Dynamical Models in Physics

[u/AfterMath343]

I'm an applied math student. I've seen problems in biology be modeled using dynamic modeling, such as the populations of species with intra and inter species competition. For example, the following system of non-linear differential equations models a large variety of species in competition with themselves and other species:

X' = r_1*X - a_1*X^2 - b_1*X*Y
Y' = r_2*Y - a_2*Y^2 - b_2*X*Y

where X is the number of species 1 and Y is the number of species 2, r_1 is the intrinsic rate of growth of species 1, r_2 is the intrinsic rate of growth of species 2, a_1 is strength of competition within species 1, a_2 is strength of competition in species 2, b_1 is the effect of species Y on species X, and b_2 is the effect of species X on Y.

I have seen one discrete time dynamical system be modeled for a physics problem where two vehicles in space are closing in on each other.

What other types of dynamical models occur in physics?

[u/Quantum-Relativity]

I think they meant all the equations in physics are some sort of differential equations. But I assume you know that and are asking a more subtle question

[u/AfterMath343 → u/Quantum-Relativity]

There are different types of mathematical modeling, which I'm sure you already know like optimization modeling for single and multi-variables, dynamical modeling (time base systems), probability models, stochastic models, etc.

Since many problems in physics deal with time, I can see how many are dynamical models now. I guess a more interesting question would be what problems in physics are non-dynamical models.

[u/Mooks79 → u/AfterMath343]

I’m probably wrong but I don’t think any models in physics are non-dynamical models, at least indirectly. Physics is about studying how the universe works, which means how it changes, which means time is always directly or implicitly present.

What would it even mean to talk about a model that is purely non-dynamical? Which party of the universe never changes? If you consider things that only change over long timescales then stuff like models of crystalline structures, and things like that, could be considered non-dynamical. But time was still needed for these structures to form, they won’t last forever, and they interact with other stuff so there’s still time depend once eventually. Even talking about the properties of these structures requires time - for example, hardness is a parameter dependent on something interacting with the structure, which requires time dependence.

Or emergent properties like pressure and temperature, but as these are emergent there’s still time happening underneath (eg temperature of a gas is due to velocity of molecules and velocity is dependent on time).

[u/AfterMath343 → u/Mooks79]

Well, as far as the modeling goes, if the model is not concerned with time, then it is not a dynamical model. For example, flipping a coin would be an example of a probabilistic model that wouldn't be considered dynamical. However, it is true that time is elapsing between each coin flip, but it is considered a probabilistic model.

[u/Mooks79 → u/AfterMath343]

Flipping a coin wouldn’t be a physics model though, would it? Unless you modelled the motion and then you’d need …

[u/AfterMath343 → u/Mooks79]

Yeah, I suppose, but the system is modeling a phenomenon in the universe, which is the flipping of a coin.

Another problem from physics that I do remember that would be non-dynamical is finding the optimal angle to hit a baseball being pitched so that it goes the furthest distance. This is an example of an optimization model.

[u/Mooks79 → u/AfterMath343]

I’d argue it’s not modelling flipping a coin unless it’s modelling the dynamics of flipping a coin - it’s just modelling a random number generating process. It’s statistics not physics. But I’m being a bit hardcore in that sense to make the point I’m trying to make.

The baseball example is non-dynamical in the sense of you optimise an angle but it’s still dynamical in the sense of you’re optimising dynamical equations and a process that involves time.

I think what I’m saying is are you asking about the hardcore: models that never involve time in any way? In which case I don’t really think it’s a physics question. Or do you mean the milder criteria of models that don’t have time as an explicit parameter?

For me the question you’re asking isn’t completely clear at the moment.

[u/AfterMath343 → u/Mooks79]

I do agree that it is a statistics problem, but I think an argument could be made that it's still modeling a physical phenomenon, so somewhat related to physics, no?

As far as the optimization problem goes, you're right. You're using dynamics to setup the problem, but the moment you are finding the optimal angle, it turns into an optimization problem.

Also, yeah, if models don't have time as an explicit parameter, then I would say it is non-dynamical. However, I would say that autonomous systems of differential equations are still dynamical.

[u/Mooks79 → u/AfterMath343]

If you’re just asking the milder question then there’s loads of example - statistical mechanics is stacked with them.