Student understanding of the Boltzmann factor

Abstract

[This paper is part of the Focused Collection on Upper Division Physics Courses.] We present results of our investigation into student understanding of the physical significance and utility of the Boltzmann factor in several simple models. We identify various justifications, both correct and incorrect, that students use when answering written questions that require application of the Boltzmann factor. Results from written data as well as teaching interviews suggest that many students can neither recognize situations in which the Boltzmann factor is applicable nor articulate the physical significance of the Boltzmann factor as an expression for multiplicity, a fundamental quantity of statistical mechanics. The specific student difficulties seen in the written data led us to develop a guided-inquiry tutorial activity, centered around the derivation of the Boltzmann factor, for use in undergraduate statistical mechanics courses. We report on the development process of our tutorial, including data from teaching interviews and classroom observations of student discussions about the Boltzmann factor and its derivation during the tutorial development process. This additional information informed modifications that improved students' abilities to complete the tutorial during the allowed class time without sacrificing the effectiveness as we have measured it. These data also show an increase in students' appreciation of the origin and significance of the Boltzmann factor during the student discussions. Our findings provide evidence that working in groups to better understand the physical origins of the canonical probability distribution helps students gain a better understanding of when the Boltzmann factor is applicable and how to use it appropriately in answering relevant questions.

Identifying student difficulties with conflicting ideas in statistical mechanics

“…We have previously reported the development and implementation of a tutorial designed to help students in an upper-division statistical mechanics course develop a robust understanding of the Boltzmann factor. [1] Here we present the results of an investigation into student understanding of the connection between the Boltzmann factor and the density of states function.…”

“…In the physical scenario above, the Boltzmann factor is proportional to the multiplicity of the reservoir as a function of the energy of the system (ω res ∝ e −E/kT ). 1 To discuss the probability of the system having a particular energy, one must consider the total multiplicity given by the product of ω sys and ω res :…”

Designing upper-division thermal physics assessment items informed by faculty perspectives of key content coverage

“…Upper-division investigations are comparatively less common, especially for statistical mechanics, though some exist. Several examples of foci for these few investigations include student reasoning surrounding the Boltzmann factor [23] and Taylor expansions in statistical mechanics [24], as well as some statistical mechanics instructional methods, such as using statistical approaches to teach entropy [25]. Some researchers have investigated student reasoning in thermal physics through interviews, reviews of student coursework, or conceptual assessments, while others have utilized tutorials.…”

Editorial: Focused Collection: PER in Upper-Division Physics Courses

“…One article, by Smith et al [36] presents evidence from written work and interviews of persistent student difficulties with heat engines and the second law. The second article by Smith, Mountcastle, and Thompson [37] focuses on student understanding of the Boltzmann factor and its connections to multiplicity and entropy. This paper discusses the implementation with multiple student populations of a tutorial that takes the form of a "guided derivation" of the Boltzmann factor and shows how such a tutorial can enhance learning.…”