The Ideal Gas in Slow Time
Abstract We continue our exploration of thermodynamics at long observational timescales, "slow time," by including turbulent dynamics leading to a condition of fluctuating local equilibrium. Averaging these fluctuations in wind speed and temperature results in a velocity distribution with heavy tails which, however, are necessarily truncated at some large molecular speed preserving all moments of the velocity distribution including the energy. This leads to an expression for the ideal gas law in slow time which as its core has the superficially familiar term 3 2 N k th\frac{3}{2}Nk\theta in addition to a term accounting for the large scale fluctuations, which is also proportional to the particle number N; th is a new temperature including thermalization of wind. The traditional temperature T no longer exists. Likewise, the additional energy term necessitates a new quantity that parallels entropy in the sense that it captures hidden degrees of freedom. Like entropy, it captures physical properties manifesting indirectly, but on scales larger than the familiar laboratory scales. We call this quantity epitropy.