  We derive statistical-mechanical speed limits on dissipation from the
classical, chaotic dynamics of many-particle systems. In one, the rate of
irreversible entropy production in the environment is the maximum speed of a
deterministic system out of equilibrium, $\bar S_e/k_B\geq 1/2\Delta t$, and
its inverse is the minimum time to execute the process, $\Delta t\geq k_B/2\bar
S_e$. Starting with deterministic fluctuation theorems, we show there is a
corresponding class of speed limits for physical observables measuring
dissipation rates. For example, in many-particle systems interacting with a
deterministic thermostat, there is a trade-off between the time to evolve
between states and the heat flux, $\bar{Q}\Delta t\geq k_BT/2$. These bounds
constrain the relationship between dissipation and time during nonstationary
process, including transient excursions from steady states.
