Fokker–Planck Equation,Molecular Friction,and Molecular Dynamics for Brownian Particle Transport near External Solid Surfaces

The Fokker–Planck (FP) equation describing the dynamics of a single Brownian particle near a fixed external surface is derived using the multiple-time-scales perturbation method, previously used by Cukier and Deutch and Nienhuis in the absence of any external surfaces, and Piasecki et al. for two Brownian spheres in a hard fluid. The FP equation includes an explicit expression for the (time-independent) particle friction tensor in terms of the force autocorrelation function and equilibrium average force on the particle by the surrounding fluid and in the presence of a fixed external surface, such as an adsorbate. The scaling and perturbation analysis given here also shows that the force autocorrelation function must decay rapidly on the zeroth-order time scale ₀, which physically requires N _Kn1, where N _Kn is the Knudsen number (ratio of the length scale for fluid intermolecular interactions to the Brownian particle length scale). This restricts the theory given here to liquid systems where N _Kn1. For a specified particle configuration with respect to the external surface, equilibrium canonical molecular dynamics (MD) calculations are conducted, as shown here, in order to obtain numerical values of the friction tensor from the force autocorrelation expression. Molecular dynamics computations of the friction tensor for a single spherical particle in the absence of a fixed external surface are shown to recover Stokes' law for various types of fluid molecule–particle interaction potentials. Analytical studies of the static force correlation function also demonstrate the remarkable principle of force-time parity whereby the particle friction coefficient is nearly independent of the fluid molecule–particle interaction potential. Molecular dynamics computations of the friction tensor for a single spherical particle near a fixed external spherical surface (adsorbate) demonstrate a breakdown in continuum hydrodynamic results at close particle–surface separation distances on the order of several molecular diameters.     

Time-Dependent Gaussian Solution for the Kostin Equation Around Classical Trajectories

The structure of time-dependent Gaussian solutions for the Kostin equation in dissipative quantum mechanics is analyzed. Expanding the generic external potential near the center of mass of the wave packet, one conclude that: the center of mass follows the dynamics of a classical particle under the external potential and a damping proportional to the velocity; the width of the wave packet satisfy a non-conservative Pinney equation. An appropriate perturbation theory is developed for the free particle case, solving the long standing problem of finding analytic expressions for square integrable solutions of the free Kostin equation. The associated Wigner function is also studied.     

Paths of spinning particles in general relativity as geodesics of an Einstein connection

This paper deals with the Papapetrou-Pirani equations of motion for a spinning test particle in general relativity. The motion of the center of mass can be represented by the geodesic equation of an affine connection that is the sum of the Christoffel connection and a tensor that depends on the Riemann-Christoffel curvature tensor, the mass of the particle, its 4-velocity, and its spin tensor. The connection is not unique, and here it is chosen to satisfy one of the basic geometrical principles of Einstein's unified field theory: The symmetric part of the fundamental tensor of the geometry is specified to be the metric tensor of general relativity. The special case of conformally flat space-times is discussed.     

Difference of Directions Between External Driving Force and MovementDirection of Center of Mass

Driven dynamics of a two-dimensional Frenkel-Kontorova model is studied in the paper. In our numerical simulations, it is found that the movement direction of the center of mass is not consistent with that of the external driving force except for some special symmetric directions at the lower driving force. Our results also indicate that the movement direction of the center of mass strongly depends on both the magnitude and the direction of the external driving force as well as the misfit angle between two layers.     

Diffusion coefficient for a Brownian particle in a periodic field of force: I. Large friction limit

The diffusion tensor for a Brownian particle in a periodic field of force is studied in the strong damping limit, in which the Smoluchowski equation is valid.A general relation between the diffusion tensor and the Smoluchowski “relaxation operator” is derived; the effect of the periodic force, at least in the simplest situation of diagonal and uniform friction, appears as a dressing of the bare particle mass to an effective tensor mass.From this the explicit form of the diffusion coefficient as a functional of the potential energy in the one-dimensional case is obtained, showing a temperature dependence which deviates at high temperatures from a simple Arrhenius behaviour.Finally, the expression for the mobility of the Brownian particle is derived, and by comparison with the expression for the diffusion coefficient the Einstein relation between diffusion and mobility is proved to be satisfied.     

The equation of motion for the trajectory of a circling particle with an overdamped circle center

Inspired by biological microorganisms swimming in circles in liquid with low Reynolds number, I developed the dynamic theory for computing the helical trajectory of a circling particle with an overdamped circle center. The equation of motion for the circling particle is a hybrid equation of deterministic terms and stochastic terms. Observing the motion of a swimming microorganism, I found the strength of stochastic fluctuations should be much smaller than that governs deterministic dynamics. This dynamic theory predicts a nonlinear transverse motion perpendicular to the direction of external force. Both the living microorganism and artificial circling particle are applicable for an experimental check of this prediction. For the convenience of easy theoretical research, I further derived the probability conservation equations based on this dynamic theory both in two-dimensional and three-dimensional space.     

The orientational pair correlation functions in a dense hard sphere fluid at long times

The two-particle contribution to the potential part of the stress tensor autocorrelation function of a dense hard sphere fluid is studied. It is shown that the long-time decay is given as the solution of a diffusion equation for the relative particle in a potential of mean force. The diffusion constant needed in order to accurately reproduce molecular dynamics results is found to be somewhat lower than the self-diffusion constant.     

The dynamical equation of the spinning electron

We obtain the relativistic and non-relativistic invariant dynamical equations of the spinning electron by means of invariance arguments. The dynamics expresses the evolution of the point charge which satisfies a fourthorder differential equation or, alternatively, by describing the evolution of both the center of mass and center of charge of the particle. These equations and the interaction between two electrons will be analyzed.     

On the phenomenological three-graviton vertex

In General Relativity, the graviton interacts in three-graviton vertex with a tensor that is not the energy-momentum tensor of the gravitational field. We consider the possibility that the graviton interacts with the definite gravitational energy-momentum tensor that we previously found in the G ² approximation. This tensor in a gauge, where nonphysical degrees of freedom do not contribute, is remarkable, because it gives positive gravitational energy density for the Newtonian center in the same manner as the electromagnetic energy-momentum tensor does for the Coulomb center. We show that the assumed three-graviton vertex does not lead to contradiction with the precession of Mercury’s perihelion. In the S-matrix approach used here, the external gravitational field has only a subsidiary role, similar to the external field in quantum electrodynamics. This approach with the assumed vertex leads to the gravitational field that cannot be obtained from a consistent gravity equation.     

Self-adaptive behavior of nunchakus-like tracer induced by active Brownian particles

We design a nunchakus-like tracer and investigate its self-adaptive behavior in an active Brownian particle (ABP) bath via systematically tuning the self-propelled capability and density of ABPs. Specifically, the nunchakus-like tracer will have a stable wedge-like shape in the ABP bath when the self-propelled force is high enough. We analyze the angle between the two arms of the tracer and the velocity of the joint point of the tracer. The angle exhibits a non-monotonic phenomenon as a function of active force. However, it increases with density of ABPs increasing monotonically. A simple linear relationship between the velocity and the self-propelled force is found under the highly active force. In other words, the joint points of the tracer diffuse and the super-diffusive behavior can make the relation between the self-propelled force and the density of ABPs persist longer. In addition, we find that the tracer can flip at high density of ABPs. Our results also suggest the new self-adaptive model research of the transport properties in a non-equilibrium medium.     

Force-driven micro-rheology

Within a microscopic formalism for the nonequilibrium response of colloidal suspensions driven by an external force, we study the active micro-rheology of a glass-forming colloidal suspensions. In this technique, a probe particle is subject to an external force, and its nonequilibrium dynamics is monitored. Strong external forcing delocalizes the particle from its nearest-neighbor cage, resulting in a pronounced force-thinning behavior of the single-particle friction. We discuss the dynamics in the vicinity of this delocalization transition, and how long-range transport is induced for a particle that is localized in the quiescent case.     

Densities of electron’s continuum in gravitational and electromagnetic fields

Relativistic dynamics of distributed mass and charge densities of the extended classical particle is considered for arbitrary gravitational and electromagnetic fields. Both geodesic and field gravitational equations can be derived by variation of the same Lagrange density in the classical action of a nonlocal particle distributed over its radial field. Vector geodesic relations for material space densities are contraction consequences of tensor gravitational equations for continuous sources and their fields. Classical four-flows of elementary material space depend on local electromagnetic fourpotentials for charged densities, as in quantum theory. Besides the Lorentz force, these potentials result in two more accelerating factors vanishing under equilibrium internal stresses within the continuous particle.     

A Brownian Particle in a Microscopic Periodic Potential

We study a model for a massive test particle in a microscopic periodic potential and interacting with a reservoir of light particles. In the regime considered, the fluctuations in the test particle’s momentum resulting from collisions typically outweigh the shifts in momentum generated by the periodic force, so the force is effectively a perturbative contribution. The mathematical starting point is an idealized reduced dynamics for the test particle given by a linear Boltzmann equation. In the limit that the mass ratio of a single reservoir particle to the test particle tends to zero, we show that there is convergence to the Ornstein–Uhlenbeck process under the standard normalizations for the test particle variables. Our analysis is primarily directed towards bounding the perturbative effect of the periodic potential on the particle’s momentum.     

Algebraic interpretation of the Yang-Mills field

Einstein's principle of general relativity is a dynamical-group approach in that all dynamics is implied by the invariance and no force is introduced (as an external, symmetry-breaking factor). In this spirit we take a Poincaré-invariant free wave equation and, deforming the Poincaré group to the de Sitter group, obtain interaction. This illustrates our algebraic approach to gauge invariance, whereby the (generalized) Maxwell tensor of the Yang-Mills field appears as structure constants of the homogeneous algebra obtained as a deformation of an inhomogeneous one, with interaction appearing via the same tensor, which plays a role corresponding to the curvature tensor in Einstein's general relativity.     

Condensate oscillations in a Penrose tiling lattice

We study the dynamics of a Bose-Einstein condensate subject to a particular Penrose tiling lattice. In such a lattice, the potential energy at each site depends on the neighbour sites, accordingly to the model introduced by Sutherland [16]. The Bose-Einstein wavepacket, initially at rest at the lattice symmetry center, is released. We observe a very complex time-evolution that strongly depends on the symmetry center (two choices are possible), on the potential energy landscape dispersion, and on the interaction strength. The condensate-width oscillates at different frequencies and we can identify large-frequency reshaping oscillations and low-frequency rescaling oscillations. We discuss in which conditions these oscillations are spatially bounded, denoting a self-trapping dynamics.     

自驱动颗粒体系中的熵力

在许多纳米复合材料体系中熵力(entropy force)是普遍存在的,但由于熵力的存在会导致纳米颗粒的凝聚从而降低其许多性能,因此在大多数情况下熵力的存在对体系并无益处,所以研究如何减小熵力对体系的影响是非常重要的.不带角速度的自驱动粒子在熵力作用下会集聚在纳米颗粒(或者纳米棒)周围,这会对纳米颗粒(或者纳米棒)产生很大的相互作用力.对于纳米颗粒,在不带角速度的自驱动粒子体系中存在着非常大的排斥力.而对于纳米棒,由于纳米棒内外的不对称性,使得两个纳米棒之间会产生吸引-排斥转变,同时这个吸引-排斥转变与纳米棒之间的距离有关.当自驱动粒子加上一个自转角速度ω之后,熵力的作用就大大减弱,纳米颗粒不再集聚.研究结果有助于对非平衡态下纳米颗粒(或纳米棒)之间熵相互作用力的认识.     

Dirac equation for the Hulthén potential within the Yukawa-type tensor interaction

Using the Nikiforov-Uvarov (NU) method, pseudospin and spin symmetric solutions of the Dirac equation for the scalar and vector Hulthén potentials with the Yukawa-type tensor potential are obtained for an arbitrary spin-orbit coupling quantum number κ. We deduce the energy eigenvalue equations and corresponding upper- and lower-spinor wave functions in both the pseudospin and spin symmetry cases. Numerical results of the energy eigenvalue equations and the upper- and lower-spinor wave functions are presented to show the effects of the external potential and particle mass parameters as well as pseudospin and spin symmetric constants on the bound-state energies and wave functions in the absence and presence of the tensor interaction.     

One-Dimensional Steady Transport by Molecular Dynamics Simulation：Non-Boltzmann Position Distribution and Non-Arrhenius Dynamical Behavior

A non-equilibrium steady state can be characterized by a nonzero but stationary flux driven by a static external force. Under a weak external force, the drift velocity is difficult to detect because the drift motion is feeble and submerged in the intense thermal diffusion. In this article, we employ an accurate method in molecular dynamics simulation to determine the drift velocity of a particle driven by a weak external force in a one-dimensional periodic potential. With the calculated drift velocity, we found that the mobility and diffusion of the particle obey the Einstein relation, whereas their temperature dependences deviate from the Arrhenius law. A microscopic hopping mechanism was proposed to explain the non-Arrhenius behavior. Moreover, the position distribution of the particle in the potential well was found to deviate from the Boltzmann equation in a non-equilibrium steady state. The non-Boltzmann behavior may be attributed to the thermostat which introduces an effective "viscous" drag opposite to the drift direction of the particle.     

双稳系统的随机能量共振和作功效率

分析了处于双稳系统中的布朗粒子与外界的周期性外力和热随机力的功、热交互作用,建立了基于Langevin方程的随机能量平衡方程.围绕着受周期力、随机力和阻尼力共同作用的Langevin方程,采用动力学和非平衡热力学相结合的方法,从以"力"为立足点转到以"能量"为研究核心,深入分析了布朗粒子沿单一轨线运动时系统与环境之间的能量交换和作功效率,揭示了双稳系统的随机能量共振现象. 关键词： 双稳系统 随机能量共振 作功效率     

Effective Dynamics of a Tracer Particle Interacting with an Ideal Bose Gas

We study a system consisting of a heavy quantum particle, called the tracer particle, coupled to an ideal gas of light Bose particles, the ratio of masses of the tracer particle and a gas particle being proportional to the gas density. All particles have non-relativistic kinematics. The tracer particle is driven by an external potential and couples to the gas particles through a pair potential. We compare the quantum dynamics of this system to an effective dynamics given by a Newtonian equation of motion for the tracer particle coupled to a classical wave equation for the Bose gas. We quantify the closeness of these two dynamics as the mean-field limit is approached (gas density ${\to \infty}$ ). Our estimates allow us to interchange the thermodynamic with the mean-field limit.