Statistical field theory

In theoretical physics, statistical field theory (SFT) is a theoretical framework that describes phase transitions.^[1] It does not denote a single theory but encompasses many models, including for magnetism, superconductivity, superfluidity,^[2] topological phase transition, wetting^[3]^[4] as well as non-equilibrium phase transitions.^[5] A SFT is any model in statistical mechanics where the degrees of freedom comprise a field or fields. In other words, the microstates of the system are expressed through field configurations. It is closely related to quantum field theory, which describes the quantum mechanics of fields, and shares with it many techniques, such as the path integral formulation and renormalization. If the system involves polymers, it is also known as polymer field theory.

In fact, by performing a Wick rotation from Minkowski space to Euclidean space, many results of statistical field theory can be applied directly to its quantum equivalent.^{[citation needed]} The correlation functions of a statistical field theory are called Schwinger functions, and their properties are described by the Osterwalder–Schrader axioms.

Statistical field theories are widely used to describe systems in polymer physics or biophysics, such as polymer films, nanostructured block copolymers^[6] or polyelectrolytes.^[7]

Notes

^ Le Bellac, Michel (1991). Quantum and Statistical Field Theory. Oxford: Clarendon Press. ISBN 978-0198539643.
^ Altland, Alexander; Simons, Ben (2010). Condensed Matter Field Theory (2nd ed.). Cambridge: Cambridge University Press. ISBN 978-0-521-76975-4.
^ Rejmer, K.; Dietrich, S.; Napiórkowski, M. (1999). "Filling transition for a wedge". Phys. Rev. E. 60 (4): 4027–4042. arXiv:cond-mat/9812115. Bibcode:1999PhRvE..60.4027R. doi:10.1103/PhysRevE.60.4027. PMID 11970240. S2CID 23431707.
^ Parry, A.O.; Rascon, C.; Wood, A.J. (1999). "Universality for 2D Wedge Wetting". Phys. Rev. Lett. 83 (26): 5535–5538. arXiv:cond-mat/9912388. Bibcode:1999PhRvL..83.5535P. doi:10.1103/PhysRevLett.83.5535. S2CID 119364261.
^ Täuber, Uwe (2014). Critical Dynamics. Cambridge: Cambridge University Press. ISBN 978-0-521-84223-5.
^ Baeurle SA, Usami T, Gusev AA (2006). "A new multiscale modeling approach for the prediction of mechanical properties of polymer-based nanomaterials". Polymer. 47 (26): 8604–8617. doi:10.1016/j.polymer.2006.10.017.
^ Baeurle SA, Nogovitsin EA (2007). "Challenging scaling laws of flexible polyelectrolyte solutions with effective renormalization concepts". Polymer. 48 (16): 4883–4899. doi:10.1016/j.polymer.2007.05.080.

Itzykson, Claude; Drouffe, Jean-Michel (1991). Statistical Field Theory. Cambridge Monographs on Mathematical Physics. Vol. I, II. Cambridge University Press. ISBN 0-521-40806-7. ISBN 0-521-40805-9
Parisi, Giorgio (1998). Statistical Field Theory. Advanced Book Classics. Perseus Books. ISBN 978-0-7382-0051-4.
Simon, Barry (1974). The P(φ)₂ Euclidean (quantum) field theory. Princeton University Press. ISBN 0-691-08144-1.
Glimm, James; Jaffe, Arthur (1987). Quantum Physics: A Functional Integral Point of View (2nd ed.). Springer. ISBN 0-387-96477-0.

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v t e Statistical mechanics
Theory	Principle of maximum entropy ergodic theory
Statistical thermodynamics	Ensembles partition functions equations of state thermodynamic potential: U H F G Maxwell relations
Models	Ferromagnetism models Ising Potts Heisenberg percolation Particles with force field depletion force Lennard-Jones potential
Mathematical approaches	Boltzmann equation H-theorem Vlasov equation BBGKY hierarchy stochastic process mean-field theory and conformal field theory
Critical phenomena	Phase transition Critical exponents correlation length size scaling
Entropy	Boltzmann Shannon Tsallis Rényi von Neumann
Applications	Statistical field theory elementary particle superfluidity Condensed matter physics Complex system chaos information theory Boltzmann machine