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The Multiplicity Function

The Multiplicity Function - molecules in a room for example...

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The Multiplicity Function Usually we are interested not in writing out a general form for all states, but are more focused on one particular state. As we saw above, sometimes there are multiple states with the same number of spins in the up position. Let N up be the number of particles in the "up" state, and N down be the number of particles in the "down" state (then N = N up + N down ). We refer to the number of states with the same values of N and N up by the function g ( N , N up ) , called the multiplicity function. For our system, g ( N , N up ) is given by the coefficient in the preceding sum: g ( N , N up ) = Notice that for very large and very small values of N up , g is small, but for N up = N down , g is a maximum. Probability You may be wondering what this has to do with thermodynamics. In large systems, we can't know exactly what each particle is going to do for all time. To describe just one state of the gas
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Unformatted text preview: molecules in a room, for example, we would need to specify over 10 26 variables, namely the position and velocity of each molecule, and that ignores internal degrees of freedom such as rotation! Therefore when we look at large systems, we need to look at probability and what states and configurations are most likely. Consider a system with g possible states. Let s be a label that indicates the state of a given system. Then let P ( s ) be the probability that the system is in state s , given by: P ( s ) = Therefore, if there are 100 states, the probability of being in any particular state is P (100) = 0.01 . Notice that the probability that the system is in some state, any state, must be one, since we are certain to find the system in one of the g states at any time. Therefore, we write: P ( s ) = 1...
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