Discrete Kinetic Models Research Articles

Towards a consistent lattice Boltzmann model for two-phase fluids

We revisit the construction of discrete kinetic models for single-component isothermal two-phase flows. Starting from a kinetic model for a non-ideal fluid, we show that, under conventional scaling, the Navier–Stokes equations with a non-ideal equation of state are recovered in the hydrodynamic limit. A scaling based on the smallness of velocity increments is then introduced, which recovers the full Navier–Stokes–Korteweg equations. The proposed model is realized on a standard lattice and validated on a variety of benchmarks. Through a detailed study of thermodynamic properties including co-existence densities, surface tension, Tolman length and sound speed, we show thermodynamic consistency, well-posedness and convergence of the proposed model. Furthermore, hydrodynamic consistency is demonstrated by verification of Galilean invariance of the dissipation rate of shear and normal modes and the study of visco-capillary coupling effects. Finally, the model is validated on dynamic test cases in three dimensions with complex geometries and large density ratios such as drop impact on textured surfaces and mercury drops coalescence.

Discrete lumping kinetic models for hydrodesulfuration and hydrocracking of a mixture of FCC feedstock and light gasoil

Discrete lumping kinetic models for hydrodesulfuration and hydrocracking of a mixture of FCC feedstock and light gasoil

Effective Finite-Difference Techniques for Estimating Sensitivities for Stochastic Biochemical Systems

Cellular environments are in essence stochastic, owing to the random character of the biochemical reaction events in a single cell. Stochastic fluctuations may substantially contribute to the dynamics of systems with small copy numbers of some biochemical species. Then, stochastic models are indispensable for properly portraying the behaviour of the system. Sensitivity analysis is one of the central tools for studying stochastic models of cellular dynamics. Here, we propose some finite-difference strategies for estimating parametric sensitivities of higher-order moments of the system state for stochastic discrete biochemical kinetic models. To reduce the variance of the sensitivity estimator, we employ various coupling techniques. The advantages of the proposed methods are illustrated in several models of biochemical systems of practical relevance.

Protein Folding Dynamics as Diffusion on a Free Energy Surface: Rate Equation Terms, Transition Paths, and Analysis of Single-Molecule Photon Trajectories.

The rates of protein (un)folding are often described as diffusion on the projection of a hyperdimensional energy landscape onto a few (ideally one) order parameters. Testing such an approximation by experiment requires resolving the reactive transition paths of individual molecules, which is now becoming feasible with advanced single-molecule spectroscopic techniques. This has also sparked the interest of theorists in better understanding reactive transition paths. Here we focus on these issues aiming to establish (i) practical guidelines for the mechanistic interpretation of transition path times (TPT) and (ii) methods to extract the free energy surface and protein dynamics from the maximum likelihood analysis of photon trajectories (MLA-PT). We represent the (un)folding rates as diffusion on a 1D free energy surface with the FRET efficiency as a reaction coordinate proxy. We then perform diffusive kinetic simulations on surfaces with two minima and a barrier, but with different shapes (curvatures, barrier height, and symmetry), coupled to stochastic simulations of photon emissions that reproduce current SM-FRET experiments. From the analysis of transition paths, we find that the TPT is inversely proportional to the barrier height (difference in free energy between minimum and barrier top) for any given surface shape, and that dividing the TPT into climb and descent segments provides key information about the barrier's symmetry. We also find that the original MLA-PT procedure used to determine the TPT from experiments underestimates its value, particularly for the cases with smaller barriers (e.g., fast folders), and we suggest a simple strategy to correct for this bias. Importantly, we also demonstrate that photon trajectories contain enough information to extract the 1D free energy surface's shape and dynamics (if TPT is >4-5-fold longer than the interphoton time) using the MLA-PT directly implemented with a diffusive free energy surface model. When dealing with real (unknown) experimental data, the comparison between the likelihoods of the free energy surface and discrete kinetic three-state models can be used to evaluate the statistical significance of the estimated free energy surface.

Mathematical modeling of the building materials production processes

We propose a new mathematical model of the process of production of building materials based on the reaction of autocatalysis. Discrete kinetic models can be used to mathematically describe this process with a different number of reagents. In the case of two reagents, the Carleman discrete kinetic system can be used to describe the autocatalysis reaction. The mathematical model proposed in this paper takes into account the terms of a higher order of smallness for a small parameter. The account of these terms allows to describe more correctly process. The numerical research of the proposed mathematical model is carried out. Numerical results are presented. The analysis and comparison of the obtained numerical results with the results of the numerical study of the kinetic system of the Carleman equations is carried out.

The influence of contrarians in the dynamics of opinion formation

In this work we consider the presence of contrarian agents in discrete 3-state kinetic exchange opinion models. The contrarians are individuals that adopt the choice opposite to the prevailing choice of their contacts, whatever this choice is. We consider binary as well as three-agent interactions, with stochastic parameters, in a fully-connected population. Our numerical results suggest that the presence of contrarians destroys the absorbing state of the original model, changing the transition to the para–ferromagnetic type. In this case, the consequence for the society is that the three opinions coexist in the population, in both phases (ordered and disordered). Furthermore, the order–disorder transition is suppressed for a sufficient large fraction of contrarians. In some cases the transition is discontinuous, and it changes to continuous before it is suppressed. Some of our results are complemented by analytical calculations based on the master equation.

Fluctuation theorems for discrete kinetic models of molecular motors

Motivated by discrete kinetic models for non-cooperative molecular motors on periodic tracks, we consider random walks (also not Markov) on quasi one dimensional (1d) lattices, obtained by gluing several copies of a fundamental graph in a linear fashion. We show that, for a suitable class of quasi-1d lattices, the large deviation rate function associated to the position of the walker satisfies a Gallavotti–Cohen symmetry for any choice of the dynamical parameters defining the stochastic walk. This class includes the linear model considered in Lacoste et al (2008 Phys. Rev. E 78 011915). We also derive fluctuation theorems for the time-integrated cycle currents and discuss how the matrix approach of Lacoste et al (2008 Phys. Rev. E 78 011915) can be extended to derive the above Gallavotti–Cohen symmetry for any Markov random walk on with periodic jump rates. Finally, we review in the present context some large deviation results of Faggionato and Silvestri (2017 Ann. Inst. Henri Poincaré 53 46–78) and give some specific examples with explicit computations.

Boundary layers for discrete kinetic models: Multicomponent mixtures, polyatomic molecules, bimolecular reactions, and quantum kinetic equations

We consider some extensions of the classical discrete Boltzmann equation to the cases of multicomponent mixtures, polyatomic molecules (with a finite number of different internal energies), and chemical reactions, but also general discrete quantum kinetic Boltzmann-like equations; discrete versions of the Nordheim-Boltzmann (or Uehling-Uhlenbeck) equation for bosons and fermions and a kinetic equation for excitations in a Bose gas interacting with a Bose-Einstein condensate. In each case we have an H-theorem and so for the planar stationary half-space problem, we have convergence to an equilibrium distribution at infinity (or at least a manifold of equilibrium distributions). In particular, we consider the nonlinear half-space problem of condensation and evaporation for these discrete Boltzmann-like equations. We assume that the flow tends to a stationary point at infinity and that the outgoing flow is known at the wall, maybe also partly linearly depending on the incoming flow. We find that the systems we obtain are of similar structures as for the classical discrete Boltzmann equation (for single species), and that previously obtained results for the discrete Boltzmann equation can be applied after being generalized. Then the number of conditions on the assigned data at the wall needed for existence of a unique solution is found. The number of parameters to be specified in the boundary conditions depends on if we have subsonic or supersonic condensation or evaporation. All our results are valid for any finite number of velocities.

Discrete Kinetic Models for Molecular Motors: Asymptotic Velocity and Gaussian Fluctuations

We consider random walks on quasi one dimensional lattices, as introduced in Faggionato and Silvestri (Random Walks on Quasi One Dimensional Lattices: Large Deviations and Fluctuation Theorems, 2014). This mathematical setting covers a large class of discrete kinetic models for non-cooperative molecular motors on periodic tracks. We derive general formulas for the asymptotic velocity and diffusion coefficient, and we show how to reduce their computation to suitable linear systems of the same degree of a single fundamental cell, with possible linear chain removals. We apply the above results to special families of kinetic models, also catching some errors in the biophysics literature.

A well-balanced and asymptotic-preserving scheme for the one-dimensional linear Dirac equation

The numerical approximation of one-dimensional relativistic Dirac wave equations is considered within the recent framework consisting in deriving local scattering matrices at each interface of the uniform Cartesian computational grid. For a Courant number equal to unity, it is rigorously shown that such a discretization preserves exactly the $$L^2$$ norm despite being explicit in time. This construction is well-suited for particles for which the reference velocity is of the order of $$c$$ , the speed of light. Moreover, when $$c$$ diverges, that is to say, for slow particles (the characteristic scale of the motion is non-relativistic), Dirac equations are naturally written so as to let a “diffusive limit” emerge numerically, like for discrete 2-velocity kinetic models. It is shown that an asymptotic-preserving scheme can be deduced from the aforementioned well-balanced one, with the following properties: it yields unconditionally a classical Schrodinger equation for free particles, but it handles the more intricate case with an external potential only conditionally (the grid should be such that $$c \Delta x\rightarrow 0$$ ). Such a stringent restriction on the computational grid can be circumvented easily in order to derive a seemingly original Schrodinger scheme still containing tiny relativistic features. Numerical tests (on both linear and nonlinear equations) are displayed.

Fundamental diagrams for kinetic equations of traffic flow

In this paper we investigate the ability of some recently introduced discrete kinetic models of vehicular traffic to catch, in their large time behavior, typical features of theoretical fundamental diagrams. Specifically, we address the so-called "spatially homogeneous problem" and, in the representative case of an exploratory model, we study the qualitative properties of its solutions for a generic number of discrete microstates. This includes, in particular, asymptotic trends and equilibria, whence fundamental diagrams originate.

Discrete kinetic models in the fluid dynamic limit

We investigate discrete kinetic models in the fluid dynamic limit described by the Euler system and the Navier–Stokes correction obtained by the Chapman–Enskog procedure. We show why reliable “small” systems can be expected only for small Mach numbers and derive a calculus for designing models for given Prandtl numbers.

Correction: Discrete Kinetic Models from Funneled Energy Landscape Simulations

There were errors in the author affiliations. The correct versions of the affiliations are available below: Nicholas P. Schafer2,5☯, Ryan M. B. Hoffman1,2☯, Anat Burger2,3, Patricio O. Craig1,2, Elizabeth A. Komives1, Peter G. Wolynes2,4* 1 Chemistry/Biochemistry, University of California San Diego, La Jolla, California, United States of America, 2 Center for Theoretical Biological Physics, University of California, San Diego and Rice University, La Jolla, CA and Houston, TX, USA 3 Physics, University of California San Diego, La Jolla, California, United States of America, 4 Chemistry, Rice University, Houston, Texas, United States of America, 5 Physics and Astronomy, Rice University, Houston, Texas, United States of America In numerous occasions in the References section, the author name Wolnes P should have been Wolynes PG.

Discrete Kinetic Models from Funneled Energy Landscape Simulations

A general method for facilitating the interpretation of computer simulations of protein folding with minimally frustrated energy landscapes is detailed and applied to a designed ankyrin repeat protein (4ANK). In the method, groups of residues are assigned to foldons and these foldons are used to map the conformational space of the protein onto a set of discrete macrobasins. The free energies of the individual macrobasins are then calculated, informing practical kinetic analysis. Two simple assumptions about the universality of the rate for downhill transitions between macrobasins and the natural local connectivity between macrobasins lead to a scheme for predicting overall folding and unfolding rates, generating chevron plots under varying thermodynamic conditions, and inferring dominant kinetic folding pathways. To illustrate the approach, free energies of macrobasins were calculated from biased simulations of a non-additive structure-based model using two structurally motivated foldon definitions at the full and half ankyrin repeat resolutions. The calculated chevrons have features consistent with those measured in stopped flow chemical denaturation experiments. The dominant inferred folding pathway has an “inside-out”, nucleation-propagation like character.

Multiscale modeling of macromolecular biosystems

In this article, we review the recent progress in multiresolution modeling of structure and dynamics of protein, RNA and their complexes. Many approaches using both physics-based and knowledge-based potentials have been developed at multiple granularities to model both protein and RNA. Coarse graining can be achieved not only in the length, but also in the time domain using discrete time and discrete state kinetic network models. Models with different resolutions can be combined either in a sequential or parallel fashion. Similarly, the modeling of assemblies is also often achieved using multiple granularities. The progress shows that a multiresolution approach has considerable potential to continue extending the length and time scales of macromolecular modeling.

Construction of Normal Discrete Velocity Models of the Boltzmann Equation

Discretization methods have been developed on the idea of replacing the original Boltzmann equation (BE) by a finite set of nonlinear hyperbolic PDEs corresponding to the densities linked to a suitable finite set of velocities. One open problem related to the discrete BE is the construction of normal (fulfilling only physical conservation laws) discrete velocity models (DVMs). In many papers on DVMs, authors postulate from the beginning that a finite velocity space with such properties is given and after that study the discrete BE. Our aim is not to study the equations for DVMs, but to discuss all possible choices of finite sets of velocities satisfying this type of restrictions. Using our previous results, i.e. the general algorithm for the construction of normal discrete kinetic models (DKMs), we develop and implement an algorithm for the particular case of DVMs of the BE and give a complete classification for models with small number n of velocities (n ≤ 10).

From generalized kinetic theory to discrete velocity modeling of vehicular traffic. A stochastic game approach

This work reports on vehicular traffic modeling by methods of the discrete kinetic theory. The purpose is to detail a reference mathematical framework for some discrete velocity kinetic models recently introduced in the literature, which proved capable of reproducing interesting traffic phenomena without using experimental information as modeling assumptions. To this end, we firstly derive a general discrete velocity kinetic framework with binary nonlocal interactions. Then, resorting to some ideas of stochastic game theory, we outline specific modeling guidelines for vehicular traffic, and finally we discuss the derivation of the above-mentioned vehicular traffic models from these mathematical structures.

Construction of Discrete Kinetic Models with Given Invariants

We consider the general problem of the construction of discrete kinetic models (DKMs) with given conservation laws. This problem was first stated by Gatignol in connection with discrete models of the Boltzmann equation (BE) and it has been addressed in the last decade by several authors. Even though a practical criterion for the non-existence of spurious conservation laws has been devised, and a method for enlarging existing physical models by new velocity points without adding non-physical invariants has been proposed, a general algorithm for the construction of all normal (physical) discrete models with assigned conservation laws, in any dimension and for any number of points, is still lacking in the literature. We introduce the most general class of discrete kinetic models and obtain a general method for the construction and classification of normal DKMs. In particular, it is proved that for any given dimension d≥2 and for any sufficiently large number N of velocities (for example, N≥6 for the planar case d=2) there exists just a finite number of distinct classes of DKMs. We apply the general method in the particular cases of discrete velocity models (DVMs) of the inelastic BE and elastic BE. Using our general approach to DKMs and our results on normal DVMs for a single gas, we develop a method for the construction of the most natural (from physical point of view) subclass of normal DVMs for binary gas mixtures. We call such models supernormal models (SNMs) (they have the property that by isolating the velocities of single gases involved in the mixture, we also obtain normal DVMs).

Molecular Motors: A Theorist's Perspective

Individual molecular motors, or motor proteins, are enzymatic molecules that convert chemical energy, typically obtained from the hydrolysis of ATP (adenosine triphosphate), into mechanical work and motion. Processive motor proteins, such as kinesin, dynein, and certain myosins, step unidirectionally along linear tracks, specifically microtubules and actin filaments, and play a crucial role in cellular transport processes, organization, and function. In this review some theoretical aspects of motor-protein dynamics are presented in the light of current experimental methods that enable the measurement of the biochemical and biomechanical properties on a single-molecule basis. After a brief discussion of continuum ratchet concepts, we focus on discrete kinetic and stochastic models that yield predictions for the mean velocity, V(F, [ATP], ...), and other observables as a function of an imposed load force F, the ATP concentration, and other variables. The combination of appropriate theory with single-molecule observations should help uncover the mechanisms underlying motor-protein function.

Exact solutions for discrete velocity models

A new class of exact solutions for discrete kinetic models is presented. It is shown that these solutions can be used to solve both initial and boundary value problems of rarefied gas dynamics.