  Non-equilibrium thermodynamics provides a general framework for understanding
non-equilibrium processes, particularly in small systems that are typically far
from equilibrium and dominated by fluctuations. However, the experimental
investigation of non-equilibrium thermodynamics remains challenging due to the
lack of approaches to precisely manipulate non-equilibrium states and dynamics.
Here, by shaping the effective potential of energy, we propose a general method
to construct a non-equilibrium steady state (NESS) with arbitrary energy
distribution. Using a well-designed energy-dependent feedback damping, the
dynamics of an optically levitated nanoparticle in vacuum is manipulated and
driven into a NESS with the desired energy distribution. Based on this
approach, a phonon laser state is constructed with an ultra-narrow linewidth of
6.40 uHz. Such an arbitrary NESS construction method provides a new approach to
manipulating the dynamics processes of micromechanical systems and paves the
way for the systematic study of non-equilibrium dynamics in interdisciplinary
research fields.
