278 research outputs found
Models of stochastic gene expression and Weyl algebra
International audienceThis paper presents a symbolic algorithm for computing the ODE systems which describe the evolution of the moments associated to a chemical reaction system, considered from a stochastic point of view. The algorithm, which is formulated in the Weyl algebra, seems more efficient than the corresponding method, based on partial derivatives. In particular, an efficient method for handling conservation laws is presented. The output of the algorithm can be used for a further investigation of the system behaviour, by numerical methods. Relevant examples are carried out
From quantum stochastic differential equations to Gisin-Percival state diffusion
Starting from the quantum stochastic differential equations of Hudson and
Parthasarathy (Comm. Math. Phys. 93, 301 (1984)) and exploiting the
Wiener-Ito-Segal isomorphism between the Boson Fock reservoir space
and
the Hilbert space , where is the Wiener probability measure of
a complex -dimensional vector-valued standard Brownian motion
, we derive a non-linear stochastic Schrodinger
equation describing a classical diffusion of states of a quantum system, driven
by the Brownian motion . Changing this Brownian motion by an
appropriate Girsanov transformation, we arrive at the Gisin-Percival state
diffusion equation (J. Phys. A, 167, 315 (1992)). This approach also yields an
explicit solution of the Gisin-Percival equation, in terms of the
Hudson-Parthasarathy unitary process and a radomized Weyl displacement process.
Irreversible dynamics of system density operators described by the well-known
Gorini-Kossakowski-Sudarshan-Lindblad master equation is unraveled by
coarse-graining over the Gisin-Percival quantum state trajectories.Comment: 28 pages, one pdf figure. An error in the multiplying factor in Eq.
(102) corrected. To appear in Journal of Mathematical Physic
Chemical Reaction Systems, Computer Algebra and Systems Biology
International audienceIn this invited paper, we survey some of the results obtained in the computer algebra team of Lille, in the domain of systems biology. So far, we have mostly focused on models (systems of equations) arising from generalized chemical reaction systems. Eight years ago, our team was involved in a joint project, with physicists and biologists, on the modeling problem of the circadian clock of the green algae Ostreococcus tauri. This cooperation led us to different algorithms dedicated to the reduction problem of the deterministic models of chemical reaction systems. More recently, we have been working more tightly with another team of our lab, the BioComputing group, interested by the stochastic dynamics of chemical reaction systems. This cooperation led us to efficient algorithms for building the ODE systems which define the statistical moments associated to these dynamics. Most of these algorithms were implemented in the MAPLE computer algebra software. We have chosen to present them through the corresponding MAPLE packages
Combinatorial Conversion and Moment Bisimulation for Stochastic Rewriting Systems
We develop a novel method to analyze the dynamics of stochastic rewriting
systems evolving over finitary adhesive, extensive categories. Our formalism is
based on the so-called rule algebra framework and exhibits an intimate
relationship between the combinatorics of the rewriting rules (as encoded in
the rule algebra) and the dynamics which these rules generate on observables
(as encoded in the stochastic mechanics formalism). We introduce the concept of
combinatorial conversion, whereby under certain technical conditions the
evolution equation for (the exponential generating function of) the statistical
moments of observables can be expressed as the action of certain differential
operators on formal power series. This permits us to formulate the novel
concept of moment-bisimulation, whereby two dynamical systems are compared in
terms of their evolution of sets of observables that are in bijection. In
particular, we exhibit non-trivial examples of graphical rewriting systems that
are moment-bisimilar to certain discrete rewriting systems (such as branching
processes or the larger class of stochastic chemical reaction systems). Our
results point towards applications of a vast number of existing
well-established exact and approximate analysis techniques developed for
chemical reaction systems to the far richer class of general stochastic
rewriting systems
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