# What is phenomenological equation and phenomenological model?

I come across these terms in some papers. My understanding is that it is an equation or model describing a phenomenon. Usually, the equations are given and claimed to be true with only some explanations and justification, but not derived from the first principle. The reason is often a bit obscure to follow, on what base they can do that with high confidence? How can they derive it in the first place? If the only purpose is to generate the expected phenomenon, how can they justify it is really the equation governing the phenomenon?

Is there good definition and explanations of these two terms? I expect few solid physics examples should make it easy to understand.

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Some examples come to my mind:

• Fourier's law of heat conduction $\vec{J} = -c\vec{\nabla}T$ in crystalline solids is a good example of a phenomenological law. It is an ampirical law easy to verify in a broad range of materials in various phases and yet, as explained in this presentation, there is no derivation of it from first principles in solids and people that do try the exercise just find stranger and stranger results. Macroscopically it is easy to make sense of it in many ways, one of the most rigorous one being the linear response theory...but this is still a phenomenological assumption.

• When in high school mechanics you look at balances of forces between the gravitational pulling from the Earth on you and the reaction from the ground so that you can stand still, you phenomenologically put the reaction of the ground because you know it is there but there is almost no way to derive it from first principle.

• The whole Landau theory of phase transitions is phenomenological as well but explains a lot of things and has been in some way rationalized by Ginzburg although we are far from first principle derivation shere.

But I think that where there might be confusion here is that phenomenology and devising models to describe what's going on in Nature is what everyday physics is mostly about while another part of Physics (the same people or others) of course tries to relate all of them in coherent framework/story.

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You could read the definition in dozens of dictionaries and encyclopedias like this http://en.wikipedia.org/wiki/Phenomenology_%28science%29

My feeling is that the term is related to the word "Phenomena" that is something that is observed. Thus, the phenomenological models and equations describe rather instrument readings than some fundamental processes behind it. For example, there is a relaxation time $\alpha$ in quantum physics which is often taken as a constant in the exponential factor $e^{-\alpha t}$ since the experimental results show nearly exponential decay of the non-equilibrium particle concentration. That is the phenomenological approach. Alternative approach is to compute this quantity using fundamental physics and taken into account scattering processes standing behind this exponential decay.

Usually, phenomenological method utilizes a kind of scientific intuition. Soon or later, most phenomenological quantities are proved by theoretical computations if they describe the experimental situation with satisfactory accuracy.

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