How Does the Kinetic Energy Sum Relate to Total Energy in a System of Particles?

Click For Summary

Discussion Overview

The discussion revolves around the relationship between the sum of kinetic energy and the total energy in a system of particles, specifically in the context of identical linear harmonic oscillators. Participants explore the implications of inequalities in energy expressions and seek clarification on the underlying principles.

Discussion Character

  • Exploratory, Technical explanation, Conceptual clarification

Main Points Raised

  • Some participants express confusion about the inequality \(\sum^{3N}_{i=1}\frac{p^2_i}{2m}\leq E\) and question whether \(E\) represents the total energy of the system.
  • One participant mentions that the second sum in the energy expression cannot be negative, implying a consideration of energy contributions from potential energy.
  • Another participant suggests that the inequality indicates the sum of kinetic energy being less than or equal to the total energy, which they find intuitive.
  • Further inquiries are made regarding the derivation of phase volume, entropy, and energy per particle for a system of harmonic oscillators, emphasizing the need for clarification on the inequality's validity.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the interpretation of the inequality or the nature of \(E\). Multiple viewpoints and questions remain unresolved.

Contextual Notes

Participants have not fully addressed the assumptions underlying the inequality, nor have they clarified the definitions of total energy and kinetic energy in this context. The discussion also lacks resolution on the mathematical steps involved in deriving the expressions mentioned.

Petar Mali
Messages
283
Reaction score
0
[tex]\sum^{3N}_{i=1}\frac{p^2_i}{2m}\leq E[/tex]

Why I can write this inequality? Is this [tex]E[/tex] energy of system?
 
Physics news on Phys.org
Petar Mali said:
[tex]\sum^{3N}_{i=1}\frac{p^2_i}{2m}\leq E[/tex]

Why I can write this inequality? Is this [tex]E[/tex] energy of system?

For system of [tex]N[/tex] identical linear harmonic oscilators of mass [tex]m[/tex], frequency [tex]\omega[/tex] find phase volume, entropy and energy per particle.

[tex]H(p,x)=\sum^{N}_{i=1}\frac{p^2_i}{2m}+\frac{1}{2}\sum^{N}_{i=1}m\omega^2x^2\leq E[/tex]

Why inequality? Can I get some explanation?

Thanks!
 
The second sum can't be negative.
 
I don't really know too much but...

[tex]\sum^{3N}_{i=1}\frac{p^2_i}{2m}\leq E[/tex]

Looks like it's saying that the sum of the kinetic energy is less than or equal to the total energy, which makes sense to me.
 

Similar threads

Undergrad Lab energy available in threshold (endothermic) reactions
  • · Replies 5 ·
Replies
5
Views
4K
Undergrad Potential energy of a system of particles
  • · Replies 7 ·
Replies
7
Views
701
Graduate The fractional energy loss of charged particle per radiation length
  • · Replies 1 ·
Replies
1
Views
2K
Graduate Why KE of Annihilation Electrons Differs?
  • · Replies 11 ·
Replies
11
Views
2K
Undergrad Energy and states of a particle in a box.
  • · Replies 3 ·
Replies
3
Views
2K
Undergrad Maximum acceleration of an alpha particle?
  • · Replies 5 ·
Replies
5
Views
2K
High School When to think of PE as property of a system vs of a particle
  • · Replies 7 ·
Replies
7
Views
2K
Undergrad P and X completely describing a system of particles
  • · Replies 2 ·
Replies
2
Views
1K
Undergrad Calculate the number of states for a particle in a box
  • · Replies 8 ·
Replies
8
Views
2K
Undergrad Total or kinetic energy in Bethe Bloch stopping power?
  • · Replies 4 ·
Replies
4
Views
2K