The velocity autocorrelation function and self-diffusion coefficient of fluids with steeply repulsive potentials

We consider the velocity autocorrelation function, vacf, or C_v(t) and self-diffusion coefficients, D, of steeply repulsive inverse power fluids (SRP) in which the particles interact with a pair potential, ? (r) = ?(σ/r)ⁿ. The C_v(t) are calculated numerically by molecular dynamics simulations. Accurate expressions for the short time expansion of C_v(t) to order O(t⁴) for n large are derived for this fluid. We propose novel expressions for C_v (t) that, for n large, spans the transition from the short time regime (expandable in even powers of time) and the longer time exponential-like regime characteristic of hard spheres. Inter alia we introduce relaxation times that characterize the duration of a collision and the decay of the velocity correlation within the mean-collision or Enskog-like relaxation time, T_E.     

Long-time behavior of the angular velocity autocorrelation function for a fluid of rough spheres

According to hydrodynamical and mode-coupling theories, the angular velocity autocorrelation function decays at long times as ₀(t/10^–14 sec)^–5/2. For rough spheres under the conditions reported here, the quantity ₀ is predicted to be 262. The molecular dynamics studies presented here yield a long-time tail of the form 230(t/10^–14 sec)^–2.38. The disagreement between theory and computer results probably arises from statistical error intrinsic to the computations.The authors are indebted to the National Science Foundation and the Computer Center of the University of Minnesota for financial support of the research reported here.     

Evidence for slowly-decaying tails in the velocity autocorrelation function of a two-dimensional Lorentz gas

Molecular dynamics calculations on a two-dimensional hard-disk Lorentz gas indicate that the velocity autocorrelation function shows a slowly-decaying tail. The decay is consistent with a $\text{1}\text{τ}{}^2$ dependence, but the coefficient is significantly greater than predicted by Ernst and Weyland.     

Long-time tails of the velocity autocorrelation functions for the triangular periodic Lorentz gas

We present numrical results on the velocity autocorrelation function (VACF)C(t)=<ν(t)·ν(0)> for the periodic Lorentz gas on a two-dimensional triangular lattice as a function of the radiusR of the hard disk scatterers on the lattice. Our results for the unbounded horizon case confirm 1/t decay of the VACF for long times (out to 100 times the mean free time between collisions) and provide strong support for the conjecture by Friedman and Martin that the 1/t decay is due to long free paths along which a moving particle does not scatter up to timet. Even after new sets of long free paths become available forR<1/4, we continue to find good agreement between numerical results and an analytically estimated 1/t decay. For the bounded horizon case , our numerical VACFs decay exponentially, although it is difficult to discriminate among pure exponential decay, exponential decay with prefactor, and stretched exponential decay.     

Lyapunov modes and time-correlation functions for two-dimensional systems

The relation between the Lyapunov modes (delocalized Lyapunov vectors) and the momentum autocorrelation function is discussed in two-dimensional hard-disk systems. We show numerical evidence that the smallest time-oscillating period of the Lyapunov modes is twice as long as the time-oscillating period of momentum autocorrelation function for both square and rectangular two-dimensional systems with hard-wall boundary conditions.     

Structures and autocorrelation functions of liquid Al and Mg modelled via Lennard-Jones potential from molecular dynamics simulation

The structures and autocorrelation functions of Al and Mg in the liquid state are investigated through the pair distribution functiong(r), the diffusion coefficients as well as the shear viscosity via the Green-Kubo and Einstein relations. From the structure and the Enskog relation we determined the frequency of collisions of atoms in the first shell ofg(r) in the systems. We also discovered that the packing fraction of Lennard-Jones liquids should be approximately half the reduced density value. This approximation is accurate to within 99%. The temperature dependence of the pair distribution function and the atomic mean square displacement are investigated by performing simulations at various experimental temperatures and corresponding densities. The structures of the systems are affected by temperature via movements of atoms in the first minimum ofg(r). The Lennard-Jones model shows that density dependence of the shear viscosity is in agreement with what is expected of simple liquids in the range of investigated temperatures and densities. In the gas limit, the Stoke-Einstein relationDη =K _BT /2πσ is grossly overestimated by Lennard-Jones model. This could not be attributed to deficiencies in the model, as other investigators using first principle method could not obtain the gas limit of the Stoke-Einstein relation.     

Universal distribution functions in two-dimensional localized systems

We find the conductance distribution function of the two-dimensional Anderson model in the strongly localized limit. The fluctuations of lng grow with lateral size as L1/3 and follow a universal distribution that depends on the type of leads. For narrow leads, it is the Tracy-Widom distribution, which appears in the problem of the largest eigenvalue of random matrices from the Gaussian unitary ensemble and in many other problems like the longest increasing subsequence of a permutation, directed polymers, or polynuclear growth. We also show that for wide leads the conductance follows a related, but different, distribution.     

角速度突变三例

分析了角速度突变的三个例子,以加深对角动量及角动量守恒的认识。     

Kinetic equations for autocorrelation functions in dilute gases

We show that in the case of a dilute gas of neutral particles kinetic equations for autocorrelation functions such as $$\left\langle {\hat f\left( {r,v,t} \right)\hat f\left( {r\prime v\prime ,t\prime } \right)} \right\rangle ,where\hat f\left( {r,v,t} \right) = \sum {_{i = 1}^N } \delta \left( {r - r_i \left( t \right)} \right)\delta \left( {v - v_i \left( {tt} \right)} \right)$$ , can be obtained in a very simple manner by the use of the truncated BBGKY hierarchy. The resulting equations correspond to the low-density limit of the results of van Leeuwen and Yip. Moreover, the derivation does not make use of the Bogoliubov adiabatic approximation, and therefore includes non-Markovian effects which can be important in describing light scattering from gases and the collisional narrowing of atomic dipole radiation. The resulting equations in the long-wavelength limit correspond to the non-Markovian Boltzmann equation for the self-correlation part and the non-Markovian, linearized Boltzmann equation for the total autocorrelation function.     

Three-mode optomechanical system for angular velocity detection

We propose a scheme for measuring the angular velocity of absolute rotation using a three-mode optomechanical system in which one mode of the two-dimensional(2 D) mechanical resonator is coupled to an optical cavity. When the total system rotates, the Coriolis force acting on the 2 D mechanical resonator due to the absolute rotation will affect the mechanical motion and thus change the phase of the output field from the cavity. The angular velocity of the absolute rotation can be estimated by monitoring the spectrum of the output field from the cavity via homodyne measurement. The minimum measurable angular velocity, which is determined by the noise spectrum, is calculated. The working range of the gyroscope for measuring angular velocity is discussed.     

Self-trapping under two-dimensional spin-orbit coupling and spatially growing repulsive nonlinearity

Frontiers of Physics - The scattering of an electron by a muon in the presence of a linearly polarized laser field is investigated in the first Born approximation. The theoretical results reveal...     

On the correlation functions of fully frustrated two-dimensional Ising systems

By mapping the triangular antiferromagnet, the Villain model and the Union-Jack model on the quantumXY-chain we show a simple way to obtain the exponent =1/2 for the decay of the spin-correlation function at the frustration points of these models. A similar procedure for the Baxter model leads to non-universal values of .     

Complex singularities of the fluid velocity autocorrelation function

There are intensive debates regarding the nature of supercritical fluids: if their evolution from liquid-like to gas-like behavior is a continuous multistage process or there is a sharp well-defined crossover. Velocity auto-correlation function Z is the established detector of evolution of fluid particles dynamics. Usually, complex singularities of correlation functions give more information. For this reason, we investigate Z in complex plane of frequencies using numerical analytic continuation. We have found that naive picture with few isolated poles fails describing Z(ω) of one-component Lennard-Jones (LJ) fluid. Instead, we see the singularity manifold forming branch cuts extending approximately parallel to the real frequency axis. That suggests LJ velocity autocorrelation function is a multivalued function of complex frequency. The branch cuts are separated from the real axis by the well-defined “gap” whose width corresponds to an important time scale of a fluid characterizing crossover of system dynamics from kinetic to hydrodynamic regime. Our working hypothesis is that the branch cut origin is related to competition between one-particle dynamics and hydrodynamics. The observed analytic structure of Z is very stable under changes in the temperature; it survives at temperatures two orders of magnitude higher than the critical one.     

Short time behavior of the velocity autocorrelation function

A systematic method for evaluating velocity correlation functions of a hard sphere fluid for short times is presented The complete contributions ～ t and t² are obtained, formally valid for all densities.     

The non-exponential behavior of the velocity autocorrelation function

The contribution to the velocity autocorrelation function from ring events is studied for low densities and t?2t₀ using a kinetic model. Agreement with computer experiments for $\text{V}\text{V}{}_0\text{= 18}$ is found to be good.