Microstates and oscillators help

In summary, microstates are the different possible energy states or configurations that a system can have, and oscillators can be used to model and understand these states. Microstates are directly related to entropy, and oscillators have practical applications in fields such as thermodynamics and quantum mechanics. Additionally, they can be used to explain macroscopic phenomena by examining the microscopic interactions within a system.
  • #1
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Homework Statement


The number of microstates of a system of N oscillators containing Q quanta of energy homework is given by

W(N,Q) = (N+Q-1)!/[(N-1)!Q!]

Show that when one further quantum is added to the system the number of microstates increases by a factor of approximately (1+N/Q), provided that N,Q>>1.


Homework Equations





The Attempt at a Solution



So W(N,Q+1) = (N+Q)!/[(N-1)!(Q+1)!]

My problem class leader showed us how to do the question but I am unsure of how he did it, so this is what he did:

W(N,Q+1)/W(N,Q)= [(N+Q)!(N-1)!Q!]/[(N-1)!(Q+1)!(N+Q-1)!]

= [(N+Q)!Q!]/[(Q+1)!(N+Q-1)!] by cancelling

= N+Q/Q+1 = 1+N/Q

I think he got to the last step by approxiamtion since N,Q>>1 but i don't see how that works. Also i don't know why there aren't factorial signs there, whether he or i forgot to write them in or whether they disappear for some reason.

Any explanations would be so helpful, thanks
 
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  • #2
karnten07 said:

Homework Statement


The number of microstates of a system of N oscillators containing Q quanta of energy homework is given by

W(N,Q) = (N+Q-1)!/[(N-1)!Q!]

Show that when one further quantum is added to the system the number of microstates increases by a factor of approximately (1+N/Q), provided that N,Q>>1.


Homework Equations





The Attempt at a Solution



So W(N,Q+1) = (N+Q)!/[(N-1)!(Q+1)!]

My problem class leader showed us how to do the question but I am unsure of how he did it, so this is what he did:

W(N,Q+1)/W(N,Q)= [(N+Q)!(N-1)!Q!]/[(N-1)!(Q+1)!(N+Q-1)!]

= [(N+Q)!Q!]/[(Q+1)!(N+Q-1)!] by cancelling

= N+Q/Q+1 = 1+N/Q

I think he got to the last step by approxiamtion since N,Q>>1 but i don't see how that works. Also i don't know why there aren't factorial signs there, whether he or i forgot to write them in or whether they disappear for some reason.

Any explanations would be so helpful, thanks


First, notice that (Q+1)!/Q! = Q+1 Do you see why?

In general, if you have (X+1)!/ X! , this is equal to X+1 (where X can be anything)

Now, note that [tex] \frac{N+Q}{Q+1} = \frac{1 + N/Q}{1+1/Q} \approx 1+N/Q - 1/Q - N/Q^2 \ldots[/tex] where I have used the binomial expansion and have neglected the corrections of order [itex]1/Q^2, N/Q^2 [/itex] and higher .
 
Last edited:
  • #3
kdv said:
First, notice that (Q+1)!/Q! = Q+! Do you see why?

In general, if you have (X+1)!/ X! , this is equal to X (where X can be anything)

Now, note that [tex] \frac{N+Q}{Q+1} = \frac{1 + N/Q}{1+1/Q} \approx 1+N/Q - 1/Q - N/Q^2 \ldots[/tex] where I have used the binomial expansion and have neglected the corrections of order [itex]1/Q^2, N/Q^2 [/itex] and higher .

Yes i see it now, thankyou so much.
 
  • #4
karnten07 said:
Yes i see it now, thankyou so much.

I made a few typos in my post (now corrected)

I meant to say (Q+1)!/Q! = Q+1

and (X+1)!/X! = X+1

Sorry for the typos.

And you are welcome.
 

1. What are microstates?

Microstates refer to the different possible arrangements or configurations that a system can have at a given time. In the context of oscillators, microstates can refer to the different possible energy states that the oscillators can have.

2. How do oscillators help in understanding microstates?

Oscillators, such as simple harmonic oscillators, can be used to model and study the energy states of a system and how they change over time. This can help in understanding the microstates of a system and the factors that influence them.

3. What is the relationship between microstates and entropy?

Microstates are directly related to entropy, which is a measure of the disorder or randomness of a system. The more microstates a system has, the higher its entropy and vice versa.

4. How can microstates and oscillators be applied in practical situations?

Microstates and oscillators have various applications in fields such as thermodynamics, statistical mechanics, and quantum mechanics. They help in understanding the behavior of physical systems and predicting their properties under different conditions.

5. Can microstates and oscillators be used to explain macroscopic phenomena?

Yes, microstates and oscillators can be used to explain macroscopic phenomena, such as phase transitions and heat transfer. By studying the microstates of a system, we can understand how macroscopic properties emerge from microscopic interactions.

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