3,604 research outputs found
How accurate are the non-linear chemical Fokker-Planck and chemical Langevin equations?
The chemical Fokker-Planck equation and the corresponding chemical Langevin
equation are commonly used approximations of the chemical master equation.
These equations are derived from an uncontrolled, second-order truncation of
the Kramers-Moyal expansion of the chemical master equation and hence their
accuracy remains to be clarified. We use the system-size expansion to show that
chemical Fokker-Planck estimates of the mean concentrations and of the variance
of the concentration fluctuations about the mean are accurate to order
for reaction systems which do not obey detailed balance and at
least accurate to order for systems obeying detailed balance,
where is the characteristic size of the system. Hence the chemical
Fokker-Planck equation turns out to be more accurate than the linear-noise
approximation of the chemical master equation (the linear Fokker-Planck
equation) which leads to mean concentration estimates accurate to order
and variance estimates accurate to order . This
higher accuracy is particularly conspicuous for chemical systems realized in
small volumes such as biochemical reactions inside cells. A formula is also
obtained for the approximate size of the relative errors in the concentration
and variance predictions of the chemical Fokker-Planck equation, where the
relative error is defined as the difference between the predictions of the
chemical Fokker-Planck equation and the master equation divided by the
prediction of the master equation. For dimerization and enzyme-catalyzed
reactions, the errors are typically less than few percent even when the
steady-state is characterized by merely few tens of molecules.Comment: 39 pages, 3 figures, accepted for publication in J. Chem. Phy
Floquet-Markov description of the parametrically driven, dissipative harmonic quantum oscillator
Using the parametrically driven harmonic oscillator as a working example, we
study two different Markovian approaches to the quantum dynamics of a
periodically driven system with dissipation. In the simpler approach, the
driving enters the master equation for the reduced density operator only in the
Hamiltonian term. An improved master equation is achieved by treating the
entire driven system within the Floquet formalism and coupling it to the
reservoir as a whole. The different ensuing evolution equations are compared in
various representations, particularly as Fokker-Planck equations for the Wigner
function. On all levels of approximation, these evolution equations retain the
periodicity of the driving, so that their solutions have Floquet form and
represent eigenfunctions of a non-unitary propagator over a single period of
the driving. We discuss asymptotic states in the long-time limit as well as the
conservative and the high-temperature limits. Numerical results obtained within
the different Markov approximations are compared with the exact path-integral
solution. The application of the improved Floquet-Markov scheme becomes
increasingly important when considering stronger driving and lower
temperatures.Comment: 29 pages, 7 figure
Adaptive finite element method assisted by stochastic simulation of chemical systems
Stochastic models of chemical systems are often analysed by solving the corresponding\ud
Fokker-Planck equation which is a drift-diffusion partial differential equation for the probability\ud
distribution function. Efficient numerical solution of the Fokker-Planck equation requires adaptive mesh refinements. In this paper, we present a mesh refinement approach which makes use of a stochastic simulation of the underlying chemical system. By observing the stochastic trajectory for a relatively short amount of time, the areas of the state space with non-negligible probability density are identified. By refining the finite element mesh in these areas, and coarsening elsewhere, a suitable mesh is constructed and used for the computation of the probability density
An accurate scheme to solve cluster dynamics equations using a Fokker-Planck approach
We present a numerical method to accurately simulate particle size
distributions within the formalism of rate equation cluster dynamics. This
method is based on a discretization of the associated Fokker-Planck equation.
We show that particular care has to be taken to discretize the advection part
of the Fokker-Planck equation, in order to avoid distortions of the
distribution due to numerical diffusion. For this purpose we use the
Kurganov-Noelle-Petrova scheme coupled with the monotonicity-preserving
reconstruction MP5, which leads to very accurate results. The interest of the
method is highlighted on the case of loop coarsening in aluminum. We show that
the choice of the models to describe the energetics of loops does not
significantly change the normalized loop distribution, while the choice of the
models for the absorption coefficients seems to have a significant impact on
it
A functional calculus for the magnetization dynamics
A functional calculus approach is applied to the derivation of evolution
equations for the moments of the magnetization dynamics of systems subject to
stochastic fields. It allows us to derive a general framework for obtaining the
master equation for the stochastic magnetization dynamics, that is applied to
both, Markovian and non-Markovian dynamics. The formalism is applied for
studying different kinds of interactions, that are of practical relevance and
hierarchies of evolution equations for the moments of the distribution of the
magnetization are obtained. In each case, assumptions are spelled out, in order
to close the hierarchies. These closure assumptions are tested by extensive
numerical studies, that probe the validity of Gaussian or non--Gaussian closure
Ans\"atze.Comment: 17 pages, 5 figure
Phase Space Representation for Open Quantum Systems within the Lindblad Theory
The Lindblad master equation for an open quantum system with a Hamiltonian
containing an arbitrary potential is written as an equation for the Wigner
distribution function in the phase space representation. The time derivative of
this function is given by a sum of three parts: the classical one, the quantum
corrections and the contribution due to the opening of the system. In the
particular case of a harmonic oscillator, quantum corrections do not exist.Comment: 19 pages, Latex, accepted for publication in Int. J. Mod. Phys.
Velocity-Dependent Friction and Diffusion for Grains in Neutral Gases, Dusty Plasmas and Active Systems
A self-consistent and universal description of friction and diffusion for
Brownian particles (grains) in different systems, as a gas with Boltzmann
collisions, dusty plasma with ion absorption by grains, and for active
particles (e.g., cells in biological systems) is suggested on the basis of the
appropriate Fokker-Planck equation. Restrictions for application of the
Fokker-Planck equation to the problem of velocity-dependent friction and
diffusion coefficients are found. General description for this coefficient is
formulated on the basis of master equation. Relation of the diffusion
coefficient in the coordinate and velocity spaces is found for active (capable
to transfer momentum to the ambient media) and passive particles in the
framework of the Fokker-Planck equation.
The problem of anomalous space diffusion is formulated on the basis of the
appropriate probability transition (PT) function. The method of partial
differentiation is avoided to construct the correct probability distributions
for arbitrary distances, what is important for applications to different
stochastic problems. Generale equation for time-dependent PT function is
formulated and discussed.
Generalized friction in the velocity space is determined and applied to
describe the friction force itself as well as the drag force in the case of a
non-zero driven ion velocity in plasmas. The negative friction due to ion
scattering on grains exists and can be realized for the appropriate
experimental conditions.Comment: 21 page
Interrelations between Stochastic Equations for Systems with Pair Interactions
Several types of stochastic equations are important in thermodynamics,
chemistry, evolutionary biology, population dynamics and quantitative social
science. For systems with pair interactions four different types of equations
are derived, starting from a master equation for the state space: First,
general mean value and (co)variance equations. Second, Boltzmann-like
equations. Third, a master equation for the configuration space allowing
transition rates which depend on the occupation numbers of the states. Fourth,
a Fokker-Planck equation and a ``Boltzmann-Fokker-Planck equation''. The
interrelations of these equations and the conditions for their validity are
worked out clearly. A procedure for a selfconsistent solution of the nonlinear
equations is proposed. Generalizations to interactions between an arbitrary
number of systems are discussed.Comment: For related work see
http://www.theo2.physik.uni-stuttgart.de/helbing.htm
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