It comes sometimes from Laplace equation.
But if such dependency, there are options :
one could use exp(-r), which is much easier in calculations
1/sqrt(r), which is the solution in fracture mechanics, and then a potential is not singular.
That will imply a geometrical mean and splittin of variables and would be more easy to handle in Quantum field theories.
Dear Lena,
Perhaps I am not fully understanding you, but your question seems without meaning. From your question I think as you believe that the potential is only a mathematical solution of the Laplace equation. Obviously, that is not true because it must have an inmediate physical interpretation through the an observable force F.
Let me to put you some examples. In electromagnetism the scalar potential V(r), is associated to an electrostatic force
F= - e \nabla V(r)
being e the electric charge and the potential
V(r)= K e/r
where K is a constant e the electric charge which produces de potential and r the distance. The same could be done for the Newton gravitation force given by
F= - m \nabla V(r)
being m the mass and the gravitational potential
V(r)= G m/r
with G the gravitational constant of gravitation. F is a force in Newtons in MKS units in both cases, while the electric potential is in volts V and the gravitational in J/kg.
Notice that if you had an exponential as potential V(r)= exp(-kr) then the force would be very different than the previous ones. It is true that you could try to approach in series, but in such a case you need to say where all the terms are that you don't need to use.
By the way, I assume that V(r)= exp(-kr) is multiplied by a constant with units of potential, because it is not possible to write only one exponential.
The 1/r potential is what is found in nature for fields in electromagnetism and gravity. 1/r dependence for radiated fields results in conservation of energy at increasing range. Looking at different fields might result in a very interesting theoretical universe but doesn't describe electromagnetics. Exponential dependence is used to describe waves (with imaginary argument), attenuation and evanescence.
In general, it comes as the fundamental solution of the Laplace operator in the 3D space.
Yes, and inside a cloud of electrons, the potential is proportional to r?
In elasticity and elsewhere in real, the space is proportional to r, for a line element of a circle. It is plane, and no source.
(When a crack, the potential to strains and stress is proportional to sqrt(r), strain has no physical scale or dimension(it is length/length), and the energy is the product thus 1/r, if I remember correctly.)
Sergey and Daniel, you are right that mathematical solution of Poisson equation with delta-function of Dirac on RHS in 3D is given by 1/r. But this is mathematical perspective. From the perspective of physicist, we should think about the range of applicability. While nobody doubts that this field is valid at long distances, the behavior in micro world can be different.
However, physicists prefer to think that this is valid even in micro distance, and add other (low-distance) potentials to describe interactions of elementary particles at micro scales. If you believe quantum physics, then nothing is defined there like in classical world. I mean Heizenberg inequality for example. But there is an interesting formal symmetry between the shapes of gravity and electrostatic potentials, in macro world.
I am wondering if Lena is trying to say something like this: When we use vacuum solutions we get the 1/r potential then built Green's functions and PT about this. If we created a fictitious medium with a different ("screened") field to act as a background, some calculations might give a nicer perturbation theory.
I have tried some analogous things myself. Not sure of the range such a method can be pushed but it is interesting.
Is this what you were thinking?
Well not in this question, but in other articles, yes. In many cases, it is not fictitious and it fits with entries in SO(3). That could be decomposed as a rotation and a stretch in material modeling, and could be a background, I'll look into it.
What do you mean by PT?
Lena, since you are using language SO(3) [this is Lie group of rotations, see https://en.wikipedia.org/wiki/Rotation_group_SO(3) ], I suspect that you have education in mathematical physics, but more biased towards mathematics.
Clifford probably means by PT perturbation theory, that is often used by mathematical physicists more biased towards physics. For better understanding, one needs to embrace more complementary field.
Yuri, thanks for the comment. I shall call it something else, if using as a Cosserat continuum. Arnold would not want his chair to be stretched, or partly forgotten?
A similar potential is used in plasma and electrolytes to find the Debye sphere, the Debye radius
I know distributions of fields in cast iron melt, from experiments, that could relate to that. If a plasma.
John von Neumann: ``Truth is much too complicated to allow anything but
approximations.''
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