  Although it has been known for half a century that the physical aging of
glasses in experiments is described well by a linear thermal-history
convolution integral over the so-called material time, the microscopic
definition and interpretation of the material time remains a mystery. We
propose that the material-time increase over a given time interval reflects the
distance traveled by the system's particles. Different possible distance
measures are discussed, starting from the standard mean-square displacement and
its inherent-state version that excludes the vibrational contribution. The
viewpoint adopted, which is inspired by and closely related to pioneering works
of Cugliandolo and Kurchan of the 1990s, implies a "geometric reversibility"
and a "unique-triangle property" characterizing the system's path in
configuration space during aging. Both of these properties are inherited from
equilibrium; they are confirmed by computer simulations of an aging binary
Lennard-Jones system. Our simulations show that the slow particles control the
material time. This motivates a "dynamic-rigidity-percolation" picture of
physical aging. The numerical data show that the material time is dominated by
the slowest particles' inherent mean-square displacement, which is conveniently
quantified by the inherent harmonic mean-square displacement. This distance
measure collapses data for potential-energy aging well in the sense that the
normalized relaxation functions following different temperature jumps are
almost the same function of the material time. Finally, the standard
Tool-Narayanaswamy linear material-time convolution integral description of
physical aging is derived from the assumption that when time is replaced by
distance in the above sense, an aging system is described by the same
expression as that of linear-response theory.
