Degrees of Freedom/Spatial Manifold

[spacebear2000]

I've been away from the forum for a while working on an interesting project developing an open source visualization system for spatial manifolds that have four dimensions. I have two primary lines of questioning that stem from this work.

1) I know that Gerris is an open source solution for modeling fluid dynamics in 2D or 3D. Can anyone speak to the extensibility of the physics or software as pertains to modeling flows in 4D spatial manifolds?

2) My understanding is that liquid water is often regarded as a prerequisite for life or living systems; this makes sense given the structure and behavior of its molecules in three-dimensional spatial manifolds. What molecule (in what state) might serve as the basis for life or living systems in a 4D spatial manifold?

To be clear, I am not asking these questions for myself or my team alone; if anyone wishes to join the effort, please check out https://osf.io/dxjeo/wiki/home/

 

[hilbert2]

It's not difficult to write a 4D version of the hydrodynamical Navier-Stokes equation, treating the velocity field $\vec{v}$ and gradient operator $\vec{\nabla}$ as four-component vectors. The problem is that in numerical simulations the addition of extra spatial coordinate makes the problem considerably more computationally expensive. The number of grid points in the discretized velocity field grows exponentially when dimensionality is increased.

Similarly, it is easy to write down the Schrödinger equation of a hydrogen atom, any other atom or some molecular species in spaces with dimension other than 3, but the quantum chemical calculations become rapidly more difficult with increasing number of dimensions. We are not able to simulate even 3D water accurately enough to reproduce its special properties (like lower density in solid than in liquid form, or negative heat expansion coefficient in a certain temperature range) in a computer simulation.

 

[spacebear2000]

@hilbert2: Thank you. From whence does the difficulty of accurately simulating 3D water arise? I know this is a naive sort of question, though I want to make sure I understand to what extent we're talking about computer processing speed and to what extent we're talking about gaps in our understanding (or some other challenging factors).

 

[hilbert2]

From whence does the difficulty of accurately simulating 3D water arise? I know this is a naive sort of question, though I want to make sure I understand to what extent we're talking about computer processing speed and to what extent we're talking about gaps in our understanding (or some other challenging factors).

It's basically only about lack of available processing speed. Even simulating a single molecule quantum mechanically can be computationally expensive, let alone simulating a macroscopic sample of water, where there's something like 10²³ molecules, each of which is strongly interacting with its nearest neighboring molecules. Calculating something like "what would the boiling point of water be if space were 4-dimensional" is nowhere near our current abilities.

Hydrodynamic 3D flow problems are possible to solve because there we treat the liquid as "continuous matter", ignoring its molecular nature at microscopic scale.

 

[spacebear2000]

Recognizing this as a computational hurdle (and not a theoretical one) is somewhat assuring, given that we can frame it as an engineering challenge. Thanks again!

 

[spacebear2000]

According to this exchange on physicsforums.com with hilbert2, hydrodynamic 3D flow problems are possible to solve because "there we treat the liquid as 'continuous matter', ignoring its molecular nature at microscopic scale...However, because the number of grid points in the discretized velocity field grows exponentially when dimensionality is increased, the computing power required to model flow in 4D is much greater than in 3D." I would like to find out how great the difference is and what may be required to overcome it.