A collection of 800 indistiguishable coins lie on a table, 720 are heads up and the rest are tails up. The table is violently shook causing many of the coins to flip. Afterward 370 are heads up and the rest are tails up. Compute the quantity Delta lnW for this situation
If n1 are heads up and n2 are tails up, then W = (n1 + n2)!/[(n1)!(n2)!]
Delta Log(W) is thus given by:
Delta Log(W) =
log(720!) + log(80!)-log(370!)- log(430!)
You can simplify this using Stirling's approximation:
log(n!) = n log(n) - n
Thank you very much
To compute the quantity Delta lnW, we need to use the formula:
Delta lnW = ln(W_final) - ln(W_initial),
where W_final is the final number of microstates (ways) and W_initial is the initial number of microstates.
In this situation, since we have 800 indistinguishable coins, the total number of possible microstates can be calculated using the combination formula:
W = C(n, k) = (n+k-1)C(k) ,
where n is the total number of coins and k is the number of heads.
1. Initial number of microstates (W_initial):
n = 800 (total number of coins)
k = 720 (number of heads initially)
W_initial = C(n, k) = (800 + 720 - 1)C(720) = 1519C720 = (1519!)/(720! * 799!)
2. Final number of microstates (W_final):
k = 370 (number of heads after shaking)
W_final = C(n, k) = (800 + 370 - 1)C(370) = 1169C370 = (1169!)/(370! * 799!)
Now, we can calculate Delta lnW:
Delta lnW = ln(W_final) - ln(W_initial) = ln[(1169!)/(370! * 799!)] - ln[(1519!)/(720! * 799!)]
Please note that calculating the exact value using factorials and logarithms may be very computationally intensive. It would be more feasible to use numerical approximation methods or logarithmic identities.
To compute the quantity ΔlnW for this situation, we need to first understand the concept of microstates and macrostates in statistical mechanics.
In this problem, we have a collection of 800 indistinguishable coins on a table. Each coin can either be heads up or tails up. The total number of possible microstates, W, represents all the different ways the coins can be arranged.
The total number of microstates, W, can be calculated using the formula:
W = (n + r - 1)! / (n! * (r - 1)!)
where n is the total number of coins and r is the number of heads up coins.
In the initial state, we have 800 coins and 720 heads up. So, we can calculate the number of microstates for the initial state, W1:
W1 = (800 + 720 - 1)! / (800! * (720 - 1)!)
After violently shaking the table, the number of heads up coins decreases to 370. So, we need to calculate the number of microstates for this final state, W2:
W2 = (800 + 370 - 1)! / (800! * (370 - 1)!)
Now, we can calculate the quantity ΔlnW using the formula:
ΔlnW = ln(W2) - ln(W1)
To compute ΔlnW, we need to calculate the logarithms of W1 and W2. However, calculating the logarithms directly using large factorials can be computationally intensive. Therefore, I will provide you with the steps to compute this value using an approximation method called Stirling's approximation.
1. Calculate the natural logarithm of W1 using Stirling's approximation:
ln(W1) = n*ln(n) - n + 0.5*ln(2πn) + O(1/n)
You can substitute the values of n = 800 and r = 720 in this equation to get the approximate value of ln(W1).
2. Calculate the natural logarithm of W2 using Stirling's approximation:
ln(W2) = n*ln(n) - n + 0.5*ln(2πn) + O(1/n)
Substitute the values of n = 800 and r = 370 in this equation to get the approximate value of ln(W2).
3. Subtract the value of ln(W1) from ln(W2) to get ΔlnW.
Again, I want to emphasize that Stirling's approximation is an approximation method and may introduce some error. It is a useful technique for calculating large factorials or exponential functions.