共查询到20条相似文献,搜索用时 15 毫秒
1.
A theoretical generalisation of the Fokker/Planck equation for atomic and molecular diffusion is compared with the results of a molecular dynamics simulation of a triatomic molecule ofC
2v symmetry. The molecular dynamics results are non-Markhovian and non-Gaussian in nature, markedly so in the case of the centre of mass linear velocityV. This may be ascertained by simulating the long-time limit of the three dimensional kinetic energy autocorrelation function <V
2(t)V
2(0)>/<V
2(0)V
2(0)>, which falls well below the theoretical Gaussian value of 3/5. By expressing the Mori continued fraction as a multidimensional Markhovian chain of differential equations and expressing this in turn as a non-Gaussian probability-diffusion equation of the Kramers/Moyal type it is possible to account for the simulation results in a qualitative fashion. 相似文献
2.
In the present article we consider a motion of a passive tracer particle, whose trajectory satisfies the Itô stochastic differential equation d
x(t) = V(t, x(t)) dt +
d
w(t), where w(·) is a Brownian motion, V is a stationary Gaussian random field with incompressible realizations independent of w(·) and >0. We prove the superdiffusive character of the motion under certain conditions on the energy spectrum of the velocity field. The result is shown both for steady (time independent) and time dependent and Markovian velocity fields. In addition, we provide explicit upper and lower bounds for the Hurst exponent of the trajectory. All previous rigorous results concerned explicitely solvable shear flows cases. 相似文献
3.
Benjamin Lindner 《Journal of statistical physics》2008,130(3):523-533
Nonequilibrium biological systems like moving cells or bacteria have been phenomenologically described by Langevin equations
of Brownian motion in which the friction function depends on the particle’s velocity in a nonlinear way. An important subclass
of such friction functions is given by power laws, i.e., instead of the Stokes friction constant γ
0 one includes a function γ(v)∼v
2α
. Here I show using a recent analytical result as well as a dimension analysis that the diffusion coefficient is proportional
to a simple power of the noise intensity D like D
(1−α)/(1+α) (independent of spatial dimension). In particular the diffusion coefficient does not depend on the noise intensity at all,
if α=1, i.e., for a cubic friction F
fric=−γ(v)v∼v
3. The exact prefactor is given in the one-dimensional case and a fit formula is proposed for the multi-dimensional problem.
All results are confirmed by stochastic simulations of the system for α=1, 2, and 3 and spatial dimension d=1, 2, and 3. Conclusions are drawn about the strong noise behavior of certain models of self-propelled motion in biology. 相似文献
4.
We consider the velocity autocorrelation function, vacf, or Cv(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 Cv(t) are calculated numerically by molecular dynamics simulations. Accurate expressions for the short time expansion of Cv(t) to order O(t4) for n large are derived for this fluid. We propose novel expressions for Cv (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, TE. 相似文献
5.
V. P. Ruban 《Journal of Experimental and Theoretical Physics》2010,111(5):776-785
We consider slow, compared to the speed of sound, motions of an ideal compressible fluid (gas) in a gravitational field in
the presence of two isentropic layers with a small specific-entropy difference between them. Assuming the flow to be potential
in each of the layers (v
1, 2 = ▿ϕ1, 2) and neglecting the acoustic degrees of freedom (div($
\bar \rho
$
\bar \rho
(z)▿ϕ1, 2) ≈ 0, where $
\bar \rho
$
\bar \rho
(z) is the average equilibrium density), we derive the equations of motion for the boundary in terms of the shape of the surface
z = η(x, y, t) itself and the difference between the boundary values of the two velocity field potentials: ψ(x, y, t) = ψ1 − ψ2. We prove the Hamilto nian structure of the derived equations specified by a Lagrangian of the form ℒ = ∫$
\bar \rho
$
\bar \rho
(η)η
t
ψdxdy − ℋ{η, ψ}. The system under consideration is the simplest theoretical model for studying internal waves in a sharply stratified
atmosphere in which the decrease in equilibrium gas density due to gas compressibility with increasing height is essentially
taken into account. For plane flows, we make a generalization to the case where each of the layers has its own constant potential
vorticity. We investigate a system with a model dependence $
\bar \rho
$
\bar \rho
(z) ∝ e
−2αz
with which the Hamiltonian ℋ{η, ψ} can be represented explicitly. We consider a long-wavelength dynamic regime with dispersion
corrections and derive an approximate nonlinear equation of the form u
t
+ auu
x
− b[−$
\hat \partial _x^2
$
\hat \partial _x^2
+ α2]1/2
u
x
= 0 (Smith’s equation) for the slow evolution of a traveling wave. 相似文献
6.
A theory is constructed for the dynamics and braking of domain walls in ferromagnets when a magnetic field is applied perpendicular
to the axis of easy magnetization (i.e., a transverse field H
⊥). The theory is valid for velocities v up to the limiting domain wall velocity v c. The Landau-Lifshitz equations in the dissipationless approximation are used to investigate the motion of domain walls and
the change in the character of the wall motion as its velocity v approaches v
c. The force acting on a domain wall due to viscous friction is calculated within the framework of generalized relaxation theory,
and the dependence of the domain wall velocity v on the forcing field H
z is investigated. Calculations of the braking force show that the contributions of various dissipation mechanisms to the friction
force have different dependences on the domain wall velocity, which affects the form of the function v=v(H
z). The shapes of the curves v(H
z) differ very markedly from one another for different values of the field H
⊥. The theory developed here can be used to describe the experimental results, in particular the almost linear behavior of
v=v(H
z) for small H
⊥ and its strongly nonlinear behavior when H
⊥∼H
a, whereas these data cannot be reconciled within the standard theory based on relaxation terms of Hilbert type.
Zh. éksp. Teor. Fiz. 112, 953–974 (September 1997) 相似文献
7.
Different initial and boundary value problems for the equation of vibrations of rods (also called Fresnel equation) are solved
by exploiting the connection with Brownian motion and the heat equation. The equation of vibrations of plates is considered
and the case of circular vibrating disks C
R
is investigated by applying the methods of planar orthogonally reflecting Brownian motion within C
R
. The analysis of the fractional version (of order ν) of the Fresnel equation is also performed and, in detail, some specific cases, like ν=1/2, 1/3, 2/3, are analyzed. By means of the fundamental solution of the Fresnel equation, a pseudo-process F(t), t>0 with real sign-varying density is constructed and some of its properties examined. The composition of F with reflecting Brownian motion B yields the law of biquadratic heat equation while the composition of F with the first passage time T
t
of B produces a genuine probability law strictly connected with the Cauchy process. 相似文献
8.
Paolo Buttà Emanuele Caglioti Carlo Marchioro 《Communications in Mathematical Physics》2003,233(3):545-569
We study the time evolution of a charged particle moving in a medium under the action of a constant electric field E. In the framework of fully Hamiltonian models, we discuss conditions on the particle/medium interaction which are necessary for the particle to reach a finite limit
velocity. We first consider the case when the charged particle is confined in an unbounded tube of ℝ3. The electric field E is directed along the symmetry axis of the tube and the particle also interacts with an infinitely many particle system.
The background system initial conditions are chosen in a set which is typical for any reasonable thermodynamic (equilibrium
or non-equilibrium) state. We prove that, for large E and bounded interactions between the charged particle and the background, the velocity v(t) of the charged particle does not reach a finite limit velocity, but it increases to infinite as: |v(t)−Et|≤C
0
(1+t), where C
0
is a constant independent of E. As a corollary we obtain that, if the initial conditions of the background system are distributed according to any Gibbs
state, then the average velocity of the charged particle diverges as time goes to infinite. This result is obtained for E large enough in comparison with the mean energy of the Gibbs state. We next study the one-dimensional case, in which the
estimates can be improved. We finally discuss, at an heuristic level, the existence of a finite limit velocity for unbounded interactions, and give some suggestions about the case of small electric fields.
Received: 7 March 2002 / Accepted: 23 September 2002 Published online: 8 January 2003
RID="*"
ID="*" Work partially supported by the GNFM-INDAM and the Italian Ministry of the University.
Communicated by J.L. Lebowitz 相似文献
9.
Conditional velocity cross correlation functions of the form <vi (0)vj (t); rij (0)> in the Lennard-Jones fluid are investigated by molecular dynamics simulation. As shown in previous work, these cross correlation functions may be related to memory functions in a similar manner as the usual velocity auto-correlation function. To compute the memory functions, a modified version of Detyna and Singer's algorithm has been used. 相似文献
10.
Matthew Babela 《Journal of Nonlinear Mathematical Physics》2017,24(1):73-78
We construct a family of integrable equations of the form vt = f(v; vx; vxx; vxxx) such that f is a transcendental function in v; vx; vxx. This family is related to the Krichever-Novikov equation by a differential substitution. Our construction of integrable equations and the corresponding differential substitutions involves geometry of a family of genus two curves and their Jacobians. 相似文献
11.
Summary In the present review of liquid dynamics studies on liquid metals are reported. Particularly the case of liquid lead is reviewed
because this case was carefully studied by neutron scattering technique,S(Q,ω) being determined at two widely different temperaturesT=623 K andT=1170 K and therefore different densities. In addition extensive supplementary MD simulations were made using a 16 384-particle
system. The simulations ranged from a determination of an effective pair potential for lead to simulation of the density correlation
functionsF(Q,t) andF
s(Q,t), as well as the longitudinal and transversal current correlation functionsJ
1(Q,t) andJ
T(Q,t). The MD simulation ?calibrated? via the experimentalS(Q) andS(Q,ω) was used to prolong the range of neutron data to draw conclusions regarding such quantities as dispersion relations for
the current correlationsJ
1(Q,t) andJ
T(Q,t), the generalized viscosity functions ν1(Q,t), ν1(Q) and νs(Q). Information regarding bulk viscosity νB(Q) is also gained. Conclusions are drawn regarding the relative importance of the derived pair potential form by comparison
to corresponding hard-sphere data. The general framework of linearized hydrodynamic equations for the macroscopic situation
transforming to visco-elastic equations of motion for finite wave-length and high frequency works well also for the case of
a continuous potential. The region of transition from simple visco-elastic to hydrodynamic behaviour is occurring at wavelengths
in the range (12÷20) ? for the cases studied. The spatial properties of the viscosity functions ν1(r), νs(r) and νB(r) are found to correlate well with the range of the radial distribution function for the liquid. The general results for liquid
lead probably have wide range of applicability to other simple liquids with similarS(Q) andg(r) properties.
The authors have agreed not to receive proofs for correction. 相似文献
12.
v 《Journal of Molecular Spectroscopy》2009,257(1):108-110
The effective correlation-free Hamiltonian and corresponding matrix elements for interacting degenerate fundamental vt(E) and combination vn(A1)+vt′(E) states in C3v molecules are derived. The Hamiltonian terms H30, H31, H32 and the recommended set of parameters following from the appropriate reduction are presented. 相似文献
13.
B. Šesták 《Czechoslovak Journal of Physics》1972,22(5):341-355
On the basis of the expansion of the distribution functionf(v, r,t) in a sum of spherical harmonics, which is equivalent to a Cartesian tensor scalar product expansion of the distribution function, i.e.,f(v, r, t)=f
0(v,r,t)+v. f
1(v,r,t)+vvf
2(v,r,t)+vvvf
3(v,r,t)+ wheref
k
(k=2, 3) arek-th order irreducible tensors, the Rosenbluth potential functions and the Fokker-Planck collision term are expanded in a similar sum. Collisions termsJ
Fk
(k=0, 1, 2) and the equations forf
k
(k=0, 1, 2) for the case of the Coulomb interactions are also determined.Technická 2, Praha 6, Czechoslovakia.The autor wishes to express his thanks to Prof. J. Kracík, DrSc. for valuable advice and suggestion. 相似文献
14.
For semiflows generated by ordinary differential equations v’=A(t)v admitting a nonuniform exponential dichotomy, we show that for any sufficiently small perturbation f there exist smooth stable and unstable manifolds for the perturbed equation v’=A(t)v+f(t,v). As an application, we establish the existence of invariant manifolds for the nonuniformly hyperbolic trajectories of a semiflow. In particular, we obtain smooth invariant manifolds for a class of vector fields that need not be C1+α for any α ∈ (0,1). To the best of our knowledge no similar statement was obtained before in the nonuniformly hyperbolic setting. We emphasize that we do not need to assume the existence of an exponential dichotomy, but only the existence of a nonuniform exponential dichotomy, with sufficiently small nonuniformity when compared to the Lyapunov exponents of the original linear equation. Furthermore, for example in the case of stable manifolds, we only need to assume that there exist negative Lyapunov exponents, while we also allow zero exponents. Our proof of the smoothness of the invariant manifolds is based on the construction of an invariant family of cones.Supported by the Center for Mathematical Analysis, Geometry, and Dynamical Systems, and through Fundação para a Ciência e a Tecnologia by Program POCTI/FEDER, Program POSI, and the grant SFRH/BPD/14404/2003. 相似文献
15.
The transport behavior of a migrating particle in a disordered medium is exhibited in the solution of a transport equation
derived from a coupled continuous time random walk (CTRW). A core aspect of CTRW is the spectrum of transitions in displacement
s and time t, ψ(s,t), that characterizes the disordered system, which determine the transport. In many applications the CTRW approach has successfully
accounted for the anomalous or non-Fickian nature of the particle plume propagation based on a power-law dependence ψ(t) in a decoupled p(s)ψ(t) approximation to ψ(s,t). For example, this power-law dependence in t derives from the complex Darcy flow fields in geological formations. Recently, the fully coupled CTRW was analyzed using
a particle tracking approach, demonstrating that the decoupled approximation is valid only for a compact distribution of s. In this paper we solve the nonlocal-in-time transport equation with a ψ(s,t) containing a power-law dependence in both s (a Lévy-like distribution) and t, which necessitates the strong s,t coupling. We show enhanced transport behavior (relative to the plume propagation behavior reported in the literature) that
derives from the rare large displacements in s (limited by the transition t). The interplay between the two coupled power laws is clearly shown in the changes in the breakthrough curves in the arrival
times, dispersion and dependence on the velocity (v=s/t) distribution. Similar enhancements are exhibited in the particle tracking results. 相似文献
16.
We study the evaporation of stars from globular clusters using the simplified Chandrasekhar model [S. Chandrasekhar, Dynamical friction. II. The rate of escape of stars from clusters and the evidence for the operation of dynamical friction, Astrophys. J. 97 (1943) 263]. This is an analytically tractable model giving reasonable agreement with more sophisticated models that require complicated numerical integrations. In the Chandrasekhar model: (i) the stellar system is assumed to be infinite and homogeneous (ii) the evolution of the velocity distribution of stars f(v,t) is governed by a Fokker-Planck equation, the so-called Kramers-Chandrasekhar equation (iii) the velocities |v| that are above a threshold value R>0 (escape velocity) are not counted in the statistical distribution of the system. In fact, high velocity stars leave the system, due to free evaporation or to the attraction of a neighboring galaxy (tidal effects). Accordingly, the total mass and energy of the system decrease in time. If the star dynamics is described by the Kramers-Chandrasekhar equation, the mass decreases to zero exponentially rapidly. Our goal is to obtain non-perturbative analytical results that complement the seminal studies of Chandrasekhar, Michie and King valid for large times t→+∞ and large escape velocities R→+∞. In particular, we obtain an exact semi-explicit solution of the Kramers-Chandrasekhar equation with the absorbing boundary condition f(R,t)=0. We use it to obtain an explicit expression of the mass loss at any time t when R→+∞. We also derive an exact integral equation giving the exponential evaporation rate λ(R), and the corresponding eigenfunction fλ(v), when t→+∞ for any sufficiently large value of the escape velocity R. For R→+∞, we obtain an explicit expression of the evaporation rate that refines the Chandrasekhar results. More generally, our results can have applications in other contexts where the Kramers equation applies, like the classical diffusion of particles over a barrier of potential (Kramers problem). 相似文献
17.
Anil Kumar Yadav Vineet K. Yadav Lallan Yadav 《International Journal of Theoretical Physics》2009,48(2):568-578
Cylindrically symmetric inhomogeneous string cosmological models are investigated in presence of string fluid as a source
of matter. To get the three types of exact solutions of Einstein’s field equations we assume A=f(x)k(t), B=g(x)ℓ(t) and C=h(x)ℓ(t). Some physical and geometric aspects of the models are discussed. 相似文献
18.
For r(0,1), let Z
r
={xR
2
|dist(x,Z
2)>r/2} and define τ
r
(x,v)=inf{t>0|x+tv∂Z
r
}. Let Φ
r
(t) be the probability that τ
r
(x,v)≥t for x and v uniformly distributed in Z
r
and §1 respectively. We prove in this paper that
as t→+∞. This result improves upon the bounds on Φ
r
in Bourgain-Golse-Wennberg [Commun. Math. Phys. 190, 491–508 (1998)]. We also discuss the applications of this result in the context of kinetic theory.
Received: 2 August 2002 / Accepted: 27 November 2002
Published online: 14 April 2003
Communicated by G. Gallavotti 相似文献
19.
We present the analytical solution to the linear evolution equation of a one component Friedmann perturbation using an equation of state of the form p = (1/3)μσ2(t), where μ is the mass density and σ(t) is the root mean square (rms) velocity in the matter dominated epoch. It is assumed that this rms velocity depends only on the time coordinate and decreases as 1/a, a being the expansion factor of the Friedmann background. The evolution equations are written for scales below the horizon using the longitudinal gauge. The general solution, in the coordinate space, of the evolution equation for the scalar mode is obtained. In the case of spherical symmetry, this solution is expressed in terms of unidimensional integrals of the initial conditions: the initial values of the Newtonian potential and its first time derivative. This perfect fluid solution is a good approximation to the evolution of warm dark matter perturbations obtained by solving the Vlasov’s equation for collisionless particles. 相似文献
20.
Vladimir Rabinovich 《Russian Journal of Mathematical Physics》2012,19(1):107-120
The propagation of electromagnetic waves issued by modulated moving sources of the form j( t,x ) = a( t )e - iw0 t [(x)\dot]0 ( t )d( x - x0 ( t ) )j\left( {t,x} \right) = a\left( t \right)e^{ - i\omega _0 t} \dot x_0 \left( t \right)\delta \left( {x - x_0 \left( t \right)} \right) is considered, where j(t, x) stands for the current density vector, x = (x
1, x
2, x
3) ∈ ℝ3 for the space variables, t ∈ ℝ for time, t → x
0(t) ∈ ℝ3 for the vector function defining the motion of the source, ω
0 for the eigenfrequency of the source, a(t) for a narrow-band amplitude, and δ for the standard δ function. Suppose that the media under consideration are dispersive. This means that the electric and magnetic permittivity
ɛ(ω), μ(ω) depends on the frequency ω. We obtain a representation of electromagnetic fields in the form of time-frequency oscillating integrals whose phase contains
a large parameter λ > 0 characterizing the slowness of the change of the amplitude a(t) and the velocity [(x)\dot]0 ( t )\dot x_0 \left( t \right) and a large distance between positions of the source and the receiver. Applying the two-dimensional stationary phase method
to the integrals, we obtain explicit formulas for the electromagnetic field and for the Doppler effects. As an application
of our approach, we consider the propagation of electromagnetic waves produced by moving source in a cold nonmagnetized plasma
and the Cherenkov radiation in dispersive media. 相似文献