the resulting formalism for performing the limit typically introduces many subtleties. When using mean-field theory to perform one's calculations, taking this limit often requires introducing extra order parameters, a property known as "replica symmetry breaking" which is closely related to ergodicity breaking and slow dynamics within disorder systems. Physical applications The replica trick is used in determining ground states of statistical mechanical systems, in the mean-field approximation. Typically, for systems in which the determination of ground state is easy, one can analyze fluctuations near the ground state. Otherwise one uses the replica method. An example is the case of a quenched disorder in a system like a spin glass with different types of magnetic links between spins, leading to many different configurations of spins having the same energy. In the statistical physics of systems with quenched disorder, any two states with the same realization of the disorder (or in case of spin glasses, with the same distribution of ferromagnetic and antiferromagnetic bonds) are called replicas of each other. For systems with quenched disorder, one typically expects that macroscopic quantities will be self-averaging, whereby any macroscopic quantity for a specific realization of the disorder will be indistinguishable from the same quantity calculated by averaging over all possible realizations of the disorder. Introducing replicas allows one to perform this average over different disorder realizations. In the case of a spin glass, we expect the free energy per spin (or any self averaging quantity) in the thermodynamic limit to be independent of the particular values of ferromagnetic and antiferromagnetic couplings between individual sites, across the lattice. So, we explicitly find the free energy as a function of the disorder parameter (in this case, parameters of the distribution of ferromagnetic and antiferromagnetic bonds) and average the free energy over all realizations of the disorder (all values of the coupling between sites, each with its corresponding probability, given by the distribution function). As free energy takes the form: where describes the disorder (for spin glasses, it describes the nature of magnetic interaction between each of the individual sites and ) and we are taking the average over all values of the couplings described in , weighted with a given distribution. To perform the averaging over the logarithm function, the replica trick comes in handy, in replacing the logarithm with its limit form mentioned above. In this case, the quantity represents the joint partition function of