Monte Carlo Approaches for Simulating a Particle at a Diffusivity Interface and the "Ito--Stratonovich 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 "Ito-Stratonovich 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 interpretations (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 (for example, two fluids with different viscosities). We consider the properties of the particle distribution for reflecting boundary conditions at the ends of the 1D interval. A discontinuity at the interface in the stationary-state particle distribution is found, except for the isothermal case, as expected. We also study the first-passage problem using absorbing boundary conditions. Good agreement is found when comparing the MC approaches against theoretical predictions as well as Brownian and Langevin dynamics simulations. Additionally, while this problem was chosen primarily to verify the algorithms, the results themselves turn out to be interesting -particularly when comparing across interpretations. For instance, we report that: 1) for some calculi, there can be more particles on the low-viscosity side at earlier times and then more particles on the highviscosity side 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 low-viscosity side in all other cases; 3) the mean first-passage time to the wall on the low-viscosity side grows as the viscosity on the high-viscosity side is increased, except for the isothermal case where it approaches a constant; 4) when the viscosity ratio is high, the first-passage-time distribution for the wall on the lower-viscosity side is much broader than for the other wall, with a power-law dependence of the former in a certain time interval whose exponent depends on the calculus; 5) the average portion of time the particle spends on a particular side can be very different from the probability to reach the wall on that side and depends significantly on how the averaging is done.

Electrokinetic Current Driven by a Viscosity Gradient

“…A fundamental mathematical ambiguity, often called the Itô-Stratonovich dilemma, arises in stochastic models where the noise is state-dependent (or multiplicative). Depending on whether one evaluates the steps' sizes based on the noise magnitude at the beginning of each step, at the end, or somewhere in between, the particle will either drift or not drift [3,9,17,26,29]. A few experiments previously concluded that the isothermal (end-of-step) rule applies to colloidal particles, based on observations of drift in the effective viscosity gradient created by the particles' proximity to a solid surface [3,15,28]; the observed drift was subtle and short-ranged because the proximity effect only stifles diffusion over distances comparable to the particle size [6].…”

“…The isothermal rule also leads to a generalization of Fick's law in which the flux, J, is related to the concentration profile, φ(x), by J(x) = −D(x) dφ(x) dx [9]. Thus, a uniform distribution of particles exhibits no net flux.…”

Application of underdamped Langevin dynamics simulations for the study of diffusion from a drug-eluting stent

“…In the latter case, one must introduce decision rules for crossing the boundary between layers, even if those layers are not separated by a membrane. A Monte-Carlo algorithm implementing such types of decision rules has been recently presented [46], but we are not aware of a similar algorithm for overdamped Brownian dynamics. This extraordinary problem is completely avoided in underdamped Langevin dynamics simulations which, if run properly, produce correct thermal diffusion between sharp interfaces.…”

Ionic current driven by a viscosity gradient

“…By adding the second term, sometimes called the spurious or noise-induced dri, we recover the results of the isothermal convention. 4,21,22 In other words, a trajectory in the isothermal convention is equivalent to one in the Itô convention with an added dri term. 21,22 3 Results of simulations with diffusivity gradients…”

“…4,21,22 In other words, a trajectory in the isothermal convention is equivalent to one in the Itô convention with an added dri term. 21,22 3 Results of simulations with diffusivity gradients…”