Brownian motion: from kinetics to hydrodynamics

Brownian motion has served as a pilot of studies in diffusion and other transport phenomena for over a century. The foundation of Brownian motion, laid by Einstein, has generally been accepted to be far from being complete since the late 1960s, because it fails to take important hydrodynamic effects into account. The hydrodynamic effects yield a time dependence of the diffusion coefficient, and this extends the ordinary hydrodynamics. However, the time profile of the diffusion coefficient across the kinetic and hydrodynamic regions is still absent, which prohibits a complete description of Brownian motion in the entire course of time. Here we close this gap. We manage to separate the diffusion process into two parts: a kinetic process governed by the kinetics based on molecular chaos approximation and a hydrodynamics process described by linear hydrodynamics. We find the analytical solution of vortex backflow of hydrodynamic modes triggered by a tagged particle. Coupling it to the kinetic process we obtain explicit expressions of the velocity autocorrelation function and the time profile of diffusion coefficient. This leads to an accurate account of both kinetic and hydrodynamic effects. Our theory is applicable for fluid and Brownian particles, even of irregular-shaped objects, in very general environments ranging from dilute gases to dense liquids. The analytical results are in excellent agreement with numerical experiments.Comment: 8pages,3figure

The hard-disk fluid revisited

The hard-disk model plays a role of touchstone for testing and developing the transport theory. By large scale molecular dynamics simulations of this model, three important autocorrelation functions, and as a result the corresponding transport coefficients, i.e., the diffusion constant, the thermal conductivity and the shear viscosity, are found to deviate significantly from the predictions of the conventional transport theory beyond the dilute limit. To improve the theory, we consider both the kinetic process and the hydrodynamic process in the whole time range, rather than each process in a seperated time scale as the conventional transport theory does. With this consideration, a unified and coherent expression free of any fitting parameters is derived succesfully in the case of the velocity autocorrelation function, and its superiority to the conventional `piecewise' formula is shown. This expression applies to the whole time range and up to moderate densities, and thus bridges the kinetics and hydrodynamics approaches in a self-consistent manner.Comment: 5 pages, 4 figure

Copy the dynamics using a learning machine

Is it possible to generally construct a dynamical system to simulate a black system without recovering the equations of motion of the latter? Here we show that this goal can be approached by a learning machine. Trained by a set of input-output responses or a segment of time series of a black system, a learning machine can be served as a copy system to mimic the dynamics of various black systems. It can not only behave as the black system at the parameter set that the training data are made, but also recur the evolution history of the black system. As a result, the learning machine provides an effective way for prediction, and enables one to probe the global dynamics of a black system. These findings have significance for practical systems whose equations of motion cannot be approached accurately. Examples of copying the dynamics of an artificial neural network, the Lorenz system, and a variable star are given. Our idea paves a possible way towards copy a living brain.Comment: 8 pages, 4 figure

Inferring Global Dynamics of a Black-Box System Using Machine Learning

We present that, instead of establishing the equations of motion, one can model-freely reveal the dynamical properties of a black-box system using a learning machine. Trained only by a segment of time series of a state variable recorded at present parameters values, the dynamics of the learning machine at different training stages can be mapped to the dynamics of the target system along a particular path in its parameter space, following an appropriate training strategy that monotonously decreases the cost function. This path is important, because along that, the primary dynamical properties of the target system will subsequently emerge, in the simple-to-complex order, matching closely the evolution law of certain self-evolved systems in nature. Why such a path can be reproduced is attributed to our training strategy. This particular function of the learning machine opens up a novel way to probe the global dynamical properties of a black-box system without artificially establish the equations of motion, and as such it might have countless applications. As an example, this method is applied to infer what dynamical stages a variable star has experienced and how it will evolve in future, by using the light curve observed presently.Comment: 17 pages, 8 figure

The 2-adic valuations of Stirling numbers of the second kind

In this paper, we investigate the 2-adic valuations of the Stirling numbers $S(n, k)$ of the second kind. We show that $v_2(S(4i, 5))=v_2(S(4i+3, 5))$ if and only if $i\not\equiv 7\pmod {32}$. This confirms a conjecture of Amdeberhan, Manna and Moll raised in 2008. We show also that $v_2(S(2^n+1, k+1))= s_2(n)-1$ for any positive integer $n$, where $s_2(n)$ is the sum of binary digits of $n$. It proves another conjecture of Amdeberhan, Manna and Moll.Comment: 9 pages. To appear in International Journal of Number Theor

Mechanical Resonance of embedded cluster

Embedded clusters, which are embedded in bulk materials and different from the surroundings in structures, should be common in materials. This paper studies resonance of such clusters. This work is stimulated by a recent experimental observation that some localized clusters behavior like fluid at the mesoscopic scale in many solid materials [Science in China(Series B). 46, 176 (2003)]. We argue that the phenomenon is just a vivid illustration of resonance of embedded clusters, driven by ubiquitous microwaves. Because the underlying mechanism is fundamental and embedded structures are usual, the phenomenon would have great significance in material physics.Comment: 6 pages, 2 figure

Modified Stokes-Einstein Relation for Small Brownian Particles

The Stokes-Einstein (SE) relation has been widely applied to quantitatively describe the Brownian motion. Notwithstanding, here we show that even for a simple fluid, the SE relation may not be completely applicable. Namely, although the SE relation could be a good approximation for a large enough Brownian particle, we find that it induces significant error for a smaller Brownian particle, and the error increases with the decrease of the Brownian particle's size, till the SE relation fails completely when the size of Brownian particle is comparable with that of a fluid molecule. The cause is rooted in the fact that the kinetic and the hydrodynamic effects depend on the size of the Brownian particle differently. By excluding the kinetic contribution to the diffusion coefficient, we propose a revised Stokes-Einstein relation and show that it expands significantly the applicable range.Comment: 3 figure

Self-consistent Thomas-Fermi Approximation for Equation of State at Subnuclear Densities

The self-consistent Thomas-Fermi approximation is an essential method for studying the non-uniform nuclear matter with relativistic mean-field theory. In this method, the nucleon distribution in the Wigner-Seitz cell is obtained self-consistently. We make a detailed comparison between the present results and previous calculations in the Thomas-Fermi approximation with a parameterized nucleon distribution that has been adopted in the widely used Shen equation of state.Comment: 4 pages, 2 figures, conferenc

Collision-Free Kinematics for Redundant Manipulators in Dynamic Scenes using Optimal Reciprocal Velocity Obstacles

We present a novel algorithm for collision-free kinematics of multiple manipulators in a shared workspace with moving obstacles. Our optimization-based approach simultaneously handles collision-free constraints based on reciprocal velocity obstacles and inverse kinematics constraints for high-DOF manipulators. We present an efficient method based on particle swarm optimization that can generate collision-free configurations for each redundant manipulator. Furthermore, our approach can be used to compute safe and oscillation-free trajectories in a few milli-seconds. We highlight the real-time performance of our algorithm on multiple Baxter robots with 14-DOF manipulators operating in a workspace with dynamic obstacles. Videos are available at https://sites.google.com/view/collision-free-kinematicsComment: 8 pages, 12 figures, 1 tabl

Thermal state truncation by using quantum scissors device

A non-Gaussian state being a mixture of the vacuum and single-photon states can be generated by truncating a thermal state in a quantum scissors device of Pegg et al. [Phys. Rev. Lett. 81 (1998) 1604]. In contrast to the thermal state, the generated state shows nonclassical property including the negativity of Wigner function. Besides, signal amplification and signal-to-noise ratio enhancement can be achieved.Comment: 6 pages, 7 figure