58 research outputs found
The Smoluchowski-Kramers limit of stochastic differential equations with arbitrary state-dependent friction
We study a class of systems of stochastic differential equations describing
diffusive phenomena. The Smoluchowski-Kramers approximation is used to describe
their dynamics in the small mass limit. Our systems have arbitrary
state-dependent friction and noise coefficients. We identify the limiting
equation and, in particular, the additional drift term that appears in the
limit is expressed in terms of the solution to a Lyapunov matrix equation. The
proof uses a theory of convergence of stochastic integrals developed by Kurtz
and Protter. The result is sufficiently general to include systems driven by
both white and Ornstein-Uhlenbeck colored noises. We discuss applications of
the main theorem to several physical phenomena, including the experimental
study of Brownian motion in a diffusion gradient.Comment: This paper has been corrected from a previous version. Author Austin
McDaniel has been added. Lemma 2 has been rewritten, Lemma 3 added, previous
version's Lemma 3 moved to Lemma 4. 20 pages, 1 figur
Social networks in the single cell
Plant mitochondrial DNA (mtDNA) can become damaged in many ways. A major repair mechanism is homologous recombination, which requires an undamaged DNA template. Presumably, this template comes from a different mitochondrion in the same cell. Plant mitochondria undergo fission and fusion to form transient networks which could allow the exchange of genetic information. To test this hypothesis, Chustecki et al. (2022) used msh1 mutants with defective DNA repair, and showed that mitochondrial interactions increased, revealing a link between the physical and genetic behavior of mitochondria
Noise-induced drift in stochastic differential equations with arbitrary friction and diffusion in the Smoluchowski-Kramers limit
We consider the dynamics of systems with arbitrary friction and diffusion.
These include, as a special case, systems for which friction and diffusion are
connected by Einstein fluctuation-dissipation relation, e.g. Brownian motion.
We study the limit where friction effects dominate the inertia, i.e. where the
mass goes to zero (Smoluchowski-Kramers limit). {Using the It\^o stochastic
integral convention,} we show that the limiting effective Langevin equations
has different drift fields depending on the relation between friction and
diffusion. {Alternatively, our results can be cast as different interpretations
of stochastic integration in the limiting equation}, which can be parametrized
by . Interestingly, in addition to the classical It\^o
(), Stratonovich () and anti-It\^o ()
integrals, we show that position-dependent , and even
stochastic integrals with arise. Our findings are
supported by numerical simulations.Comment: 11 pages, 5 figure
A homogenization theorem for Langevin systems with an application to Hamiltonian dynamics
This paper studies homogenization of stochastic differential systems. The
standard example of this phenomenon is the small mass limit of Hamiltonian
systems. We consider this case first from the heuristic point of view,
stressing the role of detailed balance and presenting the heuristics based on a
multiscale expansion. This is used to propose a physical interpretation of
recent results by the authors, as well as to motivate a new theorem proven
here. Its main content is a sufficient condition, expressed in terms of
solvability of an associated partial differential equation ("the cell
problem"), under which the homogenization limit of an SDE is calculated
explicitly. The general theorem is applied to a class of systems, satisfying a
generalized detailed balance condition with a position-dependent temperature.Comment: 32 page
Stratonovich-to-Ito transition in noisy systems with multiplicative feedback
Cataloged from PDF version of article.Intrinsically noisy mechanisms drive most physical, biological and economic phenomena. Frequently, the system's state influences the driving noise intensity (multiplicative feedback). These phenomena are often modelled using stochastic differential equations, which can be interpreted according to various conventions (for example, Ito calculus and Stratonovich calculus), leading to qualitatively different solutions. Thus, a stochastic differential equation-convention pair must be determined from the available experimental data before being able to predict the system's behaviour under new conditions. Here we experimentally demonstrate that the convention for a given system may vary with the operational conditions: we show that a noisy electric circuit shifts from obeying Stratonovich calculus to obeying Ito calculus. We track such a transition to the underlying dynamics of the system and, in particular, to the ratio between the driving noise correlation time and the feedback delay time. We discuss possible implications of our conclusions, supported by numerics, for biology and economics
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