Statistical Mechanics – Introduction

In a single cubic metre of air at room temperature and pressure, there live on the order of 1025 molecules. Each of these molecules undergoes several billion collisions with its neighbours per second. It is at the very least conceivable that, given the initial position and velocity of every single molecule in the gas, we could use Newton’s laws to propagate each coordinate forward in time and thus describe the microphysical state of the system exactly.

But the computational power necessary to process this much information would be extraordinary. To each particle we would have to append six labels, denoting its position in phase space, an action which would require at least 1012 terabytes of memory. This amount of data exceeds the present-day internet traffic accumulated over an entire year by a factor of a thousand or so.

Even if we could store this much data, the frequency at which particles collide mean that errors are promulgated very quickly. We can only ever store finitely many decimal places – that we have to truncate our data somewhere means our description is only ever an approximation. The further ahead in time we move, the less accurate our model becomes. As little as 10 collisions between particles are enough to amplify an error of one part in a billion to one of order unity.

Such extreme sensitivity to initial conditions means the motion of the particles is impossible to describe deterministically. We are inherently ignorant about the microphysical state of the system. But do we really care? What a particular particle is doing does not matter to us. We only care about the macrophysical state of the system, the one characterised by properties we can measure in a lab. These might include the system’s volume, density, energy, temperature and pressure.

The task of statistical mechanics is to provide a means of bridging the gap between the microphysics of a single particle, and the macrophysical behaviour of a system comprising very many of these particles.

In the next post, we’ll take our first step into the field of statistical mechanics by defining the Gibbs entropy.

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