Trace

Forming the trace of a system is similar to coarse graining. When the trace is formed, the information on some variables is completely lost, while coarse graining implies a partial loss of information for variables.

When the trace is formed, the macroscopic motion is non-invertible, 0 while the microscopic motion is invertible, = 0. This leads to the correct form for the second law of thermodynamics, SL-IV.