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The literature on stochastic processes (Ornstein–Uhlenbeck, Langevin) is not very clear as to the motivation behind using the Brownian motion or other types of noise in the dynamics. Are there any real-world applications of Brownian noise or other types of noise in the dynamics in macro (not particles with small dimensions) dynamical systems?

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    $\begingroup$ You could for any system of many degrees of freedom coarse grain fast interactions out with Mori-Zwanzig and end up with a stochastic system, for example. This could be done for numerical reasons or general tractability, not entirely unlike the turbulence approximations in CFD. $\endgroup$
    – alarge
    Feb 24, 2020 at 18:10
  • $\begingroup$ I agree with your comment. I am trying to find a justification for using white noise in dynamics of macro objects, such as a motor, drone kinematics or point-mass dynamical models. The control theory literature is abundant in control, planning and stability topics for such systems but do not give a clear picture as to the real-world applicability of the underlying stochastic model. $\endgroup$
    – kbakshi314
    Feb 24, 2020 at 19:27
  • $\begingroup$ I think I will accept that as the answer if you post it, @Naptzer. After scouring through the technical literature and crowdsourcing Physics SE, I have reached the conclusion that stock prices, population models, wind models for vehicle dynamics and sensor noise are the prime examples of legitimate physical models using Brownian motion for macro entities. Finally, I agree that it is relevant, although very rarely (primarily to model noise with unbounded derivatives), to use the white noise to model uncertainty in the dynamics of real-world physical objects. $\endgroup$
    – kbakshi314
    Mar 2, 2020 at 17:56

1 Answer 1

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Are there any real-world applications of Brownian noise or other types of noise in the dynamics in macro (not particles with small dimensions) dynamical systems?

There are three ways to answer your question, stemming from 'where' the noise terms should come from.

First, you can think of any kind of population dynamics, be it crowds, cells, birds, economic agents... where the noise arises in the modeling of the sub-parts of the system. One way to model all these systems is to assume some stochastic model for the agents 1, and then try to extract meaningful macro quantities of the whole thing. Some would call this 'emergent behavior'. If you'd like to dive deeper, the relevant field is probably Active Matter Theory.

Second, when dealing with already macroscopic dynamical systems in real life, one should remember that any macro object dynamic quantity (position, speed...) must be measured before being dealt with. This, in turn, implies that there is some inherent measurement noise in your system that you have to take into account. I would recommend taking a look at Kalman filters, which are a way to deal with what I just described, especially for the guidance of vehicles.

Third, when modeling macroscopic systems, some factor impacting the dynamics is either inherently "noisy" or too tricky to model exactly. E.g., taking into account the wind changes (gusts) when modeling the mechanical behavior of high metallic structures. Another example could be the number of cars and their speed on a bridge when studying the mechanical response of such infrastructure.

1 Usually, exactly as you described, i.e. some deterministic interaction or inertial terms, plus a term to account for the randomness of each agent. This can be a bird, changing direction randomly every so often (of course there might a real reason for the changes, but you don't want to end up modeling the bird's decision process).

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  • $\begingroup$ Thanks for your concise answer @Naptzer. I am familiar with filtering theory as well as large population systems exhibiting emergent behavior and I think this is a fairly comprehensive answer. I would like to add the modeling of uncertainty in the dynamics (parameters in the model or external disturbances such as wind) and noise with unbounded derivatives as example applications of the Brownian motion in modeling dynamical systems. $\endgroup$
    – kbakshi314
    Mar 4, 2020 at 16:20
  • $\begingroup$ @kb314 Thanks for the comment ! I agree 100% with adding the "modeling noisy external factors" in the dynamics, but the second point is not quite clear for real world applications... Maybe I'll let you edit the answer for this one ? And add other examples for the 3rd point, I can only think of mechanical stress on structures right now. $\endgroup$
    – Naptzer
    Mar 5, 2020 at 9:29
  • $\begingroup$ The indicative example I have is that of a mass sliding on a friction surface and encountering an ice sheet. In this case, although the derivative of the velocity cannot be unbounded (over the time interval in which the mass slides from the friction surface onto the ice), from an engineering perspective it is a jump. In this case, if one wants to design a controller which makes the mass slide at a constant speed (while being forces by other forces), it is a good idea to model this type of unbounded noise in the dynamics explicitly and design a controller which is robust to it. $\endgroup$
    – kbakshi314
    Mar 5, 2020 at 12:06

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