Assumption Of Molecular Chaos Research Articles

Wave turbulence and collective behavior models for wave equations with short- and long-range interactions

In this work, we discuss a situation which could lead to both wave turbulence and collective behavior kinetic equations. The wave turbulence kinetic models appear in the kinetic limit when the wave equations have local differential operators. Viewing wave equations on the lattice as chains of anharmonic oscillators and replacing the local differential operators (short-range interactions) by non-local ones (long-range interactions), we arrive at a new Vlasov-type kinetic model in the mean field limit under the molecular chaos assumption reminiscent of models for collective behavior in which anharmonic oscillators replace individual particles.

Reticulate collisional structure in boundary-driven granular gases.

We report a peculiar head-on collision network between two vibrating boundaries in experiments performed during a parabolic flight and in a laboratory using horizontal vibration. This structure is a new ordering, which is due to an orientation correlation between the relative position and velocity of any particle pair. It weakens the collision frequency and produces a long-range boundary effect. Moreover, we find the molecular chaos assumption is violated in a larger portion of the phase space. Using an anisotropic distribution model, we modify angular integration results and compare them to the results of the kinetic theory.

Transport coefficients of self-propelled particles: Reverse perturbations and transverse current correlations.

The reverse perturbation method [Phys. Rev. E 59, 4894 (1999)1063-651X10.1103/PhysRevE.59.4894] for shearing simple liquids and measuring their viscosity is extended to the Vicsek model (VM) of active particles [Phys. Rev. Lett. 75, 1226 (1995)PRLTAO0031-900710.1103/PhysRevLett.75.1226] and its metric-free version. The sheared systems exhibit a phenomenon that is similar to the skin effect of an alternating electric current: Momentum that is fed into the boundaries of a layer decays mostly exponentially toward the center of the layer. It is shown how two transport coefficients, i.e., the shear viscosity ν and the momentum amplification coefficient λ, can be obtained by fitting this decay with an analytical solution of the hydrodynamic equations for the VM. The viscosity of the VM consists of two parts, a kinetic and a collisional contribution. While analytical predictions already exist for the former, a novel expression for the collisional part is derived by an Enskog-like kinetic theory. To verify the predictions for the transport coefficients, Green-Kubo relations were evaluated and transverse current correlations were measured in independent simulations. Not too far to the transition to collective motion, we find excellent agreement between the different measurements of the transport coefficients. However, the measured values of ν and 1-λ are always slightly higher than the mean-field predictions, even at large mean free paths and at state points quite far from the threshold to collective motion, that is, far in the disordered phase. These findings seem to indicate that the mean-field assumption of molecular chaos is much less reliable in systems with velocity-alignment rules such as the VM, compared to models obeying detailed balance such as multiparticle collision dynamics.

Kinetic theory for financial Brownian motion from microscopic dynamics

Recent technological development has enabled researchers to study social phenomena scientifically in detail and financial markets has particularly attracted physicists since the Brownian motion has played the key role as in physics. In our previous report (arXiv:1703.06739; to appear in Phys. Rev. Lett.), we have presented a microscopic model of trend-following high-frequency traders (HFTs) and its theoretical relation to the dynamics of financial Brownian motion, directly supported by a data analysis of tracking trajectories of individual HFTs in a financial market. Here we show the mathematical foundation for the HFT model paralleling to the traditional kinetic theory in statistical physics. We first derive the time-evolution equation for the phase-space distribution for the HFT model exactly, which corresponds to the Liouville equation in conventional analytical mechanics. By a systematic reduction of the Liouville equation for the HFT model, the Bogoliubov-Born-Green-Kirkwood-Yvon hierarchal equations are derived for financial Brownian motion. We then derive the Boltzmann-like and Langevin-like equations for the order-book and the price dynamics by making the assumption of molecular chaos. The qualitative behavior of the model is asymptotically studied by solving the Boltzmann-like and Langevin-like equations for the large number of HFTs, which is numerically validated through the Monte-Carlo simulation. Our kinetic description highlights the parallel mathematical structure between the financial Brownian motion and the physical Brownian motion.

Turbulence through the Spyglass of Bilocal Kinetics.

In two recent papers we introduced a generalization of Boltzmann’s assumption of molecular chaos based on a criterion of maximum entropy, which allowed setting up a bilocal version of Boltzmann’s kinetic equation. The present paper aims to investigate how the essentially non-local character of turbulent flows can be addressed through this bilocal kinetic description, instead of the more standard approach through the local Euler/Navier–Stokes equation. Balance equations appropriate to this kinetic scheme are derived and closed so as to provide bilocal hydrodynamical equations at the non-viscous order. These equations essentially consist of two copies of the usual local equations, but coupled through a bilocal pressure tensor. Interestingly, our formalism automatically produces a closed transport equation for this coupling term.

Maxwell-Boltzmann Distribution in Solids

The velocity distribution functions of particles in one- and three-dimensional harmonic solids are investigated through molecular dynamics simulations. It is shown that, as in the case of dense fluids, these distribution functions still obey the Maxwell-Boltzmann law and the assumption of molecular chaos remains valid even at low temperatures.

Chapman–Enskog expansion for the Vicsek model of self-propelled particles

Using the standard Vicsek model, I show how the macroscopic transport equations can be systematically derived from microscopic collision rules. The approach starts with the exact evolution equation for the N-particle probability distribution and, after making the mean-field assumption of molecular chaos, leads to a multi-particle Enskog-type equation. This equation is treated by a non-standard Chapman–Enskog expansion to extract the macroscopic behavior. The expansion includes terms up to third order in a formal expansion parameter ϵ, and involves a fast time scale. A self-consistent closure of the moment equations is presented that leads to a continuity equation for the particle density and a Navier–Stokes-like equation for the momentum density. Expressions for all transport coefficients in these macroscopic equations are given explicitly in terms of microscopic parameters of the model. The transport coefficients depend on specific angular integrals which are evaluated asymptotically in the limit of infinitely many collision partners, using an analogy to a random walk. The consistency of the Chapman–Enskog approach is checked by an independent calculation of the shear viscosity using a Green–Kubo relation.

Exchange fluctuation theorem for correlated quantum systems.

We extend the exchange fluctuation theorem for energy exchange between thermal quantum systems beyond the assumption of molecular chaos, and describe the nonequilibrium exchange dynamics of correlated quantum states. The relation quantifies how the tendency for systems to equilibrate is modified in high-correlation environments. In addition, a more abstract approach leads us to a "correlation fluctuation theorem". Our results elucidate the role of measurement disturbance for such scenarios. We show a simple application by finding a semiclassical maximum work theorem in the presence of correlations. We also present a toy example of qubit-qudit heat exchange, and find that non-classical behaviour such as deterministic energy transfer and anomalous heat flow are reflected in our exchange fluctuation theorem.

Plane shear flows of frictionless spheres: Kinetic theory and 3D soft-sphere discrete element method simulations

We use existing 3D Discrete Element simulations of simple shear flows of spheres to evaluate the radial distribution function at contact that enables kinetic theory to correctly predict the pressure and the shear stress, for different values of the collisional coefficient of restitution. Then, we perform 3D Discrete Element simulations of plane flows of frictionless, inelastic spheres, sheared between walls made bumpy by gluing particles in a regular array, at fixed average volume fraction and distance between the walls. The results of the numerical simulations are used to derive boundary conditions appropriated in the cases of large and small bumpiness. Those boundary conditions are, then, employed to numerically integrate the differential equations of Extended Kinetic Theory, where the breaking of the molecular chaos assumption at volume fraction larger than 0.49 is taken into account in the expression of the dissipation rate. We show that the Extended Kinetic Theory is in very good agreement with the numerical simulations, even for coefficients of restitution as low as 0.50. When the bumpiness is increased, we observe that some of the flowing particles are stuck in the gaps between the wall spheres. As a consequence, the walls are more dissipative than expected, and the flows resemble simple shear flows, i.e., flows of rather constant volume fraction and granular temperature.

Boltzmann's H-theorem and the assumption of molecular chaos

We describe a simple dynamical model of a one-dimensional ideal gas and use computer simulations of the model to illustrate two fundamental results of kinetic theory: the Boltzmann transport equation and the Boltzmann H-theorem. Although the model is time-reversal invariant, both results predict that the behaviour of the gas is time-asymmetric. We show that the assumption of molecular chaos is a necessary condition, but not a sufficient condition, for such time-asymmetric results to correctly describe the model, and we use computer simulations to investigate the conditions under which the assumption of molecular chaos holds.

Convergence of a Stochastic Particle Approximation for Measure Solutions of the 2D Keller-Segel System

The analysis of a stochastic interacting particle scheme for the approximation of measure solutions of the parabolic-elliptic Keller–Segel system in 2D is continued. In previous work it has been shown that solutions of a regularized scheme converge to solutions of the regularized Keller–Segel system, when the number of particles tends to infinity. In the present work, the regularization is eliminated in the particle model, which requires an application of the framework of time dependent measures with diagonal defects, developed by Poupaud. The subsequent many particle limit of the BBGKY hierarchy can be solved using measure solutions of the Keller–Segel system and the molecular chaos assumption. However, a uniqueness result for the limiting hierarchy and therefore a proof of propagation of chaos is missing. Finally, the dynamics of strong measure solutions, i.e., sums of smooth distributions and Delta measures, of the particle model is discussed formally for the cases of 2 and 3 particles. The blow-up behavior for more than 2 particles is not completely understood.

Loss of Temporal Homogeneity and Symmetry in Statistical Systems: Deterministic Versus Stochastic Dynamics

A detailed analysis of deterministic (one-to-one) and stochastic (one-to-many) dynamics establishes that dS/dt > 0 is only consistent with the latter, which contains violation of temporal symmetry and homogeneity. We observe that the former only supports dS/dt = 0 and cannot give rise to Boltzmann’s molecular chaos assumption. The ensemble average is more meaningful than the temporal average, especially in non-equilibrium statistical mechanics of systems confined to disjoint phase space components, which commonly occurs at low temperatures. We propose that the stochasticity arises from extra degrees of freedom, which are not part of the system. We provide a simple resolution of the recurrence and irreversibility paradoxes.

Structure and dynamics of two-dimensional sheared granular flows

The structure and dynamics of the two-dimensional linear shear flow of inelastic disks at high area fractions are analyzed. The event-driven simulation technique is used in the hard-particle limit, where the particles interact through instantaneous collisions. The structure (relative arrangement of particles) is analyzed using the bond-orientational order parameter. It is found that the shear flow reduces the order in the system, and the order parameter in a shear flow is lower than that in a collection of elastic hard disks at equilibrium. The distribution of relative velocities between colliding particles is analyzed. The relative velocity distribution undergoes a transition from a Gaussian distribution for nearly elastic particles, to an exponential distribution at low coefficients of restitution. However, the single-particle distribution function is close to a Gaussian in the dense limit, indicating that correlations between colliding particles have a strong influence on the relative velocity distribution. This results in a much lower dissipation rate than that predicted using the molecular chaos assumption, where the velocities of colliding particles are considered to be uncorrelated.

Stochastic theory of direct simulation Monte Carlo method

A treatment of direct simulation Monte Carlo method as a Markov process with a master equation is given and the corresponding master equation is derived. A hierarchy of equations for the reduced probability distributions is derived from the master equation. An equation similar to the Boltzmann equation for single particle probability distribution is derived using assumption of molecular chaos. It is shown that starting from an uncorrelated state, the system remains uncorrelated always in the limit N→∞, where N is the number of particles. Simple applications of the formalism to direct simulation money games are given as examples to the formalism. The formalism is applied to the direct simulation of homogenous gases. It is shown that appropriately normalized single particle probability distribution satisfies the Boltzmann equation for simple gases and Wang Chang–Uhlenbeck equation for a mixture of molecular gases. As a consequence of this development we derive Birds no time counter algorithm. We extend the analysis to the inhomogeneous gases and define a new direct simulation algorithm for this case. We show that single particle probability distribution satisfies the Boltzmann equation in our algorithm in the limit N→∞, V k →0, Δt→0 where V k is the volume of kth cell. We also show that our algorithm and Bird’s algorithm approach each other in the limit N k →∞ where N k is the number of particles in the volume V k .

Particle, kinetic and fluid models for phototaxis

In this work we derive a hierarchy of new mathematical models fordescribing the motion of phototactic bacteria, i.e., bacteria thatmove towards light. These models are based on recent experimentssuggesting that the motion of such bacteria depends on theindividual bacteria, on group dynamics, and on the interactionbetween bacteria and their environment. Our first model is acollisionless interacting particle system in which we followthe location of the bacteria, their velocity, and their internalexcitation (a parameter whose role is assumed to be related tocommunication between bacteria). In this model, the light sourceacts as an external force. The resulting particle system is anextension of the Cucker-Smale flocking model. We prove that whenall particles are fully excited, their asymptotic velocity tends toan identical (pre-determined) terminal velocity. Our second model is a kinetic model for the one-particle distribution functionthat includes an internal variable representing the excitationlevel. The kinetic model is a Vlasov-type equation that is derivedfrom the particle system using the BBGKY hierarchy and molecularchaos assumption. Since bacteria tend to move in areas that werepreviously traveled by other bacteria, a surface memory effect isadded to the kinetic model as a turning operator that accounts forthe collisions between bacteria and the environment. The third andfinal model is derived as a formal macroscopic limit of the kineticmodel. It is shown to be the Vlasov-McKean equation coupledwith a reaction-diffusion equation.

Htheorem for contact forces in granular materials

A maximum entropy theorem is developed and tested for granular contact forces. Although it is idealized, describing two-dimensional packings of round, rigid, frictionless, cohesionless disks with coordination number Z=4, it appears to describe a central part of the physics present in the more general cases. The theorem does not make the strong claims of Edwards' hypothesis, nor does it rely upon Edwards' hypothesis at any point. Instead, it begins solely from the physical assumption that closed loops of grains are unable to impose strong force correlations around the loop. This statement is shown to be a generalization of Boltzmann's assumption of molecular chaos (his stosszahlansatz), allowing for the extra symmetries of granular stress propagation compared to the more limited symmetries of momentum propagation in a thermodynamic system. The theorem that follows from this is similar to Boltzmann's H theorem and is presented as an alternative to Edwards' hypothesis for explaining some granular phenomena. It identifies a very interesting feature of granular packings: if the generalized stosszahlansatz is correct, then the bulk of homogeneous granular packings must satisfy a maximum entropy condition simply by virtue of being stable, without any exploration of phase space required. This leads to an independent derivation of the contact force statistics, and these predictions have been compared to numerical simulation data in the isotropic case. The good agreement implies that the generalized stosszahlansatz is indeed accurate at least for the isotropic state of the idealized case studied here, and that it is the reductionist explanation for contact force statistics in this case.

Vacuum Radiation, Entropy, and Molecular Chaos

Vacuum radiation causes a particle to make a random walk about its dynamical trajectory. In this random walk the root mean square change in spatial coordinate is proportional to t1/2, and the fractional changes in momentum and energy are proportional to t−1/2, where t is time. Thus the exchange of energy and momentum between a particle and the vacuum tends to zero over time. At the end of a mean free path the fractional change in momentum of a particle in a gas is very small. However, at the end of the mean free path each particle undergoes an interaction that magnifies the preceding change, and the net result is that the momentum distribution of the particles in a gas is randomized in a few collision times. In this way the random action of vacuum radiation and its subsequent magnification by molecular interaction produces entropy increase. This process justifies the assumption of molecular chaos used in the Boltzmann transport equation.

Direct simulation for a homogeneous gas

A probabilistic analysis of the direct simulation of a homogeneous gas is given. A hierarchy of equations similar to the BBGKY hierarchy for the reduced probability densities is derived. By invoking the molecular chaos assumption, an equation similar to the Boltzmann equation for the single particle probability density and the corresponding H-theorem is derived.

Kinetic Theory of Granular Gases

Granular gases are composed of macroscopic bodies kept in motion by an external energy source such as a violent shaking. The behaviour of such systems is quantitatively different from that of ordinary molecular gases: due to the size of the constituents, external fields have a stronger effect on the dynamics and, more importantly, the kinetic energy of the gas is no longer a conserved quantity. The key role of the inelasticity of collisions has been correctly appreciated for about fifteen years, and the ensuing consequences in terms of phase behaviour or transport properties studied in an increasing and now vast body of literature. The purpose of this book is to help the newcomer to the field in acquiring the essential theoretical tools together with some numerical techniques. As emphasized by the authors—who were among the pioneers in the domain— the content could be covered in a one semester course for advanced undergraduates, or it could be incorporated in a more general course dealing with the statistical mechanics of dissipative systems. The book is self-contained, clear, and avoids mathematical complications. In order to elucidate the main physical ideas, heuristic points of views are sometimes preferred to a more rigorous route that would lead to a longer discussion. The 28 chapters are short; they offer exercises and worked examples, solved at the end of the book. Each part is supplemented with a relevant foreword and a useful summary including take-home messages. The editorial work is of good quality, with very few typographical errors. In spite of the title, kinetic theory stricto sensu is not the crux of the matter covered. The authors discuss the consequences of the molecular chaos assumption both at the individual particle level and in terms of collective behaviour. The first part of the book addresses the mechanics of grain collisions. It is emphasized that considering the coefficient of restitution e —a central quantity governing the inelasticity of inter-grain encounters—as velocity independent is inconsistent with the mechanical point of view. An asymptotic expression for the impact velocity dependence of e is therefore derived for visco-elastic spheres. The important inelastic Boltzmann equation is introduced in part II and the associated velocity distribution characterized for a force-free medium (so-called free cooling regime). Transport processes can then be analyzed in part III at the single particle level, and part IV from a more macroscopic viewpoint. The corresponding Chapman–Enskog-like hydrodynamic approach is worked out in detail, in a clear fashion. Finally, the tendency of granular gases to develop instabilities is illustrated in part V where the hydrodynamic picture plays a pivotal role. This book clearly sets the stage. For the sake of simplicity, the authors have discarded some subtle points, such as the open questions underlying the hydrodynamic description (why include the temperature among the hydrodynamic modes, and what about the separation of space and time scales between kinetic and hydrodynamic excitations?). Such omissions are understandable. To a certain extent however, the scope of the book is centered on previous work by the authors, and I have a few regrets. Special emphasis is put on the (variable e) visco-elastic model, which enhances the technical difficulty of the presentation. On the other hand, the important physical effects including scaling laws, hydrodynamic behaviour and structure formation, can be understood in two steps, from the results derived within the much simpler constant e model, allowing subsequently $\varepsilon$ to depend on the granular temperature. The authors justify their choice with the inconsistency of the constant e route. The improvements brought by the visco-elastic model remain to be assessed, since the rotational degrees of freedom, discarded in the book, play an important role and require due consideration of both tangential and normal restitution coefficients, that are again velocity dependent. This seems to be the price of a consistent approach, which does not lend itself to much insight. In addition, the behaviour of driven systems is not addressed, whereas in the realm of granular media, force-free systems are the exception rather than the rule. The differences between constant e and visco-elastic models is presumably less pronounced in the driven case. Study of driven systems also reveals that the rheology of granular gases is intrinsically non-Newtonian, which is a key feature. Finally, the powerful direct simulation Monte Carlo technique is not described, whereas it is an important tool, particularly relevant for the physics of the Boltzmann equation, and straightforward to implement in its simplest version. N Brilliantov and T Poschel concentrate on the (equally relevant) molecular dynamics method instead. In conclusion, the book fills a gap in the field. The companion webpage from where molecular dynamics and symbolic algebra programs can be downloaded is also useful.

Dynamic regimes of fluids simulated by multiparticle-collision dynamics

We investigate the hydrodynamic properties of a fluid simulated with a mesoscopic solvent model. Two distinct regimes are identified, the "particle regime" in which the dynamics is gaslike and the "collective regime" where the dynamics is fluidlike. This behavior can be characterized by the Schmidt number, which measures the ratio between viscous and diffusive transport. Analytical expressions for the tracer diffusion coefficient, which have been derived on the basis of a molecular-chaos assumption, are found to describe the simulation data very well in the particle regime, but important deviations are found in the collective regime. These deviations are due to hydrodynamic correlations. The model is then extended in order to investigate self-diffusion in colloidal dispersions. We study first the transport properties of heavy pointlike particles in the mesoscopic solvent, as a function of their mass and number density. Second, we introduce excluded-volume interactions among the colloidal particles and determine the dependence of the diffusion coefficient on the colloidal volume fraction for different solvent mean-free paths. In the collective regime, the results are found to be in good agreement with previous theoretical predictions based on Stokes hydrodynamics and the Smoluchowski equation.