  Near a critical point, the equilibrium relaxation time of a system diverges
and any change of control/thermodynamic parameters leads to non-equilibrium
behavior. The Kibble-Zurek problem is to determine the dynamical evolution of
the system parametrically close to its critical point when the change is
parametrically slow. The non-equilibrium behavior in this limit is controlled
entirely by the critical point and the details of the trajectory of the system
in parameter space (the protocol) close to the critical point. Together, they
define a universality class consisting of critical exponents-discussed in the
seminal work by Kibble and Zurek-and scaling functions for physical quantities,
which have not been discussed hitherto. In this article, we give an extended
and pedagogical discussion of the universal content in the Kibble-Zurek
problem. We formally define a scaling limit for physical quantities near
classical and quantum transitions for different sets of protocols. We report
computations of a few scaling functions in model Gaussian and large-N problems
and prove their universality with respect to protocol choice. We also introduce
a new protocol in which the critical point is approached asymptotically at late
times with the system marginally out of equilibrium, wherein logarithmic
violations to scaling and anomalous dimensions occur even in the simple
Gaussian problem.
