In
mathematical physics
Mathematical physics refers to the development of mathematics, mathematical methods for application to problems in physics. The ''Journal of Mathematical Physics'' defines the field as "the application of mathematics to problems in physics and t ...
and
mathematics, the continuum limit or scaling limit of a
lattice model refers to 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
Brownian motion, or pedesis (from grc, πήδησις "leaping"), is the random motion of particles suspended in a medium (a liquid or a gas).
This pattern of motion typically consists of random fluctuations in a particle's position insi ...
. Indeed, according to
Donsker's theorem, the discrete
random walk
In mathematics, a random walk is a random process that describes a path that consists of a succession of random steps on some mathematical space.
An elementary example of a random walk is the random walk on the integer number line \mathbb ...
would, in the scaling limit, approach the true
Brownian motion
Brownian motion, or pedesis (from grc, πήδησις "leaping"), is the random motion of particles suspended in a medium (a liquid or a gas).
This pattern of motion typically consists of random fluctuations in a particle's position insi ...
.
Terminology
The term ''continuum limit'' mostly finds use in the physical sciences, often in reference to models of aspects of
quantum physics
Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, q ...
, 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 theoretical physics, quantum field theory (QFT) is a theoretical framework that combines classical field theory, special relativity, and quantum mechanics. QFT is used in particle physics to construct physical models of subatomic particles a ...
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 classes
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 '' (also available online
an
''
*
H. Kleinert and V. Schulte-Frohlinde, ''Critical Properties of φ
4-Theories''
World Scientific (Singapore, 2001) Paperback '' (also availabl
online''
Lattice models
Lattice field theory
Renormalization group
Critical phenomena
Articles containing video clips
{{lattice-stub