202301200900 Problem 2.1

Suppose you flip four fair coins.

(a) Make a list of all the possible outcomes, as in Table 2.1.
(b) Make a list of all the different “macrostates” and their probabilities.
(c) Compute the multiplicity of each macrostate using the combinatorial formula 2.6, and check that these results agree with what you got by brute-force counting.

_{Extracted from D. V. Schroeder. (2000). An Introduction to Thermal Physics.}

Background.

For a fair coin, the probabilities of heads-up ( $\textrm{H}$ ) and tails-up ( $\textrm{T}$ ) are $50:50$ odds, i.e., $P(\textrm{H})=P(\textrm{T})=\displaystyle{\frac{1}{2}}$ the same.

$\begin{tabular}{cccc} Coin 1 & Coin 2 & Coin 3 & Coin 4 \\\hline H & H & H & H \\ H & H & H & T \\ H & H & T & H \\ H & H & T & T \\ H & T & H & H \\ H & T & H & T \\ H & T & T & H \\ H & T & T & T \\ T & H & H & H \\ T & H & H & T \\ T & H & T & H \\ T & H & T & T \\ T & T & H & H \\ T & T & H & T \\ T & T & T & H \\ T & T & T & T \\ \end{tabular}$

Any one possibility of the

$P_\textrm{w/ rplc}(2,4)=\text{}^2P_4=2^4=16$

permutations is called a microstate, and all (sixteen) microstates compose ensemble the probability distribution of the

$5=\mathrm{ord}(\{4\textrm{H}0\textrm{T},3\textrm{H}1\textrm{T},2\textrm{H}2\textrm{T},1\textrm{H}3\textrm{T},0\textrm{H}4\textrm{T}\})$

combinations, called the (five) macrostates. The number of microstates corresponding to a given macrostate is called the multiplicity of that macrostate. With full knowledge of the microstates of a system are its macrostates fully known; the reverse is not true.

If there are $N$ coins, the multiplicity of the macrostate with $n$ heads is