scholarly journals Mean-Field Limits: From Particle Descriptions to Macroscopic Equations

2021 ◽  
Author(s):  
José A. Carrillo ◽  
Young-Pil Choi
Keyword(s):  
Kinetic Energy ◽  
Time Derivative ◽  
Mean Field ◽  
Velocity Fields ◽  
Discrete Version ◽  
Nonlocal Interactions ◽  
Type Particle ◽  
Mean Field Limits ◽  
Dissipative Terms ◽  
Macroscopic Equations

AbstractWe rigorously derive pressureless Euler-type equations with nonlocal dissipative terms in velocity and aggregation equations with nonlocal velocity fields from Newton-type particle descriptions of swarming models with alignment interactions. Crucially, we make use of a discrete version of a modulated kinetic energy together with the bounded Lipschitz distance for measures in order to control terms in its time derivative due to the nonlocal interactions.

scholarly journals Transient Dynamics in the Random Growth and Reset Model

Entropy ◽  
2021 ◽  
Vol 23 (3) ◽  
pp. 306
Author(s):  
Tamás S. Biró ◽  
Lehel Csillag ◽  
Zoltán Néda
Keyword(s):  
Differential Equation ◽  
Mean Field ◽  
Equation System ◽  
Transient Dynamics ◽  
Type Model ◽  
Discrete Version ◽  
Differential Equation System ◽  
Practical Applications ◽  
Random Growth ◽  
Recursive Integration

A mean-field type model with random growth and reset terms is considered. The stationary distributions resulting from the corresponding master equation are relatively easy to obtain; however, for practical applications one also needs to know the convergence to stationarity. The present work contributes to this direction, studying the transient dynamics in the discrete version of the model by two different approaches. The first method is based on mathematical induction by the recursive integration of the coupled differential equations for the discrete states. The second method transforms the coupled ordinary differential equation system into a partial differential equation for the generating function. We derive analytical results for some important, practically interesting cases and discuss the obtained results for the transient dynamics.

scholarly journals Mean field limits for interacting Hawkes processes in a diffusive regime

Bernoulli ◽  
2022 ◽  
Vol 28 (1) ◽  
Author(s):  
Xavier Erny ◽  
Eva Löcherbach ◽  
Dasha Loukianova
Keyword(s):  
Mean Field ◽  
Hawkes Processes ◽  
Diffusive Regime ◽  
Mean Field Limits

STM studies of island nucleation during hyperthermal atom deposition

2002 ◽  
Vol 749 ◽  
Author(s):  
Joshua M. Pomeroy ◽  
Joel D. Brock
Keyword(s):  
Kinetic Energy ◽  
Ion Beam ◽  
Mean Field ◽  
Scaling Exponent ◽  
Island Nucleation ◽  
Precise Control ◽  
Ion Kinetic Energy ◽  
The Mean ◽  
Sputter Erosion ◽  
Hyperthermal Energy

ABSTRACTWe report fundamental changes in island nucleation dynamics as the kinetic energy of the constituent particles used for film grown is increased. A hyperthermal energy ion beam-line with precise control over ion kinetic energy was used to grow copper islands on a Cu(100) substrate. Dramatic increases in island densities were observed with increasing kinetic energy from thermal energies to 150 eV. We find that sputter erosion and the formation of adatom-vacancy pairs contribute to this increase. In addition, variations in flux and temperature suggest that the mean-field scaling exponent is sensitive to atomistic mechanisms activated by the ion beam.

scholarly journals Replica-Mean-Field Limits for Intensity-Based Neural Networks

2019 ◽  
Vol 18 (4) ◽  
pp. 1756-1797
Author(s):  
François Baccelli ◽  
Thibaud Taillefumier
Keyword(s):  
Neural Networks ◽  
Mean Field ◽  
Mean Field Limits

scholarly journals From critical behavior to catastrophic runaways: comparing sheared granular materials with bulk metallic glasses

2019 ◽  
Vol 21 (4) ◽  
Author(s):  
Alan A. Long ◽  
Dmitry V. Denisov ◽  
Peter Schall ◽  
Todd C. Hufnagel ◽  
Xiaojun Gu ◽  
...  
Keyword(s):  
Granular Materials ◽  
Metallic Glasses ◽  
Granular Media ◽  
Dynamic Change ◽  
Time Derivative ◽  
Bulk Metallic Glasses ◽  
Mean Field ◽  
Characteristic Change ◽  
Mean Field Model ◽  
Two Systems

Abstract The flow of granular materials and metallic glasses is governed by strongly correlated, avalanche-like deformation. Recent comparisons focused on the scaling regimes of the small avalanches, where strong similarities were found in the two systems. Here, we investigate the regime of large avalanches by computing the temporal profile or “shape” of each one, i.e., the time derivative of the stress-time series during each avalanche. We then compare the experimental statistics and dynamics of these shapes in granular media and bulk metallic glasses. We complement the experiments with a mean-field model that predicts a critical size beyond which avalanches turn into large runaway events. We find that this transition is reflected in a characteristic change of the peak width of the avalanche profile from broad to narrow, and we introduce a new metric for characterizing this dynamic change. The comparison of the two systems points to the same deformation mechanism in both metallic glasses and granular materials.

Development and Application of LED Illumination Color PIV

2006 ◽  
Author(s):  
Shenq-Yuh Jaw ◽  
Robert R. Hwang ◽  
K. L. Shyu
Keyword(s):  
Turbulent Flows ◽  
Channel Flow ◽  
Particle Image ◽  
Time Derivative ◽  
Digital Camera ◽  
Soap Film ◽  
Velocity Fields ◽  
Flow Measurements ◽  
Full Field ◽  
Particle Images

In this study, red, green, and blue light-emitting diodes (LED) are adopted as the light source to illuminate sequentially a two-dimensional soap film channel flow. Triple-exposure particle image is recorded on the same image frame by a 3-ccd color camera. Since the particles illuminated by the R, G, B LED will only be recorded on the R, G, B ccd-chip of the digital camera, three sequential exposure, R, G, B particle images can be obtained from separating the triple-exposure particle image. Two sequential velocity fields can be determined from the correlation analysis of the R-G and G-B sequential particle images. Time derivative of the velocity fields, and hence the evolution of the unsteady flow or the characteristics of turbulent flows can be analyzed from the two velocity fields determined. The color PIV method incorporated with the LED light has proven to be a cheap, safe, and powerful tool for the full-field flow measurements. Results of the flow past circular cylinder in the confined soap film channel flow are presented.

scholarly journals Mean-field limits of the quantum Potts model

1983 ◽  
Vol 90 (3) ◽  
pp. 373-387 ◽  
Author(s):  
A. Cant ◽  
Paul A. Pearce
Keyword(s):  
Potts Model ◽  
Mean Field ◽  
Mean Field Limits
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