Thermal Systems in Equilibrium

Name____________________________________________Date_______________

    In this laboratory session we will explore a simple thermodynamic system made up of ten particles. For our sample we will use coins which allow for us the two quantum states (here, by quantum states we merely mean that there are two discrete values the particles may take on; i.e., heads or tails for each coin. In the real world particles have the quantum property of spin - they act like they are spinning up or down.) We start with a few definitions that will make our task simpler:

DEGENERACY  =  the number of quantum states with the same energy or about the same energy.

g    =  the number of states with the same energy or an energy that falls within some small range about that value. recall in class that we demonstrated that for binary systems (coins are binary systems) that the number of states available to the system can be written as:

g = 2N

where N is the number of particles in the system. Here it represents the number of coins (10.) N is usually a large number for a real thermodynamic system. For example, N may be 1023  so that g can be a very large number. In such a case special ways to calculate such large values need to be developed.

s  =  the spin excess. With coins we simply mean the number of heads minus the number of tails. It just gives us a way of centering about a zero value in the middle. In a quantum system of thermodynamic particles we can treat each particle as if it had spin up or spin down. We will see later that this describes the real world of atomic and subatomic particles. They actually act as if they were really spinning up or spinning down. We address this phenomena when we explore the quantum world.

    If N  =  the total number of particles, then we can write:

N  =  Nh  +  Nt

or the Number of Heads plus the Number of Tails.  In our case the total Number of Heads plus the Number of Tails is 10.

    Since s is the Number of Heads minus the Number of Tails we can write:

s  =  Nh - Nt

    Also, we can write the Number of Heads as:

Nh  =  (N + s) / 2

    and the Number of Tails as:

Nt  =  (N - s) / 2

    To produce all the possible combinations that arise when we combine the ten coins, we can use the binomial expansion. We do not even attempt to do this by hand since there are 210  or 1024 different combinations. The function g has a number of distinct states with ( N/2 + s ) spin up and ( N/2 - s ) spin down states. The following is for information only:

( X + Y )N   = Y0XN  + N Y1XN-1 + ... + YNX0

For N = 10 and X and Y representing the Number of Heads and Numbers of Tails respectively:

g = 1 + 10 + 45 + 120 + 210 + 252 + 210 + 120 + 45 + 10 + 1

g = 1024 = 210 

    For real thermal systems where N is a very very large number, the generating function is impossible to calculate exactly. One approximation is the Stirling Approximation.
It essentially shows us why large N systems are so sharply peaked that any state outside the central tendency has an extremely low probability of existing. We find ourselves with a practical definition of NEVER.

PROCEDURE:

1. Take your coins and the randomizer (can with which to shake them up) and toss them onto the table. Be sure to shake it sufficiently to randomize the system. Record the Number of Heads and Number of Tails facing upwards. In practice one complements the other since there are exactly ten coins. If you know one, you know the other. Make a Table of Trail Number, Number of Heads, Number of Tails and Spin Excess s. If you do this on a spreadsheet program (such as Excel) this is easy.

2.  Repeat this until all groups in the class contribute enough data so a total of 1024 trials is run. Your instructor can help you with this.

Trial Number
# Heads
# Tails 
Spin Excess
1
5 
5 
0
2
4
6
-2
3
6
4
2
4
7
3
4
5
3
7
-4
6
5
5
0
7
6
4
2
8
4
6
-2
Table 1. Sample Summary Sheet - Excel

3. Combine your data with that of the rest of the class. There should be 1024 total trials. It is important that you carefully randomize each trail. Just collecting and slipping the coins on the table may not provide a random trial.

4. Graph both the class' 1024 trials and the theoretical outcomes as predicted by the Binomial Expansion. Plot the spin excess on the X-axis and the frequency of occurence (how many times each set of events happened)

    Comment on the usefulness of the term spin excess for describing a system.


 

5. The probability that a system is in a certain state is merely 1 divided by the  number of accessible states for the system. The we can say that the probability of the system being in a particular state where the exact combination of heads and tails is specified in order is:

P  =  1 / g  =  1 / 2N

    Thus the probability of finding a certain combination of heads and tails dependent on a specific order (like the first is heads, second tails, thrid heads, etc.) is:

1 / 1024

   or a very small number. But in a thermodynamic system, or even our coin system, we treat all the particles as if they were the same. In reality, they are the same, on the average. Then we are more interested in finding the probability of a certain state with a number of ways to get a certain spin excess or:

P(s)  =  g(N,s) / 2N

    For example, if there are 252 ways to get five heads, or zero spin excess, then it is probable:

P(0)  =  252 / 1024

or about 1 out of 4 that we would have a spin excess of zero. How does this compare with your limited number of trails for youre small group? How does it compare with the entire class and the theoretical distribution?
 

It should be recognized that the results are statistical in nature and purport to give values that can be expected on the average rather than receisely for a specific system. For a large N system the deviations from the predicted outcome will be small, very small. We can extend our discussions to entropy, temperature, etc. That is not our purpose here.