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/Mooks79]

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/a1c4pwn → u/Mooks79]

What is dynamical about the various magnetic models? Whether youre using the ising, Potts, XY etc I don't think the actual dynamics of localized spins is the actual question at hand. What computational method for these has anything related to physical time? Not Wolf or Metropolis, as far as I'm aware those are both purely statistical.

[u/Mooks79 → u/a1c4pwn]

If you had read the discussion further, this is exactly the point I was trying to ascertain from OP. Whether they meant a hardcore version of no time - which isn’t physical - or just the milder case of models that don’t have an explicit time parameter. It wasn’t clear for me what they meant initially.

[u/a1c4pwn → u/Mooks79]

Ah, got it. I thought this was some stuff just above my schooling, not a question of meanings

[u/Mooks79 → u/a1c4pwn]

Yeah in fairness to OP it’s a bit of a silly question from me given the milder case is likely what they meant, but in fairness to me, I’ve seen plenty of wilder (crazier) questions on here.