CHE485_Solutions7_2013

CHE 485: STATISTICAL MECHANICS AND THERMODYNAMICS ASSIGNMENT # 7 SOLUTIONS TOPICS: (1) FLUCTUATIONS (2) ISING MODEL

1. (Chandler 3.15) For an open multicomponent system show that 〈 (

〈

〉

)

〈

where

〉

〉 is the fluctuation from the average of number of

〉 to particles of type i and is the chemical potential of type i. Similarly relate 〈 a thermodynamic quantity. Finally, for a one component system in the grand canonical 〉 and relate this quantity to the constant volume heat capacity ensemble evaluate 〈 and compressibility. Solution: Using the given definition of we get: 〈 〉 〈 〈 〉 ( 〈 〉)〉 Expanding, taking averages and canceling out terms we get 〈

〉

〈

〉

〈 〉〈 〉

∑

(∑

) (∑

)

Here the probability distribution is given by: ∑ where Now taking derivatives of (

)

∑

(

)

∑

with respect to

and

we get

Taking the mixed derivative we get ∑ Substituting these values in the above equation we get 〈

〉

( (

〈 〉

)

(

)

)

Similarly 〈

〉

〈

〈 〉 ( 〈 〉) 〈 〉 〉 〈( 〈 〉 〈 〉 〈 〉〈 〉) 〈 〉 〉 〈( 〈 〉 〈 〉 〈 〉〈 〉 〈 〉〈 〉 〈 〉〈 〉〈 〉)〉 〈 〉 〈 〉〈 〉 〈 〉〈 〉 〈 〉〈 〉 We only have to evaluate the first expression, which is given by: 〈

〉

〈 〉

〈 〉 〈 〉

〈 〉〈 〉〈 〉

∑

So putting the other expressions we get the equation as: 〈

〉

(

〈 〉

(

)

)

For a one component system in the grand canonical ensemble the probability distribution is given by: ∑ and The fluctuation in energy is given by: 〈

〉

〈

〉

〈 〉

∑

(

)

[

〈 〉

Now Hence *

∑

〈 〉

*

Taking the second derivative we get:

Also *

〉 +

Here and temperature.

[ *

(

)

]

〈 〉

+

〈

∑

] +

〈 〉

+

〈 〉

*

* * +

〈 〉

〈 〉

+

〈 〉

*

+

〈 〉

* * +

+ 〈 〉

+

* +

are the isothermal compressibility and co-efficient of volume expansion with

2. (Chandler 3.17a) x is a random variable with distribution p(x) in a0 〈 〉 This is just using the previously shown limit of 〈 〉

.

〉 we see that Now based on the definition of 〈 〉 〈∑ when for all i, which is the state when all the spins are aligned with the magnetic field. This shows that there is not fluctuation for this ground state. Problem 5 (Chimowitz 7.3) The energy if interacting spins in a mean field model is given by: ∑ To show that for a single spin 〈 〉 given by

fluctuating in this field the mean field strength is

For a single spin we see that the energy is given by 〈 〉 The fluctuation term is zero as there are no neighboring interactions. Then partition function is given by: ∑ For a single spin there are only two states with

, so the partition function is given by:

Hence the expectation value will be given by: 〈 〉 For we get the equation as We consider the function The derivative is given by The equation has real solution only when the slope of exceeds the slope of the later is just 1. So this only requires that

for which