Continuum limit

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.

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.

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

• Universality class

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)