  We consider the steepest entropy ascent (SEA) ansatz to describe the
non-linear thermodynamic evolution of a quantum system. Recently this principle
has been dubbed the fourth law of thermodynamics (Beretta Gian Paolo. 2020 The
fourth law of thermodynamics: steepest entropy ascent). A unique global
equilibrium state exists in this context, and any other state is driven by the
maximum entropy generation principle towards this equilibrium. We study the SEA
evolution of a continuous-time quantum walker (CTQW) on a cycle graph with N
nodes. SEA solutions are difficult to find analytically. We provide an
approximate scheme to find a general single-particle evolution equation
governed by the SEA principle, whose solution produces dissipation dynamics. We
call this scheme fixed Lagrange's multiplier (FLM) method. In the Bloch sphere
representation, we find trajectories traced out by the Bloch vector within the
sphere itself. We have discussed these trajectories under various initial
conditions for the case of a qubit. A similar dissipative motion is also
observed in the case of CTQW, where probability amplitudes have been used to
characterize decoherence. Our FLM scheme shows good agreement with numerical
results. As reported in the text, in CTQW, a strong delocalization exists for
low system relaxation time.
