What is the highest power in physics equations?

[Dadface]

Sorry if a similar question has been posted before but I have been thinking about this recently.

When you scan through physics equations, quantities raised to the power of one are so common that the one is omitted.There are numerous equations where there are quantities squared but a smaller number of equations where there are quantities cubed.I can only think of one equation where there is a quantity raised to the power of four this being temperature in Stefans law, the derivation of this equation referring to T being raised to the power of five.My question is, are there any physics equations where there are higher powers?

 

[Mapes]

Probably not the biggest, but the very common http://en.wikipedia.org/wiki/Lennard%E2%80%93Jones_potential" has terms to the sixth and twelfth powers. These aren't minor approximation terms, either; the difference between the two represents the bonding between two atoms (which affects density, heat capacity, phase changes, thermal expansion, etc.). However, the higher-power term is empirical.

 

[A.T.]

My question is, are there any physics equations where there are higher powers?

Do exponential functions count?

 

[Vanadium 50]

The relation between ⁸B neutrinos and the core temperature of the sun goes as T²⁵. That's the highest power that immediately comes to mind.

 

[Dadface]

Probably not the biggest, but the very common http://en.wikipedia.org/wiki/Lennard%E2%80%93Jones_potential" has terms to the sixth and twelfth powers. These aren't minor approximation terms, either; the difference between the two represents the bonding between two atoms (which affects density, heat capacity, phase changes, thermal expansion, etc.). However, the higher-power term is empirical.

Thank you.Interesting, but I note from the reference you gave that the equation is an approximation and that there is no theoretical justification for the repulsive term.Still,I suppose it is a powerful equation and should be counted.

Do exponential functions count?

You got me there. I was thinking more in terms of equations that had a power/powers of fixed value.

 

[Dadface]

The relation between ⁸B neutrinos and the core temperature of the sun goes as T²⁵. That's the highest power that immediately comes to mind.

That is big! Mathematicians seem to deal with powers without limits but the only example I can think of is Fermats last theorem.I wonder if any of these equations have relevance to physics or the other sciences.

 

[Dadface]

So far we have:
0...Or should this be considered trivial?
1
2
3
4
5...I'm not sure if my example should be counted.
6.
Now there seems to be a gap to 12 and then an even bigger gap to 25.
I find it interesting that there might be gaps but then I can be quite dopey.I wonder if it stops at 25.

 

[zcd]

Kinetic energy is approximated at $$KE=\frac{1}{2}mv^{2}$$ for low speeds but the taylor series for KE with speed near c is given by $$\frac{1}{2}mv^{2}+\frac{3}{8c^{2}}mv^{4}+\frac{5}{16c^{4}}v^{6}+...=m\sum_{n=1}^{\infty} \frac{(2n-1)v^{2n}}{2^{2n-1}c^{2(n-1)}}$$. Does that count?

 

[Dadface]

Kinetic energy is approximated at $$KE=\frac{1}{2}mv^{2}$$ for low speeds but the taylor series for KE with speed near c is given by $$\frac{1}{2}mv^{2}+\frac{3}{8c^{2}}mv^{4}+\frac{5}{16c^{4}}v^{6}+...=m\sum_{n=1}^{\infty} \frac{(2n-1)v^{2n}}{2^{2n-1}c^{2(n-1)}}$$. Does that count?

Clever but i think it better to discount power series that converge to a final equation.You got me thinking there

 

[Cyrus]

There really isn't a limit as to how high a power something can have. It depends on what end result you get when doing model structure determination.

For example, you may have a sinusoidal signal in your data but not know it. The result is that the model structure will contain terms raised to the power of the Taylor series expansion of a sine or cosine since that is the only way in which the data can be fit without using a sine/cosine signal explicitly.

 

[Redbelly98]

Probably not the biggest, but the very common http://en.wikipedia.org/wiki/Lennard%E2%80%93Jones_potential" has terms to the sixth and twelfth powers.

And it follows that the force associated with such a potential has terms to the 7th and 13th powers.

 

[maverick_starstrider]

Why are we discounting exponenetials again (i.e. exp(x))?

 

[Redbelly98]

Why are we discounting exponenetials again (i.e. exp(x))?

The OP's question was about physical quantities being raised to some (constant) numerical power. e is not a physical quantity, and x is not a numerical power. So e^x is not within the scope of this discussion.

 

[Dadface]

There really isn't a limit as to how high a power something can have. It depends on what end result you get when doing model structure determination.

For example, you may have a sinusoidal signal in your data but not know it. The result is that the model structure will contain terms raised to the power of the Taylor series expansion of a sine or cosine since that is the only way in which the data can be fit without using a sine/cosine signal explicitly.

I would have thought that specific physics equations are used during the modelling.

And it follows that the force associated with such a potential has terms to the 7th and 13th powers.

Two more numbers to add to the list.
Thanks for your inputs everyone.They have all given me further food for thought.

 

[Cyrus]

I would have thought that specific physics equations are used during the modelling.

No, model structure determination means you don't know the model structure. That's not the same as parameter estimation, where you assume a model structure but find the parameters (coefficients).

Remember that any physical system that involve the equations of motion will act as if they have a damping term for anything second order or higher. This is why it's very unlikely to find powers much higher than 2. They will dampen out quicker and quicker.

 

[Cyrus]

Edit: Should read "Remember that any physical system that involve the equations of motion will act as if they have a damping term for anything first order or higher in the state variables.

 

[turin]

Does an infinite potential barrier count (i.e., does the limit of the power as it approaches ∞ count)? The infinite potential barrier is used often enough (e.g. in QM). Or are you looking for physical laws specific to particular processes or theories?

Also, I was thinking that there may be some density functions out there that have "exotic" powers.

 

[Dadface]

Edit: Should read "Remember that any physical system that involve the equations of motion will act as if they have a damping term for anything first order or higher in the state variables.

Does an infinite potential barrier count (i.e., does the limit of the power as it approaches ∞ count)? The infinite potential barrier is used often enough (e.g. in QM). Or are you looking for physical laws specific to particular processes or theories?

Also, I was thinking that there may be some density functions out there that have "exotic" powers.

For want of a better word I was thinking about "generic" equations, all encompassing equations which can be used as the starting point of calculations and which are independent of the system being investigated.