Multiplicity for 3 Dimensional problem

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Homework Help Overview

The discussion revolves around finding an expression for multiplicity (W) in a three-dimensional context for a chain that can move in multiple directions. The original poster is exploring how to extend the concept of multiplicity from one dimension to three dimensions, specifically considering the implications of movement in six total directions.

Discussion Character

  • Exploratory, Conceptual clarification, Mathematical reasoning

Approaches and Questions Raised

  • The original poster attempts to relate the multiplicity of a one-dimensional chain to a three-dimensional scenario. Some participants suggest considering the mathematical representation of multiplicity and how it might generalize from two directions to six. Others question the implications of changing the number of links in the chain and how that affects the expression for multiplicity.

Discussion Status

The discussion is active, with participants exploring different interpretations of the problem and offering insights into how to approach the generalization of multiplicity. There is no explicit consensus yet, but guidance has been provided regarding the mathematical framework involved.

Contextual Notes

Participants are navigating the assumptions around the number of links in the chain and the implications of dimensionality on the multiplicity expression. The original poster is seeking clarity on how to appropriately extend the concept from one dimension to three dimensions without altering the fundamental parameters of the problem.

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Homework Statement


I am having trouble finding an expression for W (multiplicity) for a chain that can move in all possible directions (3 dimensions)

Homework Equations


Multiplicity is the number of possible states over total states.


The Attempt at a Solution

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I understand for a chain that can move in 1 dimension (left or right), the multiplicity is N!/(nr!(N-nr)!). N is the number of monomers the chain is made from and nr is the number of links pointing right and nl = N - nr, is the number of links pointing left. I was wondering how to find the multiplicity when the chain can movie in 3 dimensions (6 total directions)?
 
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If I understand the problem correctly, think about this: the multiplicity for your 1D chain is the coefficient of [itex](x_{r})^{n_r}(x_{l})^{n_l}[/itex] in the expansion of
[tex](x_r + x_l)^N[/tex]
Does that suggest anything to you? Any way to generalize this from 2 directions to 6?
 
Is it possible to raise the expansion to 3N instead of N to illustrate the other possible directions?
 
Well, remember what N represents: the number of links in the chain. If you did that, you'd be getting an expression for a chain with triple the length.

What in [itex](x_r + x_l)^N[/itex] corresponds to the number of directions?
 
Oh I see, thank you
 

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