We consider particles obeying Langevin dynamics while being at known positions and having known velocities at the two end-points of a given interval. Their motion in phase space can be modeled as an Ornstein–Uhlenbeck process conditioned at the two end-points—a generalization of the Brownian bridge. Using standard ideas from stochastic optimal control we construct a stochastic differential equation (SDE) that generates such a bridge that agrees with the statistics of the conditioned process, as a degenerate diffusion. Higher order linear diffusions are also considered. In general, a time-varying drift is sufficient to modify the prior SDE and meet the end-point conditions. When the drift is obtained by solving a suitable differential Lyapunov equation, the SDE models correctly the statistics of the bridge. These types of models are relevant in controlling and modeling distribution of particles and the interpolation of density functions.
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Exact solution for the Anisotropic Ornstein–Uhlenbeck process
Active-Matter models commonly consider particles with overdamped dynamics subject
to a force (speed) with constant modulus and random direction. Some models also
include random noise in particle displacement (a Wiener process), resulting in diffusive
motion at short time scales. On the other hand, Ornstein–Uhlenbeck processes apply
Langevin dynamics to the particles’ velocity and predict motion that is not diffusive
at short time scales. Experiments show that migrating cells have gradually varying
speeds at intermediate and long time scales, with short-time diffusive behavior. While
Ornstein–Uhlenbeck processes can describe the moderate-and long-time speed variation,
Active-Matter models for over-damped particles can explain the short-time diffusive
behavior. Isotropic models cannot explain both regimes, because short-time diffusion
renders instantaneous velocity ill-defined, and prevents the use of dynamical equations
that require velocity time-derivatives. On the other hand, both models correctly describe
some of the different temporal regimes seen in migrating biological cells and must, in
the appropriate limit, yield the same observable predictions. Here we propose and solve
analytically an Anisotropic Ornstein–Uhlenbeck process for polarized particles, with
Langevin dynamics governing the particle’s movement in the polarization direction and a
Wiener process governing displacement in the orthogonal direction. Our characterization
provides a theoretically robust way to compare movement in dimensionless simulations
to movement in experiments in which measurements have meaningful space and time
units. We also propose an approach to deal with inevitable finite-precision effects in
experiments and simulations.
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- Award ID(s):
- 1720625
- NSF-PAR ID:
- 10478025
- Publisher / Repository:
- Elsevier
- Date Published:
- Journal Name:
- Physica A: Statistical Mechanics and its Applications
- Volume:
- 587
- Issue:
- C
- ISSN:
- 0378-4371
- Page Range / eLocation ID:
- 126526
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1016/j.physa.2021.126526
