Continuum limit in lattice models
An animated example of a Brownian motion -like random walk on a torus . In the scaling limit, random walk approaches the Wiener process according to Donsker's theorem . In mathematical physics and mathematics , the continuum limit or scaling limit of a lattice model characterizes its behaviour in the limit as the lattice spacing goes to zero. It is often useful to use lattice models to approximate real-world processes, such as Brownian motion . Indeed, according to Donsker's theorem , the discrete random walk would, in the scaling limit, approach the true Brownian motion .
Terminology
The term continuum limit mostly finds use in the physical sciences, often in reference to models of aspects of quantum physics , while the term scaling limit is more common in mathematical use.
Application in quantum field theory
A lattice model that approximates a continuum quantum field theory in the limit as the lattice spacing goes to zero may correspond to finding a second order phase transition of the model. This is the scaling limit of the model.
See also
References
H. E. Stanley, Introduction to Phase Transitions and Critical Phenomena
H. Kleinert , Gauge Fields in Condensed Matter , Vol. I, " SUPERFLOW AND VORTEX LINES", pp. 1–742, Vol. II, "STRESSES AND DEFECTS", pp. 743–1456, World Scientific (Singapore, 1989) ; Paperback ISBN 9971-5-0210-0 (also available online: Vol. I and Vol. II ) H. Kleinert and V. Schulte-Frohlinde, Critical Properties of φ4 -Theories , World Scientific (Singapore, 2001) ; Paperback ISBN 981-02-4658-7 (also available online )
