday to the weekend pattern. At the same time the relative humidity consistently showed diffusive character. Also the structures of the dependencies of the temperature and humidity data sets were different, as shown by the different structures of the ARIMA models which were found appropriate. In the space domain, the dynamics and dependency structure of the particular parameter were preserved. This work proposes an approach to describe the very complex conditions of indoor air and it contributes to the improvement of the representative character of microclimate monitoring.

With Monte Carlo simulations, we systematically investigate the depinning phase transition in the two-dimensional driven random-field clock model. Based on the short-time dynamic approach, we determine the transition field and critical exponents. The results show that the critical exponents vary with the form of the random-field distribution and the strength of the random fields, and the roughening dynamics of the domain interface belongs to the new subclass with $\zeta \ne {\zeta }_{\mathrm{loc}}\ne {\zeta }_s$ and ${\zeta }_{\mathrm{loc}}\ne 1$. More importantly, we find that the transition field and critical exponents change with the initial orientations of the magnetization of the two ordered domains.

The influence of grain angularity on the properties of dense flows down a rough inclined plane are investigated. Three-dimensional numerical simulations using the nonsmooth contact dynamics method are carried out with both spherical (rounded) and polyhedral (angular) grain assemblies. Both sphere and polyhedra assemblies abide by the flow start and stop laws, although much higher tilt angle values are required to trigger polyhedral grain flow. In the dense permanent flow regime, both systems show similarities in the bulk of the material (away from the top free surface and the substrate), such as uniform values of the solid fraction, inertial number and coordination number, or linear dependency of the solid fraction and effective friction coefficient with the inertial number. However, discrepancies are also observed between spherical and polyhedral particle flows. A dead (or nearly arrested) zone appears in polyhedral grain flows close to the rough bottom surface, reflected by locally concave velocity profiles, locally larger coordination number and solid fraction values, smaller inertial number values. This dead zone disappears for smooth bottom surfaces. In addition, unlike sphere assemblies, polyhedral grain assemblies exhibit significant normal stress differences, which increase close to the substrate.

The stochastic properties of the total internal energy of a dilute granular gas in the steady uniform shear flow state are investigated. A recent theory formulated for fluctuations about the homogeneous cooling state is extended by analogy with molecular systems. The theoretical predictions are compared with molecular dynamics simulation results. Good agreement is found in the limit of weak inelasticity, while systematic and relevant discrepancies are observed when the inelasticity increases. The origin of this behavior is discussed.

I reanalyze the hydrodynamic theory of fluid, polar-ordered flocks. I find new linear terms in the hydrodynamic equations which slightly modify the anisotropy, but not the scaling, of the damping of sound modes. I also find that the nonlinearities allowed in equilibrium do not stabilize long-ranged order in spatial dimensions $d=2$, in accord with the Mermin-Wagner theorem. Nonequilibrium nonlinearities $do$ stabilize long-ranged order in $d=2$, as argued by earlier work. Some of these were missed by earlier work; it is unclear whether or not they change the scaling exponents in $d=2$.

The widespread use of tweezers for measurement of ligand-DNA binding parameters is based on the McGhee–von Hippel treatment of the DNA contour and persistence length as a function of concentration. The McGhee–von Hippel approach contains the basic assumption that the binding constant $K$ is independent of the number of already bound ligands. However, the change in elasticity of DNA on binding affects the entropic part of the Gibbs free energy and, hence, the $K$ value in a concentration-dependent manner, making the whole approach inconsistent. In the present work we show that the energetic effect of DNA stiffening on noncovalent binding of small ligands is negligible with respect to the net energy of reaction, whereas the DNA stiffening on binding of large ligands must always be considered in each particular case.

We considered a multiblock molecular model of biological evolution, in which fitness is a function of the mean types of alleles located at different parts (blocks) of the genome. We formulated an infinite population model with selection and mutation, and calculated the mean fitness. For the case of recombination, we formulated a model with a multidimensional fitness landscape (the dimension of the space is equal to the number of blocks) and derived a theorem about the dynamics of initially narrow distribution. We also considered the case of lethal mutations. We also formulated the finite population version of the model in the case of lethal mutations. Our models, derived for the virus evolution, are interesting also for the statistical mechanics and the Hamilton-Jacobi equation as well.

The dynamics of a symmetric network of oscillators that are mutually coupled via multiple dynamical components with mismatched delays is studied. We find that networked oscillators experience oscillation death (OD) over a much larger domain of parameters when their different dynamical components are linked with mismatched delays than with only one delay. In particular, if the delays are mismatched by retaining a certain bias, OD is proved to be linearly stable even for very large delays for an arbitrary symmetric network. Further, we show that the minimal value of the intrinsic frequency necessary to induce OD decreases as the degree of mismatch in the coupling delays increases. The stabilizing effect of multicomponent coupling with mismatched delays is shown to be valid in networked chaotic oscillators also. The proposed coupling strategy can possibly be applied in controlling several pathological activities in neuronal systems and in engineering applications.

Stochastic rotation of rotating pairs in a hexagonal superlattice pattern is observed in a dielectric barrier discharge system. It is found that the pairs rotate with orientation and diameter randomly changing by observing a series of frames recorded by a high speed video camera. Frames recorded by a high speed framing camera with an exposure time corresponding to current pulse phases in one half cycle of the applied voltage show that one rotating spot, six small spots, and another rotating spot in one cell discharge successively. Based on this discharging sequence, forces exerted on a rotating spot are analyzed at different discharging stages in a half voltage cycle. A resultant force on a rotating spot with both magnitude and direction varied leads to the stochastic rotation.

We show that time-oscillating electric fields applied to plasmas present in flames create steady flows of gas. Ions generated within the flame move in the field and migrate a distance $\delta$ before recombining; the net flow of ions away from the flame creates a time-averaged force that drives the steady flows observed experimentally. A quantitative model describes the response of the flame and reveals how $\delta$ decreases as the frequency of the applied field increases. Interestingly, above a critical frequency, ac fields can be used to manipulate flames at a distance without the need for proximal electrodes.

In this work, we analyze the directional movement of impacting liquid drops on dual-textured solid surfaces comprising two different surface morphologies: a textured surface and a smooth surface. The dynamics of liquid drops impacting onto the junction line between the two parts of the dual-textured surfaces is studied experimentally for varying drop impact velocity. The dual-textured surfaces used here featured a variation in their textures’ geometrical parameters as well as their surface chemistry. Two types of liquid drop differing in their surface tension were used. The impact process develops a net horizontal drop velocity towards the higher-wettability surface portion and results in a bulk movement of the impacting drop liquid. The final distance moved by the impacting drop from the junction line decreases with increasing impacting drop Weber number We. A fully theoretical model, employing a balance of forces acting at the drop contact line as well as energy conservation, is formulated to determine the variation, with We, of net horizontal drop velocity and subsequent movement of the impacting drop on the dual-textured surfaces.

We use a stochastic approach to show how Taylor dispersion is affected by kinetic processes of adsorption and desorption onto surfaces. A general theory is developed, from which we derive explicitly the dispersion coefficients of canonical examples such as Poiseuille flows in planar and cylindrical geometries, in both constant and sinusoidal velocity fields. These results open the way for the measurement of adsorption and desorption rate constants using stationary flows and molecular sorting using the stochastic resonance of the adsorption and desorption processes with the oscillatory velocity field.

We calculate the binding energy of soliton molecules of an integrable nonlinear Schr$\ddot{\mathrm{o}}$dinger equation with time-dependent harmonic potential and cubic nonlinearity. Through a scaling transformation, an exact formula for the binding energy can be derived from that of the free soliton molecules in a homogeneous background. In the special case of oscillatory time dependence, sharp resonances occur at some integer and fractional multiples of the natural frequency of the molecule. Enhanced binding is obtained at these resonances and over some finite continuous range of low frequencies.

A family of models of liquid on a two-dimensional lattice (2D lattice liquid models) have been proposed as primitive models of soft-material membrane. As a first step, we have formulated them as single-component, single-layered, classical particle systems on a two-dimensional surface with no explicit viscosity. Among the family of the models, we have shown and constructed two stochastic models, a vicious walk model and a flow model, on an isotropic regular lattice and on some honeycomb lattices of various sizes. In both cases, the dynamics is governed by the nature of the frustration of the particle movements. By simulations, we have found the approximate functional form of the frustration probability and peculiar anomalous diffusions in their time-averaged mean-square displacements in the flow model. The relations to other existing statistical models and possible extensions of the models are also discussed.

The propagation speed of a starting wave, which is a wave of people's successive reactions in the relaxation process of a queue, has an essential role for pedestrians and vehicles to achieve smooth movement. For example, a queue of vehicles with appropriate headway (or density) alleviates traffic jams since the delay of reaction to start is minimized. In this paper, we have investigated the fundamental relation between the propagation speed of a starting wave and the initial density by both our mathematical model built on the stochastic cellular automata and experimental measurements. Analysis of our mathematical model implies that the relation is characterized by the power law $\alpha {\rho }^{-\beta }$ $(\beta \ne 1)$, and the experimental results verify this feature. Moreover, when the starting wave is characterized by the power law $(\beta >1)$, we have revealed the existence of optimal density, where the required time, i.e., the sum of the waiting time until the starting wave reaches the last pedestrian in a queue and his/her travel time to pass the head position of the initial queue, is minimized. This optimal density inevitably plays a significant role in achieving a smooth movement of crowds and vehicles in a queue.

In this note, we discuss possible analogies between the subcritical transition to turbulence in shear flows and the glass transition in supercooled liquids. We briefly review recent experimental and numerical results, as well as theoretical proposals, and compare the difficulties arising in assessing the divergence of the turbulence lifetime in subcritical shear flow with that encountered for the relaxation time in the study of the glass transition. In order to go beyond the purely methodological similarities, we further elaborate on this analogy and propose a simple model for the transition to turbulence, inspired by the random energy model (a standard model for the glass transition), with the aim to possibly foster yet-unexplored directions of research in subcritical shear flows.

The kinetic properties of the pressure-driven Poiseuille flow in nanochannels with and without nanoparticles were studied with a nonequilibrium molecular dynamics simulation. To allow the fluid to dissipate heat, the boundary was kept at a constant temperature. Pure fluid simulations were taken as references and also used to study the fluid-wall interfacial interaction effects. The viscosity profiles of the fluid were calculated on the basis of velocity profiles and known applied shear stress. We present the relationship between the viscosity increase and particle loading. The role of channel wall–fluid wetting properties on the flow and viscosity was also investigated.

The pair interactions $U_{ij}\left(r\right)$ determine the thermodynamics and linear transport properties of matter via the pair-distribution functions (PDFs), i.e., $g_{ij}\left(r\right)$. Great simplicity is achieved if $U_{ij}\left(r\right)$ could be directly used to predict material properties via classical simulations, avoiding many-body wave functions. Warm dense matter (WDM) is encountered in quasiequilibria where the electron temperature $T_e$ differs from the ion temperature $T_i$, as in laser-heated or in shock-compressed matter. The electron PDFs $g_{ee}\left(r\right)$ as perturbed by the ions are used to evaluate fully nonlocal exchange-correlation corrections to the free energy, using hydrogen as an example. Electron-ion potentials for ions with a bound core are discussed with Al and Si as examples, for WDM with $T_e\ne T_i$, and valid for times shorter than the electron-ion relaxation time. In some cases the potentials develop attractive regions and then become repulsive and “Yukawa-like” for higher $T_e$. These results clarify the origin of initial phonon hardening and rapid release. Pair potentials for shock-heated WDM show that phonon hardening would not occur in most such systems. Defining meaningful quasiequilibrium static transport coefficients consistent with the dynamic values is addressed. There seems to be no meaningful “static conductivity” obtainable by extrapolating experimental or theoretical $\sigma (\omega ,T_i,T_e)$ to $\omega \to 0$, unless $T_i\to T_e$ as well. Illustrative calculations of quasistatic resistivities $R(T_i,T_e)$ of laser-heated as well as shock-heated aluminum and silicon are presented using our pseudopotentials, pair potentials, and classical integral equations. The quasistatic resistivities display clear differences in their temperature evolutions, but are not the strict $\omega \to 0$ limits of the dynamic values.