Discussion_2008_01_18 - The Boltzmann distribution...

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The Boltzmann distribution Particles, as they approach equilibrium, tend toward lower energy, but thermal energy can keep some particles above their lowest energy state. The Boltzmann distribution quantifies the equilibrium distribution of particles in their possible energy states. At equilibrium, the probability that a given particle will have energy u i is Z p e kT u i i - = where Z is a normalization constant chosen such that 1 = j j p . We do not need to know Z if we are only interested in relative probabilities: - - = kT u u e p p 1 2 1 2 In the diagram above, 8 particles have potential energy u 1 and 3 have u 2 . If the system is at equilibrium, we can calculate the energy difference, u 2 -u 1 , corresponding to this population distribution. Since u 2 >u 1 , the equation gives p 2 <p 1 , as shown in the diagram (p 2 =3/11 and p 1 =8/11, assuming these are the only two energy levels). The difference between p 1 and p 2 would be particularly large at low temperature because less thermal energy would be available to keep particles in the higher potential energy state.
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