Monte Carlo Approaches for Simulating a Particle at a Diffusivity Interface and the "ItoStratonovich Dilemma"
Abstract
The possibility of different interpretations of the stochastic term (or calculi) in the overdamped Langevin equation for the motion of a particle in an inhomogeneous medium is often referred to as the "ItoStratonovich dilemma," although there is, in fact, a continuum of choices. We introduce two Monte Carlo (MC) simulation approaches for studying such systems, with both approaches giving the choice between different calculi (in particular, Ito, Stratonovich, and "isothermal"). To demonstrate these approaches, we study the diffusion on a 1D interval of a particle released at an interface (in the middle of the system) between two media where this particle has different diffusivities (e.g., two fluids with different viscosities). For reflecting boundary conditions at the ends of the interval, a discontinuity at the interface in the stationarystate particle distribution is found, except for the isothermal case, as expected. We also study the firstpassage problem using absorbing boundary conditions. Good agreement is found when comparing the MC approaches to theory as well as Brownian and Langevin dynamics simulations. Also, the results themselves turn out to be interesting. For instance: 1) for some calculi, there are more particles on the lowviscosity side (LVS) at earlier times and then fewer particles at later times; 2) there is no preference to end up on a particular wall for the Ito variant, but a bias towards the wall on the LVS in all other cases; 3) the mean firstpassage time to the wall on the LVS grows as the viscosity on the other side is increased, except for the isothermal case where it approaches a constant; 4) when the viscosity ratio is high, the firstpassagetime distribution for the wall on the LVS is much broader than for the other wall, with a powerlaw dependence of the former in a certain time interval whose exponent depends on the calculus.
 Publication:

arXiv eprints
 Pub Date:
 August 2012
 arXiv:
 arXiv:1208.5081
 Bibcode:
 2012arXiv1208.5081D
 Keywords:

 Condensed Matter  Statistical Mechanics;
 Physics  Computational Physics
 EPrint:
 31 pages, 12 figures, 1 table