Sample-path large deviations for stochastic evolutions driven by the square of a Gaussian process.

Recently, a number of physical models have emerged described by a random process with increments given by a quadratic form of a fast Gaussian process. We find that the rate function which describes sample-path large deviations for such a process can be computed from the large domain size asymptotic of a certain Fredholm determinant. The latter can be evaluated analytically using a theorem of Widom which generalizes the celebrated Szego-Kac formula to the multidimensional case. This provides a large class of random dynamical systems with timescale separation for which an explicit sample-path large-deviation functional can be found. Inspired by problems in hydrodynamics and atmosphere dynamics, we construct a simple example with a single slow degree of freedom driven by the square of a fast multivariate Gaussian process and analyze its large-deviation functional using our general results. Even though the noiseless limit of this example has a single fixed point, the corresponding large-deviation effective potential has multiple fixed points. In other words, it is the addition of noise that leads to metastability. We use the explicit answers for the rate function to construct instanton trajectories connecting the metastable states.

Microscopic simulation of membrane molecule diffusion on corralled membrane surfaces.

The current understanding of how receptors diffuse and cluster in the plasma membrane is limited. Data from single-particle tracking and laser tweezer experiments have suggested that membrane molecule diffusion is affected by the presence of barriers dividing the membrane into corrals. Here, we have developed a stochastic spatial model to simulate the effect of corrals on the diffusion of molecules in the plasma membrane. The results of this simulation confirm that a fence barrier (the ratio of the transition probability for diffusion across a boundary to that within a corral) on the order of 10(3)-10(4) recreates the experimentally measured difference in diffusivity between artificial and natural plasma membranes. An expression for the macroscopic diffusivity of receptors on corralled membranes is derived to analyze the effects of the corral parameters on diffusion rate. We also examine whether the lattice model is an appropriate description of the plasma membrane and look at three different sets of boundary conditions that describe diffusion over the barriers and whether diffusion events on the plasma membrane may occur with a physically relevant length scale. Finally, we show that to observe anomalous (two-timescale) diffusion, one needs high temporal (microsecond) resolution along with sufficiently long (more than milliseconds) trajectories.