Multiscale means either multiple length or time scales and typically refers to multiscale problems. A good example is a model for the climate, which involves knowledge on distribution of properties like e.g. temperature on many different scales. The problem with modelling is that a model often can address only a very limited range of scales, or typically one scale. It is pretty clear that in a climate model, for instance, where the atmosphere of the whole earth needs to be modeled, one cannot consider physical processes resolved down to the individual gas atom. These are typically addressed by means of upscaling. From gas kinetic theory we can derive the ideal gas law, which is then used as one of the basic ingredients for the model on the next scale.On that scale, it is of no importance where every single gas atom is and what its velocity vector is, but only what the effective pressure and temperature is. So in that sense, a multiscale model includes physics from at least two different scales, where the smaller scale is captured via effective properties. I would argue that the question  in the way you posed it, cannot be easily answered from a purely mathematical perspective, but the underlying physics needs to be considered.

Dear Steffen Berg 

Thank you for your participation; that's pretty comment, but what I want to know is:

Are there any mathematical conditions or procedure that can test the validity of certain model on different scales or you need only to apply it on the experimental data and see what you get? 

Take as an example the materials science, If you want to make a model on different scale from nano to macro, is there any procedure that can test the validity of your model?

I would say that your question can be seen from two different points of view:

1. The equations represent the same phenomenon at different scales. I think that it is rare. In this case your equations should be similar when you change the units used for the measure : your equations must be homogeneous. There are several definitions of this condition, depending of the mathematical objects whch are considered. In partuclar it excludes any exponential map.

2. We have the same phenomenon which occurs for many microsystems, which interact at a larger scale. This is probably usually the case in physics. The mathematical representation of the state of the total system must be different than for the microsystems.The results are very diffrerent, usually the interactions bring order, and there are phases, between which the system transitions.  You can have a general overview of these issues in my paper on Quantum Mechanics revisited on this site.

I am not mathematician, but in my field, the need for multiscale modeling comes usually from the fact that the macroscale models needs some collective (resultant / average) informations from the microscale models.

Mathematically, the state variables of the macroscale model are separable from the state variable of microstate model. Some parameters in the macroscale model are integral functions of one or more state variables of the microscale model.

Dear Prof Jean and Prof Talib

Thank you for your participation.

Nevertheless, The question is still opened; because the most important is the condition that you can tell according to its applicability that the equation that any one might derive can be adopted to be a multiscale model. The condition can be mathematical or may be experimental if there is someone.

Dear all;

Here is a link for a new multiscale model

https://www.researchgate.net/publication/340086146_Derivation_of_a_new_multiscale_model_I_Derivation_of_the_model_for_the_atomic_molecular_and_nano_material_scales?_sg=BTtGdgMvucPOnn9WEbJTY5jKwSve6-RyDr_J0YbHP2eXM0AfuFhCaaBv-3vQxC5Er8ItZFEgbajxmOaNzlJJSyqraqTqj6JnJ5xruDRksJf1nUUF5pk

Thank you!

Mathematics are a tool : they provide possible representation (or description) of a phenomenon, but they are not the answer by themselves. You need a bit of understanding of the objects that you study and their behavior.

Dear Jean

Of course, I fully agree with you, but sometimes the application of a certain theory faces difficulties due to some discrepancies in it. In this case, we need to widen our insight and looks for a more general scheme.

Dear Sadeem,

In quantum field theory, when computing effective potentials, one need the multiscale formalism when treating models with multiple scalar degrees of freedom. For example, in 10.1103/PhysRevD.7.1888 there are multiple couplings and multiple fields, however only one scalar degree of freedom. Nevertheless, in Article Multi-scale Renormalization Group Methods for Effective Pote...

I hope it could be helpfull. Best regards.