This formula describes the thermal entropy S of the second law of thermodynamics. Pi is the Boltzmann factor which represents the probability to find the system excited to its state i in a heat bath with temperature T with which it is in a state of equilibrium. In quantum mechanical operator form for a system with Hamiltonian H the entropy is described by the operator

ρMB =

e−H/kT Tre−H/kT

. (5.63)

This expression is called the Maxwell–Boltzmann density matrix. The entropy is then given by S = −TrρMB ln ρMB. (5.64)