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The Markov chain Monte Carlo (MCMC) method is widely used in various fields as a powerful numerical integration technique for systems with many degrees of freedom. In MCMC methods, probabilistic state transitions can be considered as a random walk in state space, and random walks allow for sampling from complex distributions. However, paradoxically, it is necessary to carefully suppress the randomness of the random walk to improve computational efficiency. By breaking detailed balance, we can create a probability flow in the state space and perform more efficient sampling along this flow. Motivated by this idea, practical and efficient nonreversible MCMC methods have been developed over the past ten years. In particular, the lifting technique, which introduces probability flows in an extended state space, has been applied to various systems and has proven more efficient than conventional reversible updates. We review and discuss several practical approaches to implementing nonreversible MCMC methods, including the shift method in the cumulative distribution and the directed-worm algorithm. [ABSTRACT FROM AUTHOR]

Motion in bounded domains is a fundamental concept in various fields, including billiard dynamics and random walks on finite lattices, and has important applications in physics, ecology, and biology. An important universal property related to the average return time to the boundary, the Mean Path Length Theorem (MPLT), has been proposed theoretically and experimentally confirmed in various contexts. We investigated a wide range of mechanisms that lead to deviations from this universal behavior, such as boundary effects, reorientation, and memory processes. This study investigates the dynamics of run-and-tumble particles within a confined two-dimensional circular domain. Through a combination of theoretical approaches and numerical simulations, we validate the MPLT under uniform and isotropic particle inflow conditions. This research demonstrates that although the MPLT is generally applicable for different step length distributions, deviations occur for non-uniform angular distributions, non-elastic boundary conditions, or memory processes. These results underline the crucial influence of boundary interactions and angular dynamics on the behavior of particles in confined spaces. Our results provide new insights into the geometry and dynamics of motion in confined spaces and contribute to a better understanding of a broad spectrum of phenomena ranging from the motion of bacteria to neutron transport. This type of analysis is crucial in situations where inhomogeneity occurs, such as multiple real-world scenarios within a limited domain. [ABSTRACT FROM AUTHOR]

Focused ultrasound has experimentally been found to enhance the diffusion of nanoparticles; our aim with this work is to study this effect closer using both experiments and non-equilibrium molecular dynamics. Measurements from single particle tracking of 40 nm polystyrene nanoparticles in an agarose hydrogel with and without focused ultrasound are presented and compared with a previous experimental study using 100 nm polystyrene nanoparticles. In both cases, we observed an increase in the mean square displacement during focused ultrasound treatment. We developed a coarse-grained non-equilibrium molecular dynamics model with an implicit solvent to investigate the increase in the mean square displacement and its frequency and amplitude dependencies. This model consists of polymer fibers and two sizes of nanoparticles, and the effect of the focused ultrasound was modeled as an external oscillating force field. A comparison between the simulation and experimental results shows similar mean square displacement trends, suggesting that the particle velocity is a significant contributor to the observed ultrasound-enhanced mean square displacement. The resulting diffusion coefficients from the model are compared to the diffusion equation for a two-time continuous time random walk. The model is found to have the same frequency dependency. At lower particle velocity amplitude values, the model has a quadratic relation with the particle velocity amplitude as described by the two-time continuous time random walk derived diffusion equation, but at higher amplitudes, the model deviates, and its diffusion coefficient reaches the non-hindered diffusion coefficient. This observation suggests that at higher ultrasound intensities in hydrogels, the non-hindered diffusion coefficient can be used. [ABSTRACT FROM AUTHOR]

Ion channels are proteins that play a significant role in physiological processes, including neuronal excitability and signal transduction. However, the precise mechanisms by which these proteins facilitate ion diffusion through cell membranes are not well understood. This is because experimental techniques to characterize ion channel activity operate on a time scale too large to understand the role of the various protein conformations on diffusion. Meanwhile, computational approaches operate on a time scale too short to rationalize the observed behavior at the microscopic scale. In this paper, we present a continuous-time random walk model that aims to bridge the scales between the atomistic models of ion channels and the experimental measurement of their conductance. We show how diffusion slows down in complex systems by using 3D lattices that map out the pore geometry of two channels: Nav1.7 and gramicidin. We also introduce spatial and dynamic site disorder to account for system heterogeneity beyond the mean field approximation. Computed diffusion coefficients show that an increase in spatial disorder slows down diffusion kinetics, while dynamic disorder has the opposite effect. Our results imply that microscopic or phenomenological models based on the potential of mean force data overlook the functional importance of protein dynamics on ion diffusion through channels. [ABSTRACT FROM AUTHOR]

We report simulation studies of 33 single intrinsically disordered proteins (IDPs) using coarse-grained bead-spring models where interactions among different amino acids are introduced through a hydropathy matrix and additional screened Coulomb interaction for the charged amino acid beads. Our simulation studies of two different hydropathy scales (HPS1, HPS2) [Dignon et al., PLoS Comput. Biol. 14, e1005941 (2018); Tesei et al. Proc. Natl. Acad. Sci. U. S. A. 118, e2111696118 (2021)] and the comparison with the existing experimental data indicate an optimal interaction parameter ϵ = 0.1 and 0.2 kcal/mol for the HPS1 and HPS2 hydropathy scales. We use these best-fit parameters to investigate both the universal aspects as well as the fine structures of the individual IDPs by introducing additional characteristics. (i) First, we investigate the polymer-specific scaling relations of the IDPs in comparison to the universal scaling relations [Bair et al., J. Chem. Phys. 158, 204902 (2023)] for the homopolymers. By studying the scaled end-to-end distances ⟨ R N 2 ⟩ / (2 L ℓ p ) and the scaled transverse fluctuations l ̃ ⊥ 2 = ⟨ l ⊥ 2 ⟩ / L , we demonstrate that IDPs are broadly characterized with a Flory exponent of ν ≃ 0.56 with the conclusion that conformations of the IDPs interpolate between Gaussian and self-avoiding random walk chains. Then, we introduce (ii) Wilson charge index (W) that captures the essential features of charge interactions and distribution in the sequence space and (iii) a skewness index (S) that captures the finer shape variation of the gyration radii distributions as a function of the net charge per residue and charge asymmetry parameter. Finally, our study of the (iv) variation of ⟨Rg⟩ as a function of salt concentration provides another important metric to bring out finer characteristics of the IDPs, which may carry relevant information for the origin of life. [ABSTRACT FROM AUTHOR]

Quantum walks are the quantum counterpart of classical random walks and have various applications in quantum information science. Polar molecules have rich internal energy structure and long coherence time and thus are considered as a promising candidate for quantum information processing. In this paper, we propose a theoretical scheme for implementing discrete-time quantum walks on a circle with dipole–dipole coupled SrO molecules. The states of the walker and the coin are encoded in the pendular states of polar molecules induced by an external electric field. We design the optimal microwave pulses for implementing quantum walks on a four-node circle and a three-node circle by multi-target optimal control theory. To reduce the accumulation of decoherence and improve the fidelity, we successfully realize a step of quantum walk with only one optimal pulse. Moreover, we also encode the walker into a three-level molecular qutrit and a four-level molecular ququart and design the corresponding optimal pulses for quantum walks, which can reduce the number of molecules used. It is found that all the quantum walks on a circle in our scheme can be achieved via optimal control fields with high fidelities. Our results could shed some light on the implementation of discrete-time quantum walks and high-dimensional quantum information processing with polar molecules. [ABSTRACT FROM AUTHOR]

How dense objects, particles, atoms, and molecules can be packed is intimately related to the properties of the corresponding hosts and macrosystems. We present results from extensive Monte Carlo simulations on maximally compressed packings of linear, freely jointed chains of tangent hard spheres of uniform size in films whose thickness is equal to the monomer diameter. We demonstrate that fully flexible chains of hard spheres can be packed as efficiently as monomeric analogs, within a statistical tolerance of less than 1%. The resulting ordered polymer morphology corresponds to an almost perfect hexagonal triangular (TRI) crystal of the p6m wallpaper group, whose sites are occupied by the chain monomers. The Flory scaling exponent, which corresponds to the maximally dense polymer packing in 2D, has a value of ν = 0.62, which lies between the limits of 0.50 (compact and collapsed state) and 0.75 (self-avoiding random walk). [ABSTRACT FROM AUTHOR]

We present a computationally efficient multiscale method for preparing equilibrated, isotropic long-chain model polymer melts. As an application, we generate Kremer–Grest melts of 1000 chains with 200 entanglements and 25 000–2000 beads/chain, which cover the experimentally relevant bending rigidities up to and beyond the limit of the isotropic–nematic transition. In the first step, we employ Monte Carlo simulations of a lattice model to equilibrate the large-scale chain structure above the tube scale while ensuring a spatially homogeneous density distribution. We then use theoretical insight from a constrained mode tube model to introduce the bead degrees of freedom together with random walk conformational statistics all the way down to the Kuhn scale of the chains. This is followed by a sequence of simulations with carefully parameterized force-capped bead–spring models, which slowly introduce the local bead packing while reproducing the larger-scale chain statistics of the target Kremer–Grest system at all levels of force-capping. Finally, we can switch to the full Kremer–Grest model without perturbing the structure. The resulting chain statistics is in excellent agreement with literature results on all length scales accessible in brute-force simulations of shorter chains. [ABSTRACT FROM AUTHOR]

We develop a multiscale simulation model for diffusion of solutes through porous triblock copolymer membranes. The approach combines two techniques: self-consistent field theory (SCFT) to predict the structure of the self-assembled, solvated membrane and on-lattice kinetic Monte Carlo (kMC) simulations to model diffusion of solutes. Solvation is simulated in SCFT by constraining the glassy membrane matrix while relaxing the brush-like membrane pore coating against the solvent. The kMC simulations capture the resulting solute spatial distribution and concentration-dependent local diffusivity in the polymer-coated pores; we parameterize the latter using particle-based simulations. We apply our approach to simulate solute diffusion through nonequilibrium morphologies of a model triblock copolymer, and we correlate diffusivity with structural descriptors of the morphologies. We also compare the model's predictions to alternative approaches based on simple lattice random walks and find our multiscale model to be more robust and systematic to parameterize. Our multiscale modeling approach is general and can be readily extended in the future to other chemistries, morphologies, and models for the local solute diffusivity and interactions with the membrane. [ABSTRACT FROM AUTHOR]

Under ambient atmospheric conditions, a thin film of water wets many solid surfaces, including insulators, ice, and salt. The film thickness as well as its transport behavior sensitively depend on the surrounding humidity. Understanding this intricate interplay is of the highest relevance for water transport through porous media, particularly in the context of soil salinization induced by evaporation. Here, we use molecular simulations to evaluate the transport properties of thin water films on prototypical salt and soil interfaces, namely NaCl and silica solid surfaces. Our results show two distinct regimes for water transport: at low water coverage, the film permeance scales linearly with the adsorbed amount, in agreement with the activated random walk model. For thicker water films, the permeance scales as the adsorbed amount to the power of 3, in line with the Stokes equation. By comparing results obtained for silica and NaCl surfaces, we find that, at low water coverage, water permeance at the silica surface is considerably lower than at the NaCl surface, which we attribute to difference in hydrogen bonding. We also investigate the effect of atomic surface defects on the transport properties. Finally, in the context of water transport through the porous material, we determine the humidity-dependent crossover between a vapor-dominated and a thin film-dominated transport regimes depending on the pore size. [ABSTRACT FROM AUTHOR]

The chain walking (CW) polymerization technique has the unique property of a movable catalyst synthesizing its own path by creating branch-on-branch structures. By successive attachment of monomers, the resulting architecture ranges from dendritic to linear growth depending on the walking rate, which is defined by the ratio of walking steps and reaction events of the catalyst. The transition regime is characterized by local dendritic sub-structures (dendritic blobs) and a global linear chain feature forming a dendritic bottle-brush. A scaling model for structures obtained by CW catalysis is presented and validated by computer simulation relating the extensions of CW structures to the catalyst's walking ability. The limiting case of linear (low walking rate) and dendritic growth (high walking rate) is recovered, and the latter is shown to bear analogies to the Barabási–Albert graph and Bernoulli growth random walk. We could quantify the size of the dendritic blob as a function of the walking rate by using spectral properties of the connectivity matrix of the simulated macromolecules. This allows us to fit the numerical constants in the scaling approach. We predict that independent of the underlying chemical process, all CW polymerization syntheses involving a highly mobile catalyst ultimately result in bottle-brush structures whose properties depend on a unique parameter: the walking rate. [ABSTRACT FROM AUTHOR]

Pesticides benefit agriculture by increasing crop yield, quality, and security. However, pesticides may inadvertently harm bees, which are valuable as pollinators. Thus, candidate pesticides in development pipelines must be assessed for toxicity to bees. Leveraging a dataset of 382 molecules with toxicity labels from honey bee exposure experiments, we train a support vector machine (SVM) to predict the toxicity of pesticides to honey bees. We compare two representations of the pesticide molecules: (i) a random walk feature vector listing counts of length-L walks on the molecular graph with each vertex- and edge-label sequence and (ii) the Molecular ACCess System (MACCS) structural key fingerprint (FP), a bit vector indicating the presence/absence of a list of pre-defined subgraph patterns in the molecular graph. We explicitly construct the MACCS FPs but rely on the fixed-length-L random walk graph kernel (RWGK) in place of the dot product for the random walk representation. The L-RWGK-SVM achieves an accuracy, precision, recall, and F1 score (mean over 2000 runs) of 0.81, 0.68, 0.71, and 0.69, respectively, on the test data set—with L = 4 being the mode optimal walk length. The MACCS-FP-SVM performs on par/marginally better than the L-RWGK-SVM, lends more interpretability, but varies more in performance. We interpret the MACCS-FP-SVM by illuminating which subgraph patterns in the molecules tend to strongly push them toward the toxic/non-toxic side of the separating hyperplane. [ABSTRACT FROM AUTHOR]

A fundamental way to analyze complex multidimensional stochastic dynamics is to describe it as diffusion on a free energy landscape—free energy as a function of reaction coordinates (RCs). For such a description to be quantitatively accurate, the RC should be chosen in an optimal way. The committor function is a primary example of an optimal RC for the description of equilibrium reaction dynamics between two states. Here, additive eigenvectors (addevs) are considered as optimal RCs to address the limitations of the committor. An addev master equation for a Markov chain is derived. A stationary solution of the equation describes a sub-ensemble of trajectories conditioned on having the same optimal RC for the forward and time-reversed dynamics in the sub-ensemble. A collection of such sub-ensembles of trajectories, called stochastic eigenmodes, can be used to describe/approximate the stochastic dynamics. A non-stationary solution describes the evolution of the probability distribution. However, in contrast to the standard master equation, it provides a time-reversible description of stochastic dynamics. It can be integrated forward and backward in time. The developed framework is illustrated on two model systems—unidirectional random walk and diffusion. [ABSTRACT FROM AUTHOR]

A Brownian bridge is a continuous random walk conditioned to end in a given region by adding an effective drift to guide paths toward the desired region of phase space. This idea has many applications in chemical science where one wants to control the endpoint of a stochastic process—e.g., polymer physics, chemical reaction pathways, heat/mass transfer, and Brownian dynamics simulations. Despite its broad applicability, the biggest limitation of the Brownian bridge technique is that it is often difficult to determine the effective drift as it comes from a solution of a Backward Fokker–Planck (BFP) equation that is infeasible to compute for complex or high-dimensional systems. This paper introduces a fast approximation method to generate a Brownian bridge process without solving the BFP equation explicitly. Specifically, this paper uses the asymptotic properties of the BFP equation to generate an approximate drift and determine ways to correct (i.e., re-weight) any errors incurred from this approximation. Because such a procedure avoids the solution of the BFP equation, we show that it drastically accelerates the generation of conditioned random walks. We also show that this approach offers reasonable improvement compared to other sampling approaches using simple bias potentials. [ABSTRACT FROM AUTHOR]

We propose the replica permutation with solute tempering (RPST) by combining the replica-permutation method (RPM) and the replica exchange with solute tempering (REST). Temperature permutations are performed among more than two replicas in RPM, whereas temperature exchanges are performed between two replicas in the replica-exchange method (REM). The temperature transition in RPM occurs more efficiently than in REM. In REST, only the temperatures of the solute region, the solute temperatures, are exchanged to reduce the number of replicas compared to REM. Therefore, RPST is expected to be an improved method taking advantage of these methods. For comparison, we applied RPST, REST, RPM, and REM to two amyloid-β(16–22) peptides in explicit water. We calculated the transition ratio and the number of tunneling events in the temperature space and the number of dimerization events of amyloid-β(16–22) peptides. The results indicate that, in RPST, the number of replicas necessary for frequent random walks in the temperature and conformational spaces is reduced compared to the other three methods. In addition, we focused on the dimerization process of amyloid-β(16–22) peptides. The RPST simulation with a relatively small number of replicas shows that the two amyloid-β(16–22) peptides form the intermolecular antiparallel β-bridges due to the hydrophilic side-chain contact between Lys and Glu and hydrophobic side-chain contact between Leu, Val, and Phe, which stabilizes the dimer of the peptides. [ABSTRACT FROM AUTHOR]

Finite Markov chains, memoryless random walks on complex networks, appear commonly as models for stochastic dynamics in condensed matter physics, biophysics, ecology, epidemiology, economics, and elsewhere. Here, we review exact numerical methods for the analysis of arbitrary discrete- and continuous-time Markovian networks. We focus on numerically stable methods that are required to treat nearly reducible Markov chains, which exhibit a separation of characteristic timescales and are therefore ill-conditioned. In this metastable regime, dense linear algebra methods are afflicted by propagation of error in the finite precision arithmetic, and the kinetic Monte Carlo algorithm to simulate paths is unfeasibly inefficient. Furthermore, iterative eigendecomposition methods fail to converge without the use of nontrivial and system-specific preconditioning techniques. An alternative approach is provided by state reduction procedures, which do not require additional a priori knowledge of the Markov chain. Macroscopic dynamical quantities, such as moments of the first passage time distribution for a transition to an absorbing state, and microscopic properties, such as the stationary, committor, and visitation probabilities for nodes, can be computed robustly using state reduction algorithms. The related kinetic path sampling algorithm allows for efficient sampling of trajectories on a nearly reducible Markov chain. Thus, all of the information required to determine the kinetically relevant transition mechanisms, and to identify the states that have a dominant effect on the global dynamics, can be computed reliably even for computationally challenging models. Rare events are a ubiquitous feature of realistic dynamical systems, and so the methods described herein are valuable in many practical applications. [ABSTRACT FROM AUTHOR]

Microscopic mechanisms of natural processes are frequently understood in terms of random walk models by analyzing local particle transitions. This is because these models properly account for dynamic processes at the molecular level and provide a clear physical picture. Recent theoretical studies made a surprising discovery that in complex systems, the symmetry of molecular forward/backward transition times with respect to local bias in the dynamics may be broken and it may take longer to go downhill than uphill. The physical origins of these phenomena remain not fully understood. Here, we explore in more detail the microscopic features of the symmetry breaking in the forward/backward transition times by analyzing exactly solvable discrete-state stochastic models. In particular, we consider a specific case of two random walkers on a four-site periodic lattice as the way to represent the general systems with multiple pathways. It is found that the asymmetry in transition times depends on several factors that include the degree of deviation from equilibrium, the particle crowding, and methods of measurements of dynamic properties. Our theoretical analysis suggests that the asymmetry in transition times can be explored experimentally for determining the important microscopic features of natural processes by quantitatively measuring the local deviations from equilibrium and the degrees of crowding. [ABSTRACT FROM AUTHOR]

Computations, which would have been intractable just a few years ago, are now possible on desktop workstations. Such is the case for the application of the Self-Consistent-Phonon (SCP) approximation to large monolayer clusters on structured surfaces, combining a SCP approach to the system dynamics with a random walk approach to finding the optimum positions of the adsorbed atoms. This combination of techniques enables the investigation of the stability, structure, and dynamics of incommensurate adsorbed monolayers at low temperatures. We refer to this approach as the Direct-Space–Self-Consistent-Phonon framework. We present the application of this framework to the study of rare-gas and molecular hydrogen adsorbates on the graphite basal-plane surface and (for xenon) the Pt(111) surface. The largest cluster size consists of 4096 particles, a system that is large enough to examine incommensurate phases without significant adverse boundary effects. The existence of "pseudo-gaps" in the phonon spectrum of nearly commensurate monolayers is demonstrated, and the implication of such "pseudo-gaps" for the determination of the location of any commensurate ↔ incommensurate phase transition is explored. The stability of striped incommensurate structures vs hexagonal incommensurate structures is examined. The inherent difficulties of using this approach for the highly quantum monolayer solids is shown to generate some particular problems. Nevertheless, we demonstrate that this approach to the stability, structure, and dynamics of quantum monolayer solids is a very useful tool in the theorist's arsenal. By implication, this approach should also be useful in the study of adsorption on graphene and carbon nanotubes at low temperatures. [ABSTRACT FROM AUTHOR]

A fractional Fokker–Planck equation based on the continuous time random walk Ansatz is written via the Langevin equations for the dynamics of a dipole interacting with its surroundings, as represented by a cage of dipolar molecules. This equation is solved in the frequency domain using matrix continued fractions, thus yielding the linear dielectric response for extensive ranges of damping, dipole moment ratio, and cage–dipole inertia ratio, and hence the complex susceptibility. The latter comprises a low frequency band with width depending on the anomalous parameter and a far infrared (THz) band with a comb-like structure of peaks. Several physical consequences of the model relevant to anomalous diffusion in the presence of interactions are discussed. The entire calculation may be regarded as an extension of the cage model interpretation of the dynamics of polar molecules to anomalous diffusion, taking into account inertial effects. [ABSTRACT FROM AUTHOR]

We develop two novel transition path sampling (TPS) algorithms for harvesting ensembles of rare event trajectories using non-equilibrium dynamics. These methods have the advantage that no predefined reaction coordinate is needed. Instead, an instantaneous reaction coordinate is based on the current path. Constituting a Monte Carlo random walk in trajectory space, the algorithms can be viewed as bridging between the original TPS methodology and the Rosenbluth based forward flux sampling methodology. We illustrate the new methods on toy models undergoing equilibrium and non-equilibrium dynamics, including an active Brownian particle system. For the latter, we find that transitions between steady states occur via states that are locally ordered but globally disordered. [ABSTRACT FROM AUTHOR]

Coarse-graining of an asymmetric nearest-neighbor discrete-time random walk on a one-dimensional lattice allows one to describe this random walk as biased one-dimensional diffusion. The latter is characterized by two parameters: the drift velocity and diffusivity. There is a general expression giving the drift velocity as a function of the parameters determining the random walk. However, a corresponding expression for the diffusivity is known only for the particular case where the random walk escapes from the lattice site at every time step. In this work, we generalize this result and derive an expression for the diffusivity, assuming that the random walk does not necessarily leave the site, and therefore, its mean lifetime on the site can be longer than the time step. [ABSTRACT FROM AUTHOR]

Conventional molecular dynamics simulations are incapable of sampling many important interactions in biomolecular systems due to their high dimensionality and rough energy landscapes. To observe rare events and calculate transition rates in these systems, enhanced sampling is a necessity. In particular, the study of ligand-protein interactions necessitates a diverse ensemble of protein conformations and transition states, and for many systems, this occurs on prohibitively long time scales. Previous strategies such as WExplore that can be used to determine these types of ensembles are hindered by problems related to the regioning of conformational space. Here, we propose a novel, regionless, enhanced sampling method that is based on the weighted ensemble framework. In this method, a value referred to as "trajectory variation" is optimized after each cycle through cloning and merging operations. This method allows for a more consistent measurement of observables and broader sampling resulting in the efficient exploration of previously unexplored conformations. We demonstrate the performance of this algorithm with the N-dimensional random walk and the unbinding of the trypsin-benzamidine system. The system is analyzed using conformation space networks, the residence time of benzamidine is confirmed, and a new unbinding pathway for the trypsin-benzamidine system is found. We expect that resampling of ensembles by variation optimization will be a useful general tool to broadly explore free energy landscapes. [ABSTRACT FROM AUTHOR]

In this article we present a computational model for the simulation of self-propelled anisotropic bacteria. To this end we use a self-propelled particle model and augment it with a statistical algorithm for the run-and-tumble motion. We derive an equation for the distribution of reorientations of the bacteria that we use to analyze the statistics of the random walk and that allows us to tune the behavior of our model to the characteristics of an E. coli bacterium. We validate our implementation in terms of a single swimmer and demonstrate that our model is capable of reproducing E. coli's run-and-tumble motion with excellent accuracy. [ABSTRACT FROM AUTHOR]

Porous media characterized by a hierarchy of length scales are ubiquitous in industry and nature, and include carbonate rocks, cements, heterogeneous catalysts, and biological cells. Nuclear magnetic resonance (NMR) is a popular tool for studying liquid-saturated porous materials, where the spin relaxation rate is generally considered proportional to pore size. However, in porous granular media, the relaxation rate is modified by diffusion between the intraparticle and interparticle pores. The observed relaxation rates do not reflect the pore size under such conditions. Deconvolving the various contributions of surface relaxation, geometry, and diffusion is nontrivial, and forward models are a powerful technique for elucidating the underlying pore structure. Various forward models have been proposed previously, including analytic solutions and random walk simulations. Here, a finite element method is adopted to simulate the diffusion of nuclear magnetization in a coupled pore geometry. We validate our model against existing solutions and use the simulations to determine the surface relaxivity of powdered silica by matching experimental results. The finite element approach is more versatile than other modeling methods, allowing direct visualization of the diffusing magnetization and being trivially extensible to multidimensional NMR exchange experiments. [ABSTRACT FROM AUTHOR]

In this paper, we show that ensembles of well-structured and unstructured proteins can be distinguished by borrowing concepts from non-equilibrium statistical mechanics. For this purpose, we represent proteins by two different polymer models and interpret the resulting polymer configurations as random walks of a diffusing particle in space. The first model is the trace of the Cα-atoms along the protein main chain, and the second is their projections onto the protein axis. The resulting trajectories are subsequently analyzed using the theory of the generalized Langevin equation. Velocities are replaced by displacements relating consecutive points on the discrete protein axes and equilibrium ensemble averages by averages over appropriate protein structure ensembles. The resulting displacement autocorrelation functions resemble those of the velocity autocorrelation functions of simple liquids and display a minimum, which can be related to the lengths of secondary structure elements. This minimum is clearly more pronounced for well-structured proteins than for unstructured ones, and the corresponding memory function displays a slower decay, indicating a stronger "folding memory." [ABSTRACT FROM AUTHOR]

The first return time (FRT) is the time it takes a random walker to first return to its original site, and the global first passage time (GFPT) is the first passage time for a random walker to move from a randomly selected site to a given site. We find that in finite networks, the variance of FRT, Var(FRT), can be expressed as Var(FRT) = 2⟨FRT⟩⟨GFPT⟩ − ⟨FRT⟩2 − ⟨FRT⟩, where ⟨·⟩ is the mean of the random variable. Therefore a method of calculating the variance of FRT on general finite networks is presented. We then calculate Var(FRT) and analyze the fluctuation of FRT on regular branched networks (i.e., Cayley tree) by using Var(FRT) and its variant as the metric. We find that the results differ from those in such other networks as Sierpinski gaskets, Vicsek fractals, T-graphs, pseudofractal scale-free webs, (u, v ) flowers, and fractal and non-fractal scale-free trees. [ABSTRACT FROM AUTHOR]

Trimolecular reaction models are investigated in the compartment-based (lattice-based) framework for stochastic reaction-diffusion modeling. The formulae for the first collision time and the mean reaction time are derived for the case where three molecules are present in the solution under periodic boundary conditions. For the case of reflecting boundary conditions, similar formulae are obtained using a computer-assisted approach. The accuracy of these formulae is further verified through comparison with numerical results. The presented derivation is based on the first passage time analysis of Montroll [J. Math. Phys. 10, 753 (1969)]. Montroll’s results for two-dimensional lattice-based random walks are adapted and applied to compartment-based models of trimolecular reactions, which are studied in one-dimensional or pseudo one-dimensional domains. [ABSTRACT FROM AUTHOR]

We study the anomalous dynamics of a biased "hungry" (or "greedy") random walk on a percolating cluster. The model mimics chemotaxis in a porous medium: In close resemblance to the 1980s arcade game PAC-MAN®, the hungry random walker consumes food, which is initially distributed in the maze, and biases its movement towards food-filled sites. We observe that the mean-squared displacement of the process follows a power law with an exponent that is different from previously known exponents describing passive or active microswimmer dynamics. The change in dynamics is well described by a dynamical exponent that depends continuously on the propensity to move towards food. It results in slower differential growth when compared to the unbiased random walk. [ABSTRACT FROM AUTHOR]

Ultrafast two-dimensional infrared spectroscopy of a thiocyanate vibrational probe (SCN-) was used to investigate local dynamics in alkylimidazolium bis-[trifluoromethylsulfonyl]imide ionic liquids ([Imn,1][Tf2N], n = 2, 4, 6) at temperatures from 5 to 80 °C. The rate of frequency fluctuations reported by SCN- increases with increasing temperature and decreasing alkyl chain length. Temperature-dependent correlation times scale proportionally to temperature-dependent bulk viscosities of each ionic liquid studied. A multimode Brownian oscillator model demonstrates that very low frequency (<10 cm-1) modes primarily drive the observed spectral diffusion and that these modes broaden and blue shift on average with increasing temperature. An Arrhenius analysis shows activation barriers for local motions around the probe between 5.5 and 6.5 kcal/mol that are very similar to those for translational diffusion of ions. [Im6,1][Tf2N] shows an unexpected decrease in activation energy compared to [Im4,1][Tf2N] that may be related to mesoscopically ordered polar and nonpolar domains. A model of dynamics on a rugged potential energy landscape provides a unifying description of the observed Arrhenius behavior and the Brownian oscillator model of the low frequency modes. [ABSTRACT FROM AUTHOR]

We consider a generalized-ensemble algorithm for coarse-grained simulations of biomolecules which allows the thermodynamic behavior of two or more sequences to be determined in a single multisequence run. By carrying out a random walk in sequence space, the method also enhances conformational sampling. Escape from local energy minima is accelerated by visiting sequences for which the minima are more shallow or absent. We test the method on an intermediate-resolution coarse-grained model for protein folding with 3 amino acid types and explore the potential for a largescale coverage of sequence space by applying the method to sets of more than 1000 sequences. The resulting thermodynamic data are used to analyze the structures and stability properties of sequences covering the space between folds with different secondary structures. [ABSTRACT FROM AUTHOR]

We study the winding angles of random and self-avoiding walks (SAWs) on square and cubic lattices with number of steps N ranging up to 107. We show that the mean square winding angle ⟨ɵ2⟩ of random walks converges to the theoretical form when N → ∞. For self-avoiding walks on the square lattice, we show that the ratio ⟨ɵ4⟩/⟨ɵ2⟩2 converges slowly to the Gaussian value 3. For self-avoiding walks on the cubic lattice, we find that the ratio ⟨ɵ4⟩/⟨ɵ2⟩2 exhibits non-monotonic dependence on N and reaches a maximum of 3.73(1) for N ≈ 104. We show that to a good approximation, the square winding angle of a self-avoiding walk on the cubic lattice can be obtained from the summation of the square change in the winding angles of ln N independent segments of the walk, where the ith segment contains 2i steps.We find that the square winding angle of the ith segment increases approximately as i0.5, which leads to an increase of the total square winding angle proportional to (ln N)1.5. [ABSTRACT FROM AUTHOR]

Diffusive processes are often represented using stochastic random walk frameworks. The amount of time taken for an individual in a random walk to intersect with an absorbing boundary is a fundamental property that is often referred to as the particle lifetime, or the first passage time. The mean lifetime of particles in a random walk model of diffusion is related to the amount of time required for the diffusive process to reach a steady state. Mathematical analysis describing the mean lifetime of particles in a standard model of diffusion without crowding is well known. However, the lifetime of agents in a random walk with crowding has received much less attention. Since many applications of diffusion in biology and biophysics include crowding effects, here we study a discrete model of diffusion that incorporates crowding. Using simulations, we show that crowding has a dramatic effect on agent lifetimes, and we derive an approximate expression for the mean agent lifetime that includes crowding effects. Our expression matches simulation results very well, and highlights the importance of crowding effects that are sometimes overlooked. [ABSTRACT FROM AUTHOR]

The motion of cells and molecules through biological environments is often hindered by the presence of other cells and molecules. A common approach to modeling this kind of hindered transport is to examine the mean squared displacement (MSD) of a motile tracer particle in a lattice-based stochastic random walk in which some lattice sites are occupied by obstacles. Unfortunately, stochastic models can be computationally expensive to analyze because we must average over a large ensemble of identically prepared realizations to obtain meaningful results. To overcome this limitation we describe an exact method for analyzing a lattice-based model of the motion of an agent moving through a crowded environment. Using our approach we calculate the exact MSD of the motile agent. Our analysis confirms the existence of a transition period where, at first, the MSD does not follow a power law with time. However, after a sufficiently long period of time, the MSD increases in proportion to time. This latter phase corresponds to Fickian diffusion with a reduced diffusivity owing to the presence of the obstacles. Our main result is to provide a mathematically motivated, reproducible, and objective estimate of the amount of time required for the transport to become Fickian. Our new method to calculate this crossover time does not rely on stochastic simulations. [ABSTRACT FROM AUTHOR]

Continuous-time random walks (CTRWs) on discrete state spaces, ranging from regular lattices to complex networks, are ubiquitous across physics, chemistry, and biology. Models with coarse-grained states (for example, those employed in studies of molecular kinetics) or spatial disorder can give rise to memory and non-exponential distributions of waiting times and first-passage statistics. However, existing methods for analyzing CTRWs on complex energy landscapes do not address these effects. Here we use statistical mechanics of the nonequilibrium path ensemble to characterize first-passage CTRWs on networks with arbitrary connectivity, energy landscape, and waiting time distributions. Our approach can be applied to calculating higher moments (beyond the mean) of path length, time, and action, as well as statistics of any conservative or non-conservative force along a path. For homogeneous networks, we derive exact relations between length and time moments, quantifying the validity of approximating a continuous-time process with its discrete-time projection. For more general models, we obtain recursion relations, reminiscent of transfer matrix and exact enumeration techniques, to efficiently calculate path statistics numerically. We have implemented our algorithm in PathMAN (Path Matrix Algorithm for Networks), a Python script that users can apply to their model of choice. We demonstrate the algorithm on a few representative examples which underscore the importance of non-exponential distributions, memory, and coarse-graining in CTRWs. [ABSTRACT FROM AUTHOR]

Random walks including non-nearest-neighbor jumps appear in many real situations such as the diffusion of adatoms and have found numerous applications including PageRank search algorithm; however, related theoretical results are much less for this dynamical process. In this paper, we present a study of mixed random walks in a family of fractal scale-free networks, where both nearest-neighbor and next-nearest-neighbor jumps are included. We focus on trapping problem in the network family, which is a particular case of random walks with a perfect trap fixed at the central high-degree node. We derive analytical expressions for the average trapping time (ATT), a quantitative indicator measuring the efficiency of the trapping process, by using two different methods, the results of which are consistent with each other. Furthermore, we analytically determine all the eigenvalues and their multiplicities for the fundamental matrix characterizing the dynamical process. Our results show that although next-nearest-neighbor jumps have no effect on the leading scaling of the trapping efficiency, they can strongly affect the prefactor of ATT, providing insight into better understanding of random-walk process in complex systems. [ABSTRACT FROM AUTHOR]

Subdiffusion in crowded environment such as movement of macromolecule in a living cell has often been observed experimentally. The primary reason for subdiffusion is volume exclusion by the crowder molecules. However, other effects such as hydrodynamic interaction may also play an important role. Although there are a large number of computer simulation studies on understanding molecular crowding, there is a lack of theoretical models that can be connected to both experiment and simulation. In the current work, we have formulated a one-dimensional correlated random walk model by connecting this to the motion in a crowded environment. We have found the exact solution of the probability distribution function of the model by solving it analytically. The parameters of our model can be obtained either from simulation or experiment. It has been shown that this analytical model captures some of the general features of diffusion in crowded environment as given in the previous literature and its prediction for transient subdiffusion closely matches the observations of a previous study of computer simulation of Escherichia coli cytoplasm. It is likely that this model will open up more development of theoretical models in this area. [ABSTRACT FROM AUTHOR]

Automatically generating chemical reaction pathways is a significant computational challenge, particularly in the case where a given chemical system can exhibit multiple reactants and products, as well as multiple pathways connecting these. Here, we outline a computational approach to allow automated sampling of chemical reaction pathways, including sampling of different chemical species at the reaction end-points. The key features of this scheme are (i) introduction of a Hamiltonian which describes a reaction "string" connecting reactant and products, (ii) definition of reactant and product species as chemical connectivity graphs, and (iii) development of a scheme for updating the chemical graphs associated with the reaction end-points. By performing molecular dynamics sampling of the Hamiltonian describing the complete reaction pathway, we are able to sample multiple different paths in configuration space between given chemical products; by periodically modifying the connectivity graphs describing the chemical identities of the end-points we are also able to sample the allowed chemical space of the system. Overall, this scheme therefore provides a route to automated generation of a "roadmap" describing chemical reactivity. This approach is first applied to model dissociation pathways in formaldehyde, H2CO, as described by a parameterised potential energy surface (PES). A second application to the HCo(CO)3 catalyzed hydroformylation of ethene (oxo process), using density functional tight-binding to model the PES, demonstrates that our graph-based approach is capable of sampling the intermediate paths in the commonly accepted catalytic mechanism, as well as several secondary reactions. Further algorithmic improvements are suggested which will pave the way for treating complex multi-step reaction processes in a more efficient manner. [ABSTRACT FROM AUTHOR]

Various structural and dynamical properties of a network are encoded in the eigenvalues of walk matrix describing random walks on the network. In this paper, we study the spectra of walk matrix of the Koch network, which displays the prominent scale-free and small-world features. Utilizing the particular architecture of the network, we obtain all the eigenvalues and their corresponding multiplicities. Based on the link between the eigenvalues of walk matrix and random target access time defined as the expected time for a walker going from an arbitrary node to another one selected randomly according to the steady-state distribution, we then derive an explicit solution to the random target access time for random walks on the Koch network. Finally, we corroborate our computation for the eigenvalues by enumerating spanning trees in the Koch network, using the connection governing eigenvalues and spanning trees, where a spanning tree of a network is a subgraph of the network, that is, a tree containing all the nodes. [ABSTRACT FROM AUTHOR]

Single linear polymer chains in dilute solutions under good solvent conditions are studied by Monte Carlo simulations with the pruned-enriched Rosenbluth method up to the chain length N~O(104). Based on the standard simple cubic lattice model (SCLM) with fixed bond length and the bond fluctuation model (BFM) with bond lengths in a range between 2 and √10, we investigate the conformations of polymer chains described by self-avoiding walks on the simple cubic lattice, and by random walks and non-reversible random walks in the absence of excluded volume interactions. In addition to flexible chains, we also extend our study to semiflexible chains for different stiffness controlled by a bending potential. The persistence lengths of chains extracted from the orientational correlations are estimated for all cases. We show that chains based on the BFM are more flexible than those based on the SCLM for a fixed bending energy. The microscopic differences between these two lattice models are discussed and the theoretical predictions of scaling laws given in the literature are checked and verified. Our simulations clarify that a different mapping ratio between the coarse-grained models and the atomistically realistic description of polymers is required in a coarse-graining approach due to the different crossovers to the asymptotic behavior. [ABSTRACT FROM AUTHOR]

We have used random walk (RW) numerical simulations to investigate the influence of the geometry on the statistics of the electron residence time τr in a trap-limited diffusion process through semiconductor nano-articles. This is an important parameter in coarse-grained modeling of charge carrier transport in nano-structured semiconductor films. The traps have been distributed randomly on the surface (r² model) or through the whole particle (r³ model) with a specified density. The trap energies have been taken from an exponential distribution and the traps release time is assumed to be a stochastic variable. We have carried out (RW) simulations to study the effect of coordination number, the spatial arrangement of the neighbors and the size of nano-particles on the statistics of τr. It has been observed that by increasing the coordination number n, the average value of electron residence time, ...r rapidly decreases to an asymptotic value. For a fixed coordination number n, the electron's mean residence time does not depend on the neighbors' spatial arrangement. In other words, ...r is a porosity-dependence, local parameter which generally varies remarkably from site to site, unless we are dealing with highly ordered structures. We have also examined the effect of nano-particle size d on the statistical behavior of ...r. Our simulations indicate that for volume distribution of traps, ...r scales as d². For a surface distribution of traps ...r increases almost linearly with d. This leads to the prediction of a linear dependence of the diffusion coefficient D on the particle size d in ordered structures or random structures above the critical concentration which is in accordance with experimental observations. [ABSTRACT FROM AUTHOR]

We study the hydrogen diffusion in liquid aluminum alloys through extensive ab initio molecular dynamics simulations. At the microscopic scale, we show that the hydrogen motion is characterized by a broad distribution of spatial jumps that does not correspond to a Brownian motion. To determine the self-diffusion coefficient of hydrogen in liquid aluminum alloys, we use a generalized continuous time random walk model recently developed to describe the hydrogen diffusion in pure aluminum. In particular, we show that the model successfully accounts the effects of alloying elements on the hydrogen diffusion in agreement with experimental features. [ABSTRACT FROM AUTHOR]

We use maximal entropy random walk (MERW) to study the trapping problem in dendrimers modeled by Cayley trees with a deep trap fixed at the central node. We derive an explicit expression for the mean first passage time from any node to the trap, as well as an exact formula for the average trapping time (ATT), which is the average of the source-to-trap mean first passage time over all nontrap starting nodes. Based on the obtained closed-form solution for ATT, we further deduce an upper bound for the leading behavior of ATT, which is the fourth power of ln N, where N is the system size. This upper bound is much smaller than the ATT of trapping depicted by unbiased random walk in Cayley trees, the leading scaling of which is a linear function of N. These results show that MERW can substantially enhance the efficiency of trapping performed in dendrimers. [ABSTRACT FROM AUTHOR]

Efficiently controlling the diffusion process is crucial in the study of diffusion problem in complex systems. In the sense of random walks with a single trap, mean trapping time (MTT) and mean diffusing time (MDT) are good measures of trapping efficiency and diffusion efficiency, respectively. They both vary with the location of the node. In this paper, we analyze the effects of node's location on trapping efficiency and diffusion efficiency of T-fractals measured by MTT and MDT. First, we provide methods to calculate the MTT for any target node and the MDT for any source node of T-fractals. The methods can also be used to calculate the mean first-passage time between any pair of nodes. Then, using the MTT and the MDT as the measure of trapping efficiency and diffusion efficiency, respectively, we compare the trapping efficiency and diffusion efficiency among all nodes of T-fractal and find the best (or worst) trapping sites and the best (or worst) diffusing sites. Our results show that the hub node of T-fractal is the best trapping site, but it is also the worst diffusing site; and that the three boundary nodes are the worst trapping sites, but they are also the best diffusing sites. Comparing the maximum of MTT and MDT with their minimums, we find that the maximum of MTT is almost 6 times of the minimum of MTT and the maximum of MDT is almost equal to the minimum for MDT. Thus, the location of target node has large effect on the trapping efficiency, but the location of source node almost has no effect on diffusion efficiency. We also simulate random walks on T-fractals, whose results are consistent with the derived results. [ABSTRACT FROM AUTHOR]

We investigate the obstructed motion of tracer (test) particles in crowded environments by carrying simulations of two-dimensional Gaussian random walk in model fibrinogen monolayers of different orientational ordering. The fibrinogen molecules are significantly anisotropic and therefore they can form structures where orientational ordering, similar to the one observed in nematic liquid crystals, appears. The work focuses on the dependence between level of the orientational order (degree of environmental crowding) of fibrinogen molecules inside a layer and non-Fickian character of the diffusion process of spherical tracer particles moving within the domain. It is shown that in general particles motion is subdiffusive and strongly anisotropic, and its characteristic features significantly change with the orientational order parameter, concentration of fibrinogens, and radius of a diffusing probe. [ABSTRACT FROM AUTHOR]

We study the conformational properties of heteropolymers containing two types of monomers A and B, modeled as self-attracting self-avoiding random walks on a regular lattice. Such a model can describe in particular the sequences of hydrophobic and hydrophilic residues in proteins [K. F. Lau and K. A. Dill, Macromolecules 22, 3986 (1989)] and polyampholytes with oppositely charged groups [Y. Kantor and M. Kardar, Europhys. Lett. 28, 169 (1994)]. Treating the sequences of the two types of monomers as quenched random variables, we provide a systematic analysis of possible generalizations of this model. To this end we apply the pruned-enriched Rosenbluth chain-growth algorithm, which allows us to obtain the phase diagrams of extended and compact states coexistence as function of both the temperature and fraction of A and B monomers along the heteropolymer chain. [ABSTRACT FROM AUTHOR]

Designing optimal structure favorable to diffusion and effectively controlling the trapping process are crucial in the study of trapping problem--random walks with a single trap. In this paper, we study the trapping problem occurring on unweighted and weighted networks, respectively. The networks under consideration display the striking scale-free, small-world, and modular properties, as observed in diverse real-world systems. For binary networks, we concentrate on three cases of trapping problems with the trap located at a peripheral node, a neighbor of the root with the least connectivity, and a farthest node, respectively. For weighted networks with edge weights controlled by a parameter, we also study three trapping problems, in which the trap is placed separately at the root, a neighbor of the root with the least degree, and a farthest node. For all the trapping problems, we obtain the analytical formulas for the average trapping time (ATT) measuring the efficiency of the trapping process, as well as the leading scaling of ATT. We show that for all the trapping problems in the binary networks with a trap located at different nodes, the dominating scalings of ATT reach the possible minimum scalings, implying that the networks have optimal structure that is advantageous to efficient trapping. Furthermore, we show that for trapping in the weighted networks, the ATT is controlled by the weight parameter, through modifying which, the ATT can behave superlinearly, linearly, sublinearly, or logarithmically with the system size. This work could help improving the design of systems with efficient trapping process and offers new insight into control of trapping in complex systems. [ABSTRACT FROM AUTHOR]

An integro-differential diffusion equation with linear force, based on the continuous time random walk model, is considered. The equation generalizes the ordinary and fractional diffusion equations. Analytical expressions for transition probability density, mean square displacement, and intermediate scattering function are presented. The mean square displacement and intermediate scattering function can fit well the simulation data of the temperature-dependent translational dynamics of nitrogen atoms of elastin for a wide range of temperatures and various scattering vectors. Moreover, the numerical results are also compared with those of a fractional diffusion equation. [ABSTRACT FROM AUTHOR]

The concept of continuous-time random walks (CTRW) is a generalization of ordinary random walk models, and it is a powerful tool for investigating a broad spectrum of phenomena in natural, engineering, social, and economic sciences. Recently, several theoretical approaches have been developed that allowed to analyze explicitly dynamics of CTRW at all times, which is critically important for understanding mechanisms of underlying phenomena. However, theoretical analysis has been done mostly for systems with a simple geometry. Here we extend the original method based on generalized master equations to analyze all-time dynamics of CTRW models on complex networks. Specific calculations are performed for models on lattices with branches and for models on coupled parallel-chain lattices. Exact expressions for velocities and dispersions are obtained. Generalized fluctuations theorems for CTRW models on complex networks are discussed. [ABSTRACT FROM AUTHOR]

A generalized Klein-Kramers equation for a particle interacting with an external field is proposed. The equation generalizes the fractional Klein-Kramers equation introduced by Barkai and Silbey [J. Phys. Chem. B 104, 3866 (2000)]. Besides, the generalized Klein-Kramers equation can also recover the integro-differential Klein-Kramers equation for continuous-time random walk; this means that it can describe the subdiffusive and superdiffusive regimes in the long-time limit. Moreover, analytic solutions for first two moments both in velocity and displacement (for force-free case) are obtained, and their dynamic behaviors are investigated. [ABSTRACT FROM AUTHOR]

We provide an explicit formula for the global mean first-passage time (GMFPT) for random walks in a general graph with a perfect trap fixed at an arbitrary node, where GMFPT is the average of mean first-passage time to the trap over all starting nodes in the whole graph. The formula is expressed in terms of eigenvalues and eigenvectors of Laplacian matrix for the graph. We then use the formula to deduce a tight lower bound for the GMFPT in terms of only the numbers of nodes and edges, as well as the degree of the trap, which can be achieved in both complete graphs and star graphs. We show that for a large sparse graph, the leading scaling for this lower bound is proportional to the system size and the reciprocal of the degree for the trap node. Particularly, we demonstrate that for a scale-free graph of size N with a degree distribution P(d) ∼ d-γ characterized by γ, when the trap is placed on a most connected node, the dominating scaling of the lower bound becomes N1-1/γ, which can be reached in some scale-free graphs. Finally, we prove that the leading behavior of upper bounds for GMFPT on any graph is at most N3 that can be reached in the bar-bell graphs. This work provides a comprehensive understanding of previous results about trapping in various special graphs with a trap located at a specific location. [ABSTRACT FROM AUTHOR]