205 research outputs found
Distribution of the Oscillation Period in the Underdamped One Dimensional Sinai Model
We consider the Newtonian dynamics of a massive particle in a one dimemsional
random potential which is a Brownian motion in space. This is the zero
temperature nondamped Sinai model. As there is no dissipation the particle
oscillates between two turning points where its kinetic energy becomes zero.
The period of oscillation is a random variable fluctuating from sample to
sample of the random potential. We compute the probability distribution of this
period exactly and show that it has a power law tail for large period, P(T)\sim
T^{-5/3} and an essential singluarity P(T)\sim \exp(-1/T) as T\to 0. Our exact
results are confirmed by numerical simulations and also via a simple scaling
argument.Comment: 9 pages LateX, 2 .eps figure
Quenching and Propagation of Combustion Without Ignition Temperature Cutoff
We study a reaction-diffusion equation in the cylinder , with combustion-type reaction term without
ignition temperature cutoff, and in the presence of a periodic flow. We show
that if the reaction function decays as a power of larger than three as
and the initial datum is small, then the flame is extinguished -- the
solution quenches. If, on the other hand, the power of decay is smaller than
three or initial datum is large, then quenching does not happen, and the
burning region spreads linearly in time. This extends results of
Aronson-Weinberger for the no-flow case. We also consider shear flows with
large amplitude and show that if the reaction power-law decay is larger than
three and the flow has only small plateaux (connected domains where it is
constant), then any compactly supported initial datum is quenched when the flow
amplitude is large enough (which is not true if the power is smaller than three
or in the presence of a large plateau). This extends results of
Constantin-Kiselev-Ryzhik for combustion with ignition temperature cutoff. Our
work carries over to the case , when
the critical power is , as well as to certain non-periodic flows
Langevin Equation for the Density of a System of Interacting Langevin Processes
We present a simple derivation of the stochastic equation obeyed by the
density function for a system of Langevin processes interacting via a pairwise
potential. The resulting equation is considerably different from the
phenomenological equations usually used to describe the dynamics of non
conserved (Model A) and conserved (Model B) particle systems. The major feature
is that the spatial white noise for this system appears not additively but
multiplicatively. This simply expresses the fact that the density cannot
fluctuate in regions devoid of particles. The steady state for the density
function may however still be recovered formally as a functional integral over
the coursed grained free energy of the system as in Models A and B.Comment: 6 pages, latex, no figure
Control of Multi-level Voltage States in a Hysteretic SQUID Ring-Resonator System
In this paper we study numerical solutions to the quasi-classical equations
of motion for a SQUID ring-radio frequency (rf) resonator system in the regime
where the ring is highly hysteretic. In line with experiment, we show that for
a suitable choice of of ring circuit parameters the solutions to these
equations of motion comprise sets of levels in the rf voltage-current dynamics
of the coupled system. We further demonstrate that transitions, both up and
down, between these levels can be controlled by voltage pulses applied to the
system, thus opening up the possibility of high order (e.g. 10 state),
multi-level logic and memory.Comment: 8 pages, 9 figure
The Measure-theoretic Identity Underlying Transient Fluctuation Theorems
We prove a measure-theoretic identity that underlies all transient
fluctuation theorems (TFTs) for entropy production and dissipated work in
inhomogeneous deterministic and stochastic processes, including those of Evans
and Searles, Crooks, and Seifert. The identity is used to deduce a tautological
physical interpretation of TFTs in terms of the arrow of time, and its
generality reveals that the self-inverse nature of the various trajectory and
process transformations historically relied upon to prove TFTs, while necessary
for these theorems from a physical standpoint, is not necessary from a
mathematical one. The moment generating functions of thermodynamic variables
appearing in the identity are shown to converge in general only in a vertical
strip in the complex plane, with the consequence that a TFT that holds over
arbitrary timescales may fail to give rise to an asymptotic fluctuation theorem
for any possible speed of the corresponding large deviation principle. The case
of strongly biased birth-death chains is presented to illustrate this
phenomenon. We also discuss insights obtained from our measure-theoretic
formalism into the results of Saha et. al. on the breakdown of TFTs for driven
Brownian particles
Quantum response of dephasing open systems
We develop a theory of adiabatic response for open systems governed by
Lindblad evolutions. The theory determines the dependence of the response
coefficients on the dephasing rates and allows for residual dissipation even
when the ground state is protected by a spectral gap. We give quantum response
a geometric interpretation in terms of Hilbert space projections: For a two
level system and, more generally, for systems with suitable functional form of
the dephasing, the dissipative and non-dissipative parts of the response are
linked to a metric and to a symplectic form. The metric is the Fubini-Study
metric and the symplectic form is the adiabatic curvature. When the metric and
symplectic structures are compatible the non-dissipative part of the inverse
matrix of response coefficients turns out to be immune to dephasing. We give
three examples of physical systems whose quantum states induce compatible
metric and symplectic structures on control space: The qubit, coherent states
and a model of the integer quantum Hall effect.Comment: Article rewritten, two appendices added. 16 pages, 2 figure
Pathwise Sensitivity Analysis in Transient Regimes
The instantaneous relative entropy (IRE) and the corresponding instanta-
neous Fisher information matrix (IFIM) for transient stochastic processes are
pre- sented in this paper. These novel tools for sensitivity analysis of
stochastic models serve as an extension of the well known relative entropy rate
(RER) and the corre- sponding Fisher information matrix (FIM) that apply to
stationary processes. Three cases are studied here, discrete-time Markov
chains, continuous-time Markov chains and stochastic differential equations. A
biological reaction network is presented as a demonstration numerical example
Scaling, renormalization and statistical conservation laws in the Kraichnan model of turbulent advection
We present a systematic way to compute the scaling exponents of the structure
functions of the Kraichnan model of turbulent advection in a series of powers
of , adimensional coupling constant measuring the degree of roughness of
the advecting velocity field. We also investigate the relation between standard
and renormalization group improved perturbation theory. The aim is to shed
light on the relation between renormalization group methods and the statistical
conservation laws of the Kraichnan model, also known as zero modes.Comment: Latex (11pt) 43 pages, 22 figures (Feynman diagrams). The reader
interested in the technical details of the calculations presented in the
paper may want to visit:
http://www.math.helsinki.fi/mathphys/paolo_files/passive_scalar/passcal.htm
Bayesian inference of biochemical kinetic parameters using the linear noise approximation
Background
Fluorescent and luminescent gene reporters allow us to dynamically quantify changes in molecular species concentration over time on the single cell level. The mathematical modeling of their interaction through multivariate dynamical models requires the deveopment of effective statistical methods to calibrate such models against available data. Given the prevalence of stochasticity and noise in biochemical systems inference for stochastic models is of special interest. In this paper we present a simple and computationally efficient algorithm for the estimation of biochemical kinetic parameters from gene reporter data.
Results
We use the linear noise approximation to model biochemical reactions through a stochastic dynamic model which essentially approximates a diffusion model by an ordinary differential equation model with an appropriately defined noise process. An explicit formula for the likelihood function can be derived allowing for computationally efficient parameter estimation. The proposed algorithm is embedded in a Bayesian framework and inference is performed using Markov chain Monte Carlo.
Conclusion
The major advantage of the method is that in contrast to the more established diffusion approximation based methods the computationally costly methods of data augmentation are not necessary. Our approach also allows for unobserved variables and measurement error. The application of the method to both simulated and experimental data shows that the proposed methodology provides a useful alternative to diffusion approximation based methods
Model Uncertainty, Ambiguity and the Precautionary Principle: Implications for Biodiversity Management
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