identical systems. REM: the easiest replica problem The random energy model (REM) is one of the simplest models of statistical mechanics of disordered systems, and probably the simplest model to show the meaning and power of the replica trick to the level 1 of replica symmetry breaking. The model is especially suitable for this introduction because an exact result by a different procedure is known, and the replica trick can be proved to work by crosschecking of results. See also The cavity method is an alternative method, often of simpler use than the replica method, for studying disordered mean-field problems. It has been devised to deal with models on locally tree-like graphs. Another alternative method is the supersymmetric method. The use of the supersymmetry method provides a mathematical rigorous alternative to the replica trick, but only in non-interacting systems. See for example the book: Also, it has been demonstrated that the Keldysh technique provides a viable alternative to the replica approach. Remarks The above identity is easily understood via Taylor expansion: References S Edwards (1971), "Statistical mechanics of rubber". In Polymer networks: structural and mechanical properties, (eds A. J. Chompff & S. Newman). New York: Plenum Press, ISBN 978-1-4757-6210-5. M Mezard, G Parisi & M Virasoro, "Spin Glass Theory and Beyond", World Scientific, 1987 Papers on Spin Glasses Books on Spin Glasses References to other approaches Statistical mechanics