Introduction to Statistical Physics, Spring 2015Instructor: Yih-Yuh Chen

Assignment 5

The due date for this assignment is Wednesday 12:00 pm, June 3, 2015.

1. (Non-negativeness of the specific heat of all matter.)

(a) From the expression for the internal energy in canonical ensemble,

X

please show that

=

X2

X

!X

!= 2 2

(b) If we write P , then please expand 2 to showthat 2 2 is non-negative.

(c) Why is it that the above shows that the isochoric specific heat of a system isalways positive?

2. Consider two weakly interacting systems which can exchange particles. Classical ther-modynamics dictates that their chemical potentials satisfy 1 = 2, with

= ( )

For a volume of ideal gas placed on the ground we will assume that its entropy is givenby a certain function : ( ). If the gas container is moved to a placewhere each particle in it gains an extra potential energy , then we we expect that = ( ). Now we consider two containers of ideal gas, one placed on theground and the other raised to some height above the ground. Then, = . Ifthe Number 2 container is at a higher place and there is a very small tube connectingthe two containers so that thermodynamic equilibrium has been reached, then we musthave

(1 1 1)1 = ( 0 2 2)

2 0=22

(0 2 2) 0

0=22

!

assuming that the number of particles inside the tube is very small and thus can beignored.

1

(a) Classical thermodynamics together with the assumption that the entropy is anextensive quantity tells us that

= ln + ln

+ constant

=

ln

+ ln

+ constant

Please show that 22 =

11

(1)Notice that we arrive at this result using pure classical thermodynamics.

(b) From microcanonical ensemble we expect that

ln 32 !3

Would you derive the same result of Eqn.(1) had you used this expression for ?(c) From microcanonical ensemble we know that, for the whole system, the total

entropy is

= ln" X1+2

1 (1 1 1) 2 (2 2 2)#

Assuming that the sum above is dominated by the terms near 1 = 1 and2 = 2 which maximize 1 (1 1 1) 2 (2 2 2), please show that thisleads to the familiar assertion that

1 = 2 = (

)

(d) Suppose = , with = 1 km. Please evaluate numerically the Boltzmannfactor exp ( ) for a nitrogen molecule at = 300 K.

3. If we throw a fair dice many times, then the expectation value of the number of dotsyou obtain should be

6X=1

= 1 16+ 2 1

6+ + 6 1

6= 35

(a) Suppose the expectation value is 3.0 (= 35 05) for some loaded dice, and youwish to use the maximum entropy formalism to assign the best probabilisticdistribution {} to the dice. Please show that

= P

for some number .2

(b) Find the numerical value of .(c) Suppose the expectation value is 4.0 (= 35 + 05) for another loaded dice, and

you wish to use the maximum entropy formalism to assign the best probabilisticdistribution {} to the dice. Please find the numerical value of for this dice.

4. Carter Problem 15-6.

5. Carter Problem 16-6.

3