A general potential energy for a multi-particle system

So, as suggested in the original thread, the potential energy between three particles is parameterised by the distances between each of the three particles, and the angle between two of the lines joining the three particles. This is certainly true of the two particles potential energy function ##V_2 = V_2(r_{ij})##, as you pointed out above. So, I would say that this is a generalisation of the pairwise potential energy function to three particles. In summary, for a system of two or more particles, it is customary to define potential energy functions ##V_{ij}## between pairs of particles, so that the total conservative force (not necessarily total) on any given particle is $$\mathbf{F}_i = \
  • #1
etotheipi
For a system of two or more particles, it is customary to define potential energy functions ##V_{ij}## between pairs of particles, so that the total conservative force (not necessarily total) on any given particle is $$\mathbf{F}_i = \sum_{j\neq i} -\nabla_i V_{ij}$$as a sum over all other particles ##j##, evaluated at the position of ##i##. Also, ##V_{ij} = V_{ij}(\mathbf{r}_i - \mathbf{r}_j)## depends only on the relative separation of both particles.

I wondered if it were ever possible to find a total potential energy function, which would have the property that if ##V = V(\mathbf{r}_1, \mathbf{r}_2, \dots \mathbf{r}_n)##, then that same force ##\mathbf{F}_i## would be $$\mathbf{F}_i = -\nabla_i V$$ for any choice of particle ##i##? ##\nabla## is a linear operator so it seems reasonable to suggest ##V = V_{12} + V_{13} + \dots + V_{n-1, n}##, but I'm not sure if it is possible to evaluate the derivatives of e.g. the potential energy between particles 3 and 4, at the position of particle 2 (e.g. does ##\nabla_2 V_{34}## make sense?), although this would have to be zero.

Thanks!
 
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  • #2
I would say no because of this pairwise description which you have stated above.Maybe some combinatorial arguments are missing to expand the concept to include also such cases.
 
  • #3
From the two equations of OP, It V should be
[tex] V = \frac{1}{2}\sum_{i} \sum_{j,j \neq i}V_{ij}[/tex]
As for your concern
[tex]\nabla_iV_{jk}=0[/tex]
for ##i \neq j,k##
 
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  • #4
anuttarasammyak said:
From the two equations of OP, It V should be
[tex] V = \frac{1}{2}\sum_{i} \sum_{j,j \neq i}V_{ij}[/tex]
As for your concern
[tex]\nabla_iV_{jk}=0[/tex]
for ##i \neq j,k##

I agree, this is what I meant by ##V = V_{12} + V_{13} + \dots + V_{n-1, n}##.

If I consider a 3 particle system s.t. ##V = V_{12} + V_{13} + V_{23}## then $$-\nabla_{\mathbf{r}_1}V = -\nabla_{\mathbf{r}_1}V_{12} -\nabla_{\mathbf{r}_1}V_{13} -\nabla_{\mathbf{r}_1}V_{23} = \mathbf{F}_{12} + \mathbf{F}_{13} + \mathbf{0} = \mathbf{F}_1$$ So in fact I think it does work that ##\mathbf{F}_i = -\nabla_i V##
 
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  • #5
etotheipi said:
For a system of two or more particles, it is customary to define potential energy functions ##V_{ij}## between pairs of particles, so that the total conservative force (not necessarily total) on any given particle is $$\mathbf{F}_i = \sum_{j\neq i} -\nabla_i V_{ij}$$as a sum over all other particles ##j##, evaluated at the position of ##i##. Also, ##V_{ij} = V_{ij}(\mathbf{r}_i - \mathbf{r}_j)## depends only on the relative separation of both particles.

I wondered if it were ever possible to find a total potential energy function, which would have the property that if ##V = V(\mathbf{r}_1, \mathbf{r}_2, \dots \mathbf{r}_n)##, then that same force ##\mathbf{F}_i## would be $$\mathbf{F}_i = -\nabla_i V$$ for any choice of particle ##i##? ##\nabla## is a linear operator so it seems reasonable to suggest ##V = V_{12} + V_{13} + \dots + V_{n-1, n}##, but I'm not sure if it is possible to evaluate the derivatives of e.g. the potential energy between particles 3 and 4, at the position of particle 2 (e.g. does ##\nabla_2 V_{34}## make sense?), although this would have to be zero.

Thanks!
Indeed you are right. The two-body form is only quite common (e.g., in the astronomical many-body problem). In general nothing fundamental (i.e., symmetry principles of Newtonian mechanics) forbid more complicated ##N##-body forces, and indeed that's what has to be taken into account in the nuclear many-body problem. Here are some references:

https://www.physicsforums.com/threa...y-when-carefully-analysed.979739/post-6263544
 
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  • #6
@vanhees71 thanks for the reference. There seem to be two distinct ideas here. The first is of a system of N particles interacting via only two body forces s.t. $$V = \frac{1}{2}\sum_{i} \sum_{j,j \neq i}V_{ij}$$with the corresponding relation of $$\mathbf{F}_i = -\nabla_i V = -\sum_{j \neq i} \nabla_i V_{ij}$$and the second idea is one of N particles interacting via an N body force for which ##V = V(\mathbf{r}_1, \mathbf{r}_2, \dots, \mathbf{r}_N)## and cannot be broken down into pairs ##V_{ij}##.

It's reasonable perhaps to generalise to a system of ##N## particles interacting via ##k## body forces such that ##V = V(\mathbf{r}_1, \mathbf{r}_2, \dots, \mathbf{r}_N) = V_{1,2, \dots, k} + V_{1,3, \dots, k+1} + \dots + V_{N-k+1, N-k+2, \dots, N}##.
 
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  • #7
I found another nice explanation here about many body potential energies, such that in a system of interacting particles $$V_{TOT} = \sum_i^N V_1(\vec{r}_i) + \sum_{i<j}^N V_2(\vec{r}_i, \vec{r}_j) + \sum_{i<j<k}^N V_3(\vec{r}_i, \vec{r}_j, \vec{r}_k) + \dots$$ such that ##\vec{F}_i = -\nabla_{\vec{r}_i} V_{TOT}##, and ##V_2##, ##V_3##, etc. arise due to 2/3/etc. particle interactions respectively. Very interesting stuff!

Often any given three particle potential energy is parameterised by ##V_3 = V_3(r_{ij}, r_{ik}, \theta_{ijk})##.
 
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Related to A general potential energy for a multi-particle system

1. What is a general potential energy for a multi-particle system?

A general potential energy for a multi-particle system is a mathematical representation of the energy stored in a system of multiple particles due to their interactions with each other. It takes into account the positions, masses, and charges of all the particles in the system and can be used to predict their behavior.

2. How is a general potential energy for a multi-particle system calculated?

A general potential energy for a multi-particle system is typically calculated using the laws of physics, such as Newton's laws of motion and Coulomb's law. These laws allow for the determination of the forces between particles, which can then be used to calculate the potential energy of the system.

3. What are some common examples of multi-particle systems with potential energy?

Some common examples of multi-particle systems with potential energy include atoms in a molecule, planets in a solar system, and particles in a solid or liquid. In each of these systems, the particles interact with each other and store potential energy based on their positions and properties.

4. How does potential energy affect the behavior of a multi-particle system?

Potential energy plays a crucial role in determining the behavior of a multi-particle system. As particles move and interact with each other, their potential energy changes, which can lead to changes in their positions and velocities. This can result in phenomena such as chemical reactions, planetary orbits, and phase transitions in materials.

5. Can a general potential energy for a multi-particle system be negative?

Yes, a general potential energy for a multi-particle system can be negative. This indicates that the particles in the system are attracted to each other and have the potential to move closer together. On the other hand, a positive potential energy indicates that the particles are repelling each other and have the potential to move further apart.

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