  In the field of non-equilibrium phase transitions, the Kibble-Zurek mechanism
(KZM) is undoubtedly an important discovery, pointing out that some universal
scaling rules are applied to a wide range of physical systems from quantum to
the cosmos in complex non-equilibrium continuous phase transitions. However,
except for some scaling relations in specific cases, the algebraic-based KZM
can not provide further details on the topological defect generation laws. In
this work, we propose an analytical-based KZM that can accurately predict
topological defect generation for a given quenching condition. Compared with
the conventional KZM, our theory is more accurate and more widely applicable,
especially in non-linear quenching conditions and inhomogeneous structural
systems, where it has more obvious advantages. Our work reveals a fundamental
and intrinsic mechanism in non-equilibrium phase transitions, blazing a new
trail for accurate description and manipulation of topological defect
generation.
