21,887 research outputs found
On the "generalized Generalized Langevin Equation"
In molecular dynamics simulations and single molecule experiments,
observables are usually measured along dynamic trajectories and then averaged
over an ensemble ("bundle") of trajectories. Under stationary conditions, the
time-evolution of such averages is described by the generalized Langevin
equation. In contrast, if the dynamics is not stationary, it is not a priori
clear which form the equation of motion for an averaged observable has. We
employ the formalism of time-dependent projection operator techniques to derive
the equation of motion for a non-equilibrium trajectory-averaged observable as
well as for its non-stationary auto-correlation function. The equation is
similar in structure to the generalized Langevin equation, but exhibits a
time-dependent memory kernel as well as a fluctuating force that implicitly
depends on the initial conditions of the process. We also derive a relation
between this memory kernel and the autocorrelation function of the fluctuating
force that has a structure similar to a fluctuation-dissipation relation. In
addition, we show how the choice of the projection operator allows to relate
the Taylor expansion of the memory kernel to data that is accessible in MD
simulations and experiments, thus allowing to construct the equation of motion.
As a numerical example, the procedure is applied to Brownian motion initialized
in non-equilibrium conditions, and is shown to be consistent with direct
measurements from simulations
Mode-coupling theory of sheared dense granular liquids
Mode-coupling theory (MCT) of sheared dense granular liquids %in the vicinity
of jamming transition is formulated. Starting from the Liouville equation of
granular particles, the generalized Langevin equation is derived with the aid
of the projection operator technique. The MCT equation for the density
correlation function obtained from the generalized Langevin equation is almost
equivalent to MCT equation for elastic particles under the shear. It is found
that there should be the plateau in the density correlation function.Comment: 22 pages, 2 figure. to be published in Progress of Theoretical
Physics. to be published in Progress of Theoretical Physic
Brownian markets
Financial market dynamics is rigorously studied via the exact generalized
Langevin equation. Assuming market Brownian self-similarity, the market return
rate memory and autocorrelation functions are derived, which exhibit an
oscillatory-decaying behavior with a long-time tail, similar to empirical
observations. Individual stocks are also described via the generalized Langevin
equation. They are classified by their relation to the market memory as heavy,
neutral and light stocks, possessing different kinds of autocorrelation
functions
Numerical study of ergodicity for the overdamped Generalized Langevin Equation with fractional noise
The Generalized Langevin Equation, in history, arises as a natural fix for
the rather traditional Langevin equation when the random force is no longer
memoryless. It has been proved that with fractional Gaussian noise (fGn) mostly
considered by biologists, the overdamped Generalized Langevin equation
satisfying fluctuation-dissipation theorem can be written as a fractional
stochastic differential equation (FSDE). While the ergodicity is clear for
linear forces, it remains less transparent for nonlinear forces. In this work,
we present both a direct and a fast algorithm respectively to this FSDE model.
The strong orders of convergence are proved for both schemes, where the role of
the memory effects can be clearly observed. We verify the convergence theorems
using linear forces, and then present the ergodicity study of the double well
potentials in both 1D and 2D setups
Theory of the Relativistic Brownian Motion. The (1+1)-Dimensional Case
We construct a theory for the 1+1-dimensional Brownian motion in a viscous
medium, which is (i) consistent with Einstein's theory of special relativity,
and (ii) reduces to the standard Brownian motion in the Newtonian limit case.
In the first part of this work the classical Langevin equations of motion,
governing the nonrelativistic dynamics of a free Brownian particle in the
presence of a heat bath (white noise), are generalized in the framework of
special relativity. Subsequently, the corresponding relativistic Langevin
equations are discussed in the context of the generalized Ito (pre-point
discretization rule) vs. the Stratonovich (mid-point discretization rule)
dilemma: It is found that the relativistic Langevin equation in the
Haenggi-Klimontovich interpretation (with the post-point discretization rule)
is the only one that yields agreement with the relativistic Maxwell
distribution. Numerical results for the relativistic Langevin equation of a
free Brownian particle are presented.Comment: see cond-mat/0607082 for an improved theor
Equilibration problem for the generalized Langevin equation
We consider the problem of equilibration of a single oscillator system with
dynamics given by the generalized Langevin equation. It is well-known that this
dynamics can be obtained if one considers a model where the single oscillator
is coupled to an infinite bath of harmonic oscillators which are initially in
equilibrium. Using this equivalence we first determine the conditions necessary
for equilibration for the case when the system potential is harmonic. We then
give an example with a particular bath where we show that, even for parameter
values where the harmonic case always equilibrates, with any finite amount of
nonlinearity the system does not equilibrate for arbitrary initial conditions.
We understand this as a consequence of the formation of nonlinear localized
excitations similar to the discrete breather modes in nonlinear lattices.Comment: 5 pages, 2 figure
- …