2,500 research outputs found
Generalized Fokker-Planck equation, Brownian motion, and ergodicity
Microscopic theory of Brownian motion of a particle of mass in a bath of
molecules of mass is considered beyond lowest order in the mass ratio
. The corresponding Langevin equation contains nonlinear corrections to
the dissipative force, and the generalized Fokker-Planck equation involves
derivatives of order higher than two. These equations are derived from first
principles with coefficients expressed in terms of correlation functions of
microscopic force on the particle. The coefficients are evaluated explicitly
for a generalized Rayleigh model with a finite time of molecule-particle
collisions. In the limit of a low-density bath, we recover the results obtained
previously for a model with instantaneous binary collisions. In general case,
the equations contain additional corrections, quadratic in bath density,
originating from a finite collision time. These corrections survive to order
and are found to make the stationary distribution non-Maxwellian.
Some relevant numerical simulations are also presented
Stochastic dynamics beyond the weak coupling limit: thermalization
We discuss the structure and asymptotic long-time properties of coupled
equations for the moments of a Brownian particle's momentum derived
microscopically beyond the lowest approximation in the weak coupling parameter.
Generalized fluctuation-dissipation relations are derived and shown to ensure
convergence to thermal equilibrium at any order of perturbation theory.Comment: 6+ page
Relaxation Phenomena in a System of Two Harmonic Oscillators
We study the process by which quantum correlations are created when an
interaction Hamiltonian is repeatedly applied to a system of two harmonic
oscillators for some characteristic time interval. We show that, for the case
where the oscillator frequencies are equal, the initial Maxwell-Boltzmann
distributions of the uncoupled parts evolve to a new equilibrium
Maxwell-Boltzmann distribution through a series of transient Maxwell-Boltzmann
distributions. Further, we discuss why the equilibrium reached when the two
oscillator frequencies are unequal, is not a thermal one. All the calculations
are exact and the results are obtained through an iterative process, without
using perturbation theory.Comment: 22 pages, 6 Figures, Added contents, to appear in PR
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
Stochastic oscillations in models of epidemics on a network of cities
We carry out an analytic investigation of stochastic oscillations in a
susceptible-infected-recovered model of disease spread on a network of
cities. In the model a fraction of individuals from city commute
to city , where they may infect, or be infected by, others. Starting from a
continuous time Markov description of the model the deterministic equations,
which are valid in the limit when the population of each city is infinite, are
recovered. The stochastic fluctuations about the fixed point of these equations
are derived by use of the van Kampen system-size expansion. The fixed point
structure of the deterministic equations is remarkably simple: a unique
non-trivial fixed point always exists and has the feature that the fraction of
susceptible, infected and recovered individuals is the same for each city
irrespective of its size. We find that the stochastic fluctuations have an
analogously simple dynamics: all oscillations have a single frequency, equal to
that found in the one city case. We interpret this phenomenon in terms of the
properties of the spectrum of the matrix of the linear approximation of the
deterministic equations at the fixed point.Comment: 13 pages, 7 figure
Fluctuation spectrum of quasispherical membranes with force-dipole activity
The fluctuation spectrum of a quasi-spherical vesicle with active membrane
proteins is calculated. The activity of the proteins is modeled as the proteins
pushing on their surroundings giving rise to non-local force distributions.
Both the contributions from the thermal fluctuations of the active protein
densities and the temporal noise in the individual active force distributions
of the proteins are taken into account. The noise in the individual force
distributions is found to become significant at short wavelengths.Comment: 9 pages, 2 figures, minor changes and addition
On Nonlinear Diffusion with Multiplicative Noise
Nonlinear diffusion is studied in the presence of multiplicative noise. The
nonlinearity can be viewed as a ``wall'' limiting the motion of the diffusing
field. A dynamic phase transition occurs when the system ``unbinds'' from the
wall. Two different universality classes, corresponding to the cases of an
``upper'' and a ``lower'' wall, are identified and their critical properties
are characterized. While the lower wall problem can be understood by applying
the knowledge of linear diffusion with multiplicative noise, the upper wall
problem exhibits an anomaly due to nontrivial dynamics in the vicinity of the
wall. Broad power-law distribution is obtained throughout the bound phase.Comment: 4 pages, LaTeX, text and figures also available at
http://matisse.ucsd.edu/~hw
Field theoretic formulation of a mode-coupling equation for colloids
The only available quantitative description of the slowing down of the
dynamics upon approaching the glass transition has been, so far, the
mode-coupling theory, developed in the 80's by G\"otze and collaborators. The
standard derivation of this theory does not result from a systematic expansion.
We present a field theoretic formulation that arrives at very similar
mode-coupling equation but which is based on a variational principle and on a
controlled expansion in a small dimensioneless parameter. Our approach applies
to such physical systems as colloids interacting via a mildly repulsive
potential. It can in principle, with moderate efforts, be extended to higher
orders and to multipoint correlation functions
Cluster approximations for infection dynamics on random networks
In this paper, we consider a simple stochastic epidemic model on large
regular random graphs and the stochastic process that corresponds to this
dynamics in the standard pair approximation. Using the fact that the nodes of a
pair are unlikely to share neighbors, we derive the master equation for this
process and obtain from the system size expansion the power spectrum of the
fluctuations in the quasi-stationary state. We show that whenever the pair
approximation deterministic equations give an accurate description of the
behavior of the system in the thermodynamic limit, the power spectrum of the
fluctuations measured in long simulations is well approximated by the
analytical power spectrum. If this assumption breaks down, then the cluster
approximation must be carried out beyond the level of pairs. We construct an
uncorrelated triplet approximation that captures the behavior of the system in
a region of parameter space where the pair approximation fails to give a good
quantitative or even qualitative agreement. For these parameter values, the
power spectrum of the fluctuations in finite systems can be computed
analytically from the master equation of the corresponding stochastic process.Comment: the notation has been changed; Ref. [26] and a new paragraph in
Section IV have been adde
Steady-state fluctuations of a genetic feedback loop:an exact solution
Genetic feedback loops in cells break detailed balance and involve
bimolecular reactions; hence exact solutions revealing the nature of the
stochastic fluctuations in these loops are lacking. We here consider the master
equation for a gene regulatory feedback loop: a gene produces protein which
then binds to the promoter of the same gene and regulates its expression. The
protein degrades in its free and bound forms. This network breaks detailed
balance and involves a single bimolecular reaction step. We provide an exact
solution of the steady-state master equation for arbitrary values of the
parameters, and present simplified solutions for a number of special cases. The
full parametric dependence of the analytical non-equilibrium steady-state
probability distribution is verified by direct numerical solution of the master
equations. For the case where the degradation rate of bound and free protein is
the same, our solution is at variance with a previous claim of an exact
solution (Hornos et al, Phys. Rev. E {\bf 72}, 051907 (2005) and subsequent
studies). We show explicitly that this is due to an unphysical formulation of
the underlying master equation in those studies.Comment: 31 pages, 3 figures. Accepted for publication in the Journal of
Chemical Physics (2012
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