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Statistical field theory

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Framework to describe phase transitions

Intheoretical physics,statistical field theory (SFT) is a theoretical framework that describes systems with many degrees of freedom, particularly nearphase transitions.^[1] It does not denote a single theory but encompasses many models, including formagnetism,superconductivity,superfluidity,^[2]topological phase transition,wetting^[3]^[4] as well as non-equilibrium phase transitions.^[5] A SFT is any model instatistical mechanics where thedegrees of freedom comprise afield or fields. In other words, themicrostates of the system are expressed through field configurations. It is closely related toquantum field theory, which describes thequantum mechanics of fields, and shares with it many techniques, such as thepath integral formulation andrenormalization.If the system involves polymers, it is also known aspolymer field theory.

In fact, by performing aWick rotation fromMinkowski space toEuclidean space, many results of statistical field theory can be applied directly to its quantum equivalent.^{[citation needed]} Thecorrelation functions of a statistical field theory are calledSchwinger functions, and their properties are described by theOsterwalder–Schrader axioms.

Statistical field theories are widely used to describe systems inpolymer physics orbiophysics, such aspolymer films, nanostructured blockcopolymers^[6] orpolyelectrolytes.^[7]

^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 ofpolymer-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.

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 withforce field depletion force Lennard-Jones potential
Mathematical approaches	Boltzmann equation H-theorem Vlasov equation BBGKY hierarchy stochastic process mean-field theory andconformal 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

Mathematical software	Arbitrary-precision arithmetic Finite element analysis Tensor software Interactive geometry software Optimization software Statistical software Numerical-analysis software Numerical libraries Solvers

Algebraic structures	Algebra of physical space Particle physics and representation theory Feynman integral Poisson algebra Quantum group Renormalization group Spacetime algebra Superalgebra Supersymmetry algebra

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