Why are steps not visible in the continuum limit of the Frenkel Kontorova model?

  • Thread starter LagrangeEuler
  • Start date
  • Tags
    Model
In summary, the model discussed in the conversation is a discrete model of many point particles. However, a continuum limit can be taken, which results in the sine-Gordon equation. This leads to a wave equation for the string and in the continuum limit, it is not possible to see steps in the velocity function, as shown in FIG 1 in the referenced paper. The reason for this is because the model is discrete and in the continuum limit, the steps disappear.
  • #1
LagrangeEuler
717
20
http://www.iop.kiev.ua/~obraun/fk-intro.htm#model

Question? Why this model is discreet? As far as I see particles can have any position?

Why then is written in the text then In the continuum-limit approximation we come to the sine-Gordon (SG) equation?
 
Physics news on Phys.org
  • #2
It's discrete in the sense that it's a model of many point particles. You can take a continuum limit. E.g., as the most simple case you can model a string first as a chain of harmonically coupled point particles. The continuum limit leads to the wave equation for the string.
 
  • #3
Thanks a lot for the answer.

And for example in this paper

http://fmc.unizar.es/people/juanjo/papers/falo93.pdf

steps appear because model is discrete. In continuum limit it would be impossible to see steps in the ##v(F)## (FIG 1). Is it possible to explain why in continuum limit is impossible to see steps?
 
Last edited by a moderator:

FAQ: Why are steps not visible in the continuum limit of the Frenkel Kontorova model?

What is the Frenkel Kontorova model?

The Frenkel Kontorova model is a mathematical model used to study the dynamics of a chain of particles, also known as atoms or molecules, connected by springs and placed on a periodic potential. It was developed in the mid-20th century by Yakov Frenkel and Theodor Kontorova to understand the behavior of solid-state materials, particularly crystals.

How does the model describe the motion of particles?

The model assumes that the particles are connected by elastic springs and are subjected to a periodic potential, which can be thought of as a series of hills and valleys. The particles can move freely along the potential, but are constrained by the springs. The model takes into account the interactions between neighboring particles, as well as the external forces acting on them, to describe their motion over time.

What are the applications of the Frenkel Kontorova model?

The Frenkel Kontorova model has been used to study a wide range of physical phenomena, including the behavior of dislocations in crystals, the dynamics of charge density waves in materials, and the movement of atoms on surfaces. It has also been applied to fields outside of materials science, such as biology and economics.

What are the limitations of the Frenkel Kontorova model?

Like any mathematical model, the Frenkel Kontorova model has its limitations. It is based on several simplifying assumptions, such as the particles being connected by linear springs and the potential being perfectly periodic. These assumptions may not accurately reflect the real-world conditions of certain materials, and can lead to discrepancies between the model's predictions and experimental observations.

How has the Frenkel Kontorova model contributed to our understanding of materials?

The Frenkel Kontorova model has provided valuable insights into the behavior of materials at a microscopic level. It has helped researchers understand the mechanisms behind phenomena such as plastic deformation and phase transitions, and has guided the development of new materials with specific properties. Its applicability to a wide range of systems has made it a useful tool for studying the behavior of materials in various conditions and environments.

Similar threads

A How Do Kinks and Particles Interact in the Frenkel-Kontorova Model?
Replies
1
Views
930
I Exploring ΛCDM Model: Cosmology Intertwined in 4 Pre-Prints
Replies
1
Views
2K
I Why Can We Still Observe the Cosmic Microwave Background?
Replies
7
Views
2K
Correspondence principle for a step potential
Replies
6
Views
3K
BH can't evaporate in real surroundings (George Ellis)
Replies
6
Views
2K
Uniqueness of deSitter spacetime and the Standard Model (hints from two papers)
Replies
3
Views
3K
I Modeling the effects of GW and the "Earth Frame"
2
Replies
66
Views
6K
Statistical mechanics, confused about an approximation and limits of integration
Replies
1
Views
1K
I Some questions about Cosmic Microwave Background radiation
Replies
13
Views
3K
Which Morelia audios are available?
Replies
6
Views
3K

Hot Threads

Recent Insights

Back
Top