Mathematical Conversations

Boltzmann defined the entropy, \(S(M)\), of a macroscopic system in a macrostate \(M\) as the "log of the volume of phase space" corresponding to the system being in \(M\). This definition was extended by von Neumann to quantum systems as "the log of the dimension of the linear subspace of the Hilbert space" corresponding to the system being in \(M\). This not only provides a formula for computing \(S(M)\), which agrees with the thermodynamic entropy when the system is in local thermal equilibrium, but also explains the origin of the time-asymmetric second law in time-symmetric microscopic laws. It shows in particular that if there is a deterministic autonomous equation describing the time evolution of a macrostate \(M(t)\) of an isolated system, be it hydrodynamic or kinetic, e.g. the Boltzmann equation for the empirical density in the 6-dimensional position-velocity space of a dilute gas, it must give an \(S(M(t))\) which is monotone non-decreasing in \(t\).

About Mathematical Conversations: We meet in Harry's Bar at 6pm, where free drinks are provided. After 20 minutes, we move to the Dilworth room, where the speaker gives a 20-minute talk, followed by 15 minutes of discussion with the audience. After that we return to the bar for further discussions.