# Deriving the Ideal Gas Law: A Statistical Story

The ideal gas law is $PV=NkT$, where $P$ is the pressure, $V$ is the volume, $N$ is the number of particles, $k=1.38\times10^{-23}\,\mathrm{m}^2\,\mathrm{kg}\,\mathrm{s}^{-2}\,\mathrm{K}^{-1}$, and $T$ is the temperature. It constitutes one of the simplest and most applied “equations of states” in all of physics, and is (or will become) incredibly familiar to any student of not only physics but also related fields such as chemistry and various engineering disciplines.

Recently, I’ve come to appreciate the ideal gas law as a very good illustrative example of what statistical mechanics is capable of. In the eighteenth and nineteenth (during the industrial revolution), the field of thermodynamics arose in pursuit of answers to how much energy can be extracted from systems. At the time, the notion of temperature was hotly debated, with one ostensibly reasonable theory being that the flow of a fluid, called “caloric,” facilitated heat transfer. It should be appreciated just how many important results could be derived by our predecessors without precise knowledge of the microscopic physics involved.

Statistical mechanics arises from a desire to understand how thermodynamics arises from microscopic physics, either classical or quantum. While classical thermodynamics is able to achieve a great deal, it is philosophically difficult to reconcile the microscopic and macroscopic worlds without some statistical framework that can translate the former to the latter. Moreover, it allows us to determine the thermodynamic behavior of new systems that are hard to reason about a priori—how, for example, should we expect the stars in a star cluster to behave? What about a complicated, novel quantum field theory?

In this article, I will present a number of different derivations of the ideal gas law within the framework of statistical mechanics. In the process, I hope to motivate the different ensembles of statistical mechanics (which ultimately just encode the behavior of possibly large systems for which only a small amount of bulk information is known). In the process, I hope that an overview of these derivations will be a useful introduction for current physics students (or other interested people) to a notoriously (but arguably needlessly) opaque field.

# COVID-19 in the USA, Measured in Tragedies

The last few months have been devastating to our societies, our economies, and our ability to get out of bed before attending meetings. In anticipation of the tumultuous days that inevitably lie immediately ahead, I wanted to get out some thoughts about the pandemic. COVID-19 is, among many other things, a collection of “big” numbers—more than 40 million cases worldwide, almost 1.2 million deaths, and so on. At least for me, when the numbers get this big, I have trouble visualizing them, and need a kind of visual aid. This post is going to be mostly USA-centric, simply because (1) I live here and (2) our collective response to COVID-19 has been comically egregious. I hope the interested reader can proportionately scale their anguish to the global scale.

# Tales from Berkeley Undergraduate Physics and Astronomy

After recently having graduated with a physics and astrophysics degree at the University of California, Berkeley, I can finally fulfill a long-awaited desire to write down a reflection of my college experiences with some sort of satisfying completeness. In writing this reflection, I hope to achieve two broad goals: (1) to view my experiences with respect to some overarching lessons in summary of how I have developed as both a researcher and a person, and (2) to provide some anecdotal advice and expectations for students hoping to follow a similar path through undergraduate.

As for the former point, when looking back over a significant chapter of one’s life, a question begs to be asked: “what was it all for?” I find that it can be incredibly rewarding to try to understand what exactly there was to be gleaned out of a personal experience, mistake, or struggle. Under a composition of such meditations, I can hope to learn precisely how my overall college experience has shaped me from the shy, anxious freshman to the graduating senior today.

As for the latter point, when making big decisions in one’s life, one often hopes to optimize over the space of all possible futures. In practice, I have learned that the closest thing one can do to this is to learn from the experiences of mentoring figures. I would characterize my experiences at Berkeley as overwhelmingly positive, but I also must emphasize that this is only one of many perspectives, with a unique combination of privileges and struggles which surely differs from person to person.

Over the years, after every semester/summer, I have attempted to succinctly capture the overarching “lesson” within a single word. This is, of course, extremely reductive, perhaps overly so at times, but I think that such a summarizing “term” can provide some useful context for further insights. For the purposes of actually writing down my experiences, it also yields a very convenient partition of this essay in some vaguely coherent fashion. Sometimes the summarizing term is academic, reflecting some scientific theme, and other times it is a more general lesson about how I interact with others and view the world that I live in.

I started my journey at Berkeley in the fall of 2016, unsure of what to expect.