This paper presents three approaches to providing undergraduate students with a conceptual understanding of the Boltzmann distribution: (1) a simple logical argument for why it is described by an exponential function, (2) a computational experiment to demonstrate and validate this result, and (3) a computer simulation of a laboratory experiment that allows this result to be observed. Together, these three perspectives complement one another to broaden students' understanding and prepare them for more formal, complete treatments. These examples illustrate how the convergence of theoretical, computational, and experimental approaches, applied to a single physical problem, contribute to a deeper and more unified understanding of statistical systems than could otherwise be had using any one of the methods alone.
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More precisely, this is the probability of the system occupying a particular microstate with this value of energy. If there is more than one microstate with the same energy, then the probability of the system having this energy is the product of this probability times the number of microstates that have that energy (i.e., the “degeneracy”).
Feynman used Q for the partition function in 1972, but today most sources use Z, as we do here.
For systems with continuous degrees of freedom—e.g., an ideal gas—this sum is replaced with an integral.