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Based on the prior work of Chahl and Gopalakrishnan (2019) to infer particle-ion collision time distributions using a Langevin Dynamics (LD) approach, we develop a model for the non-dimensional diffusion charging collision kernel β_i or H that is applicable for 0≤Ψ_E≤60,0≤Ψ_I/Ψ_E ≤1,Kn_D≤2000 (defined in the main text). The developed model for β_i for attractive Coulomb and image potential interactions, along with the model for β_i for repulsive Coulomb and image potential interactions from Gopalakrishnan et al. (2013b), is tested against published diffusion charging experimental data. Current state of the art charging models, Fuchs (1963) and Wiedensohler (1988) regression for bipolar charging, are also evaluated and discussed. Comparisons reveal that the LD-based model accurately describes unipolar fractions for 10 – 100 nm particles measured in air (Adachi et al., 1985), nitrogen and argon but not in helium (Adachi et al., 1987). Fuchs model and the LD-based model yield similar predictions in the experimental conditions considered, except in helium. In the case of bipolar charging, the LD-based model captures the experimental trends quantitatively (within ±20%) across the entire size range of 4 – 40 nm producing superior agreement than Wiedensohler’s regression. The latter systematically underpredicts charge fraction below ~20 nm in air (by up to 40%) for the data presented in Adachi et al. (1985). Comparison with the data of Gopalakrishnan et al. (2015), obtained in UHP air along with measurements of the entire ion mass-mobility distribution, shows excellent agreement with the predictions of the LD-based model. This demonstrates the capability to accommodate arbitrary ion populations in any background gas, when such data is available. Wiedensohler’s regression, derived for bipolar charging in air using average ion mass-mobility, also describes the data reasonably well in the conditions examined. However, both models failed to capture the fraction of singly and doubly charged particles in carbon dioxide warranting further investigation.
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Breaking Reversibility Accelerates Langevin Dynamics for Non-Convex Optimization
Langevin dynamics (LD) has been proven to be a powerful technique for optimizing a non-convex objective as an efficient algorithm to find local minima while eventually visiting a global minimum on longer time-scales. LD is based on the first-order Langevin diffusion which is reversible in time. We study two variants that are based on non-reversible Langevin diffusions: the underdamped Langevin dynamics (ULD) and the Langevin dynamics with a non-symmetric drift (NLD). Adopting the techniques of Tzen et al. (2018) for LD to non-reversible diffusions, we show that for a given local minimum that is within an arbitrary distance from the initialization, with high probability, either the ULD trajectory ends up somewhere outside a small neighborhood of this local minimum within a recurrence time which depends on the smallest eigenvalue of the Hessian at the local minimum or they enter this neighborhood by the recurrence time and stay there for a potentially exponentially long escape time. The ULD algorithm improves upon the recurrence time obtained for LD in Tzen et al. (2018) with respect to the dependency on the smallest eigenvalue of the Hessian at the local minimum. Similar results and improvements are obtained for the NLD algorithm. We also show that non-reversible variants can exit the basin of attraction of a local minimum faster in discrete time when the objective has two local minima separated by a saddle point and quantify the amount of improvement. Our analysis suggests that non-reversible Langevin algorithms are more efficient to locate a local minimum as well as exploring the state space.
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- PAR ID:
- 10256959
- Date Published:
- Journal Name:
- Proceedings of Machine Learning Research
- Volume:
- 33
- ISSN:
- 2640-3498
- Page Range / eLocation ID:
- 7850 - 17862
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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