A disorder is quenched when it is not evolving at the timescale of the evolution of the state-variables, and we can place them in the Hamiltonian as fixed objects. Though, the disorder takes different values at different realizations, following an underlying probability distribution. Spin-glasses have this quenched disorder. Compared to that, structural glasses have self-generated disorder during the dynamics that evolves at the same time-scale of the state-variables.

Disorder creates frustration—all the interacting pairs cannot choose to be in a satisfied state simultaneously. That leads to movements in the state variables in arbitrary directions that can cause slowing down of the relaxation process that should gradually bring the system to equilibrium. That is how meta-stable states appear.

Given the disorder is in the Hamiltonian, it appears analytically also in the partition function and in the expression of the free energy—hence, it appear in all other physical properties that can be derived from the free energy. But, how can different instances of the same material have different physical properties? The quantities must be self-averaging at the thermodynamic limit, when the system grows sufficiently larger. That inspires the analysis to take an average over the probability distribution of the quenched disorder, when building the theory of such a system, to get disorder-independent values of the physical quantities.

When each component of a thermodynamic system interact with only a finite number of other components, it is called finite dimensional system. One can argue that, surface to volume ratio vanishes for an infinite system. Hence, for a finite-dimensional system, the free energy being extensive on the number of components, it should show a central tendency.

Here comes the tricky part.

In the calculation, one needs to do a way more difficult integration (involving logarithm) in quenched average, in compared to the annealed average. In the annealed average, both the disorder and the state variables are moving fast at the same timescale and we want to find an disorder-averaged partition function and calculate the free energy corresponding to that. In the quenched average, instead, free energies are different for different realizations fo the systems—we calculate first the individual free energies from the respective partition functions, and then we average it over the disorder.

Here comes the need of the so-called replica trick, to avoid the integration of log in the quenched averaging.