Distinguishing Mathematical Consistency from Physical Realizability

[someonewholikesstuff]

In many physical models, lower-dimensional manifolds are mathematically self-consistent, but dynamically incomplete unless augmented by additional parameters (for example, time for change, or external structures that allow evolution).


This suggests a distinction between mathematical consistency and physical realizability: a model may be well-defined internally, yet require extra structure to represent dynamics or observable processes. I am interested in whether this distinction is already implicit in standard physical frameworks, or whether articulating it in terms of dimensional dependence offers any conceptual clarification.

 

[Astronuc]

In many physical models, lower-dimensional manifolds are mathematically self-consistent, but dynamically incomplete unless augmented by additional parameters (for example, time for change, or external structures that allow evolution).


This suggests a distinction between mathematical consistency and physical realizability: a model may be well-defined internally, yet require extra structure to represent dynamics or observable processes. I am interested in whether this distinction is already implicit in standard physical frameworks, or whether articulating it in terms of dimensional dependence offers any conceptual clarification.

One poses an interesting problem. Can one provide an example?

In the first paragraph, one mentions "time for change" and "external structures that allow evolution", so that implies a time-dependent problem. Then again, one might have a steady-state time-dependent problem, ih which state variables and boundary conditions change slowly or with some being static, or one might have a transient problem with rapidly changing state variables or boundary conditions, or some combination.

Then there is the question/challenge of how well the system is modeled, or how robust is the model, to capture all possible outcomes. There is a spectrum of empirical vs mechanistic based computation efforts, and it is often a matter of time and economics regarding the solution, as well as accounting for what one does not know.

 

[wrobel]

I must confess I reported the thread as AI generated. But moderators said that I was wrong. Nevertheless I am sure it is:)

Phrases like that

lower-dimensional manifolds are mathematically self-consistent, but dynamically incomplete

sound scientifically considerable but actually they are just senseless sequences of words.
Anybody can ensure that by googling "dynamically incomplete manifold"

 

[Astronuc]

dynamically incomplete manifold

Maybe it's an AI term? I googled on the phrase and it came back with a variety of results, including scientific papers, e.g., Variational manifold learning from incomplete data

or L2-Harmonic Forms on Incomplete Riemannian Manifolds ...

 

[someonewholikesstuff]

One poses an interesting problem. Can one provide an example?

In the first paragraph, one mentions "time for change" and "external structures that allow evolution", so that implies a time-dependent problem. Then again, one might have a steady-state time-dependent problem, ih which state variables and boundary conditions change slowly or with some being static, or one might have a transient problem with rapidly changing state variables or boundary conditions, or some combination.

Then there is the question/challenge of how well the system is modeled, or how robust is the model, to capture all possible outcomes. There is a spectrum of empirical vs mechanistic based computation efforts, and it is often a matter of time and economics regarding the solution, as well as accounting for what one does not know.

Thank you for the clarification — that helps sharpen the question.


A simple example I had in mind is a purely spatial 2-D configuration (e.g., a static geometric manifold or field snapshot) which is mathematically self-consistent but does not, by itself, encode change. Introducing time as an additional parameter allows the same configuration to support dynamics (e.g., evolution of fields, boundary conditions, or state variables). In that sense, time is not required for mathematical consistency but is required for physical evolution.


Similarly, reduced-dimension models in physics (such as 1-D or 2-D effective models) can be internally consistent and predictive within their domain, but their applicability relies on higher-dimensional structures or external parameters that are held fixed or averaged over. The “external structure” is not invoked as a new physical dimension, but as part of the modeling assumptions that enable evolution or interaction.


My question is therefore not about necessity in a logical sense, but about the distinction between:

• self-consistent mathematical description, and
• physical realizability and dynamical completeness within a model.

I’m trying to understand how physicists formally draw that line when constructing or interpreting models, especially when simplifying dimensionality.

 

[someonewholikesstuff]

I must confess I reported the thread as AI generated. But moderators said that I was wrong. Nevertheless I am sure it is:)

Phrases like that

sound scientifically considerable but actually they are just senseless sequences of words.
Anybody can ensure that by googling "dynamically incomplete manifold"

Hey, I actually put real time into writing that, you just probably think any type of advanced language is AI—that idea is honestly ludicrous. I’m genuinely interested in dimensional science, and I don’t just throw words together for fun.

 

[PeroK]

Hey, I actually put real time into writing that. I’m genuinely interested in dimensional science, and I don’t just throw words together for fun.

You do, however, throw lots of words together to say something simple. For example:

A simple example I had in mind is a purely spatial 2-D configuration (e.g., a static geometric manifold or field snapshot) which is mathematically self-consistent but does not, by itself, encode change. Introducing time as an additional parameter allows the same configuration to support dynamics (e.g., evolution of fields, boundary conditions, or state variables). In that sense, time is not required for mathematical consistency but is required for physical evolution.

All that says is that we need a time parameter to model physical processes.

And:

Similarly, reduced-dimension models in physics (such as 1-D or 2-D effective models) can be internally consistent and predictive within their domain, but their applicability relies on higher-dimensional structures or external parameters that are held fixed or averaged over. The “external structure” is not invoked as a new physical dimension, but as part of the modeling assumptions that enable evolution or interaction.

I'm not sure what you mean by that, but it looks like words put together because they sound good.

The technical term that you seem to be missing is degrees of freedom:

https://en.wikipedia.org/wiki/Degrees_of_freedom_(physics_and_chemistry)

 

[wrobel]

Hey, I actually put real time into writing that,

Let's be honest: you do not have regular education in math\phys. And all of that pretentious words can not hide that.
I am sure everybody will be happy to help you if you come here with questions from the textbooks you read but not with your homemade terminologies and theories.

 

[Astronuc]

physical realizability and dynamical completeness within a model.

See this discussion - https://www.physicsforums.com/threads/physical-realization-of-a-system.789471/post-4961600 - concerning the term "realizable".

physical realizability and dynamical completeness within a model.

What does one understand the quote phrase to mean?

Many systems are complex, and we develop simple models to represent the behavior of such a system. We cannont model every atom, or electron, so we develop simple mathematical models to model or simulation the observed behavior of the system. Such system could be simple electronic device, e.g., a resistor, capacitor, inductor, transistor, thermocouple, thermoelectric device, a circuit, or complex combination of aforementioned devices and circuits. Mechanically, one could simulation a single crystal of a pure metal, an alloy, an intermetallic compound, a ceramic, or some combination in a single crystal (same grain orientation) or a population of grains/crystals (over a range of nanometers to microns to mm to cm to m). We often refer to atomistic or nanoscale, mesoscale (intermediate) to engineering scale. One might model an interface of an electronic device, e.g., a gate, or a corrosion layer that evolves over time.

Within each system one is faced with a choice of how detailed a model must be to replicate whatever behavior is of interest. Models must be consistent. We apply continuity equations, conservation of mass, momentum and energy, i.e., we account for energy in, energy out, and energy present within a system's boundary,

We can divide a system into blocks or elements, and apply the same set of equations within each element. We run a simuation, then compare the mathemeatical results (values ) to meansurements (or observations) we make to see if the results are consistent.

We model dynamic behavior over short or long term time scales. One might model turbulence in a flow over (around) a wing/foil or body of an aricraft, ship or car, and calculate the forces on the structure in order to understand the behavior/performance of the object of interest, which may include fluid-structure interaction. One might be interested in lift and/or drag, but also fatigue of the structure or corrosion of the surface. Calculations (mathematical models) may be conducted on a variety of scales simultaneously (over the same time step) in order to reproduce the observed behavior.

In a solid model, one will need mathematical (or numerical) models of thermophysical properties (e.g., thermal conductivity, density) and thermomechanical properties (e.g., elastic modulus, tensile strength, etc), all of which are function of temperature and composition of the material, all of which may change with time. With each preoperty (variable), there is some undertainty (one cannot know precisely what a given property/variable is at a given time), but over time, if one's numerical simulation produces an observation (measurement), then that's a reasonable outcome.

Numerical modeling has become more sophisticated with improvements in computation hardware and software, as well as more measurements and better data, but we are far from having modeled everyting we could. On the other hand, some programs have been very successful (Voyager spacecraft), while at the same time, there have been some spectacular failures.