# Boltzmann's Entropy and the Time Evolution of Macroscopic Systems

 Mathematical Conversations Topic: Boltzmann's Entropy and the Time Evolution of Macroscopic Systems Speaker: Joel Lebowitz Affiliation: Rutgers University; Member, School of Mathematics Date: Wednesday, January 29 Time/Room: 6:00pm - 7:00pm/Dilworth Room

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. Website: http://www.math.ias.edu/~nicks/conversations.html