Discrete energy distribution in a simple system

Probability Level pending

You have a system of 5 identical, but distinguishable atoms (you can imagine that each of them has their specific name). The total energy of this system is 5e and all of this energy is distributed somehow in the system. The atoms can take only discrete values of energy: 0, e, 2e ... 5e.

How many different configurations could exist, taking into account that each atom is distinguishable? (You can also try to find a general formula for N atoms and with an energy of Ne)

130 126 98 144

Ioan Sabin Taranu by Ioan Sabin Taranu , 24, Sweden

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2 solutions

Henry U
Nov 8, 2018

By the method of Stars and Bars there are

( n + k − 1 k − 1 ) \displaystyle \binom {n+k-1} {k-1}

ways to split up the energy.

k = number of atoms = 5 k = \text{number of atoms} = 5

n = total energy of the system = 5 n = \text{total energy of the system} = 5

( n + k − 1 k − 1 ) = ( 9 4 ) = 126 \displaystyle \binom {n+k-1} {k-1} = \binom 9 4 = \boxed{126}

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