In
mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, the Kardar–Parisi–Zhang (KPZ) equation is a non-linear
stochastic partial differential equation
Stochastic partial differential equations (SPDEs) generalize partial differential equations via random force terms and coefficients, in the same way ordinary stochastic differential equations generalize ordinary differential equations.
They hav ...
, introduced by
Mehran Kardar,
Giorgio Parisi
Giorgio Parisi (born 4 August 1948) is an Italian theoretical physicist, whose research has focused on quantum field theory, statistical mechanics
In physics, statistical mechanics is a mathematical framework that applies statistical methods ...
, and Yi-Cheng Zhang in 1986.
It describes the temporal change of a height field
with spatial coordinate
and time coordinate
:
:
Here,
is
white
White is the lightest color and is achromatic (having no chroma). It is the color of objects such as snow, chalk, and milk, and is the opposite of black. White objects fully (or almost fully) reflect and scatter all the visible wa ...
Gaussian noise
Carl Friedrich Gauss (1777–1855) is the eponym of all of the topics listed below.
There are over 100 topics all named after this German mathematician and scientist, all in the fields of mathematics, physics, and astronomy. The English eponymo ...
with average
and
second moment
,
, and
are parameters of the model, and
is the dimension.
In one spatial dimension, the KPZ equation corresponds to a stochastic version of
Burgers' equation
Burgers' equation or Bateman–Burgers equation is a fundamental partial differential equation and convection–diffusion equation occurring in various areas of applied mathematics, such as fluid mechanics, nonlinear acoustics, gas dynamics, and ...
with field
via the substitution
.
Via the
renormalization group
In theoretical physics, the renormalization group (RG) is a formal apparatus that allows systematic investigation of the changes of a physical system as viewed at different scales. In particle physics, it reflects the changes in the underlying p ...
, the KPZ equation is conjectured to be the
field theory of many
surface growth models, such as the
Eden model, ballistic deposition, and the weakly asymmetric single step solid on solid process (SOS) model. A rigorous proof has been given by Bertini and Giacomin in the case of the SOS model.
KPZ universality class
Many
interacting particle system
In probability theory, an interacting particle system (IPS) is a stochastic process (X(t))_ on some configuration space \Omega= S^G given by a site space, a countably-infinite-order graph G and a local state space, a compact metric space S ...
s, such as the totally
asymmetric simple exclusion process, lie in the KPZ
universality class
In statistical mechanics, a universality class is a collection of mathematical models which share a single scale-invariant limit under the process of renormalization group flow. While the models within a class may differ dramatically at finite sc ...
. This class is characterized by the following
critical exponents
Critical exponents describe the behavior of physical quantities near continuous phase transitions. It is believed, though not proven, that they are universal, i.e. they do not depend on the details of the physical system, but only on some of its g ...
in one spatial dimension (1 + 1 dimension): the roughness exponent
, growth exponent
, and dynamic exponent
. In order to check if a growth model is within the KPZ class, one can calculate the ''width'' of the surface:
:
where
is the mean surface height at time
and
is the size of the system. For models within the KPZ class, the main properties of the surface
can be characterized by the
Family
Family (from ) is a Social group, group of people related either by consanguinity (by recognized birth) or Affinity (law), affinity (by marriage or other relationship). It forms the basis for social order. Ideally, families offer predictabili ...
–
Vicsek scaling relation
Scaling may refer to:
Science and technology
Mathematics and physics
* Scaling (geometry), a linear transformation that enlarges or diminishes objects
* Scale invariance, a feature of objects or laws that do not change if scales of length, ener ...
of the
roughness
:
with a scaling function
satisfying
:
In 2014, Hairer and Quastel showed that more generally, the following KPZ-like equations lie within the KPZ universality class:
:
where
is any even-degree
polynomial
In mathematics, a polynomial is a Expression (mathematics), mathematical expression consisting of indeterminate (variable), indeterminates (also called variable (mathematics), variables) and coefficients, that involves only the operations of addit ...
.
A family of processes that are conjectured to be universal limits in the (1+1) KPZ universality class and govern the long time fluctuations are the
Airy processes and the
KPZ fixed point.
Solving the KPZ equation
Due to the
nonlinearity
In mathematics and science, a nonlinear system (or a non-linear system) is a system in which the change of the output is not proportional to the change of the input. Nonlinear problems are of interest to engineers, biologists, physicists, mathe ...
in the equation and the presence of space-time
white noise
In signal processing, white noise is a random signal having equal intensity at different frequencies, giving it a constant power spectral density. The term is used with this or similar meanings in many scientific and technical disciplines, i ...
, solutions to the KPZ equation are known to not be
smooth or regular, but rather '
fractal
In mathematics, a fractal is a Shape, geometric shape containing detailed structure at arbitrarily small scales, usually having a fractal dimension strictly exceeding the topological dimension. Many fractals appear similar at various scale ...
' or '
rough.' Even without the nonlinear term, the equation reduces to the
stochastic heat equation, whose solution is not
differentiable
In mathematics, a differentiable function of one real variable is a function whose derivative exists at each point in its domain. In other words, the graph of a differentiable function has a non- vertical tangent line at each interior point in ...
in the space variable but satisfies a
Hölder condition
In mathematics, a real or complex-valued function on -dimensional Euclidean space satisfies a Hölder condition, or is Hölder continuous, when there are real constants , , such that
, f(x) - f(y) , \leq C\, x - y\, ^
for all and in the do ...
with exponent less than 1/2. Thus, the nonlinear term
is
ill-defined
In mathematics, a well-defined expression or unambiguous expression is an expression whose definition assigns it a unique interpretation or value. Otherwise, the expression is said to be ''not well defined'', ill defined or ''ambiguous''. A func ...
in a classical sense.
In 2013,
Martin Hairer made a breakthrough in solving the KPZ equation by an extension of the
Cole–Hopf transformation The Cole–Hopf transformation is a change of variables that allows to transform a special kind of parabolic partial differential equations (PDEs) with a quadratic nonlinearity into a linear heat equation. In particular, it provides an explicit fo ...
and constructing approximations using
Feynman diagrams
In theoretical physics, a Feynman diagram is a pictorial representation of the mathematical expressions describing the behavior and interaction of subatomic particles. The scheme is named after American physicist Richard Feynman, who introduced ...
. In 2014, he was awarded the
Fields Medal
The Fields Medal is a prize awarded to two, three, or four mathematicians under 40 years of age at the International Congress of Mathematicians, International Congress of the International Mathematical Union (IMU), a meeting that takes place e ...
for this work on the KPZ equation, along with
rough paths theory and
regularity structures. There were 6 different analytic
self-similar solutions found for the (1+1) KPZ equation with different analytic noise terms.
Physical derivation
The following non-rigorous derivation is from
and.
[ ] Suppose we want to describe a
surface growth by some
partial differential equation
In mathematics, a partial differential equation (PDE) is an equation which involves a multivariable function and one or more of its partial derivatives.
The function is often thought of as an "unknown" that solves the equation, similar to ho ...
. Let
represent the height of the surface at position
and time
. Their values are continuous. We expect that there would be a sort of
smoothening mechanism. Then the simplest equation for the surface growth may be taken to be the
diffusion equation
The diffusion equation is a parabolic partial differential equation. In physics, it describes the macroscopic behavior of many micro-particles in Brownian motion, resulting from the random movements and collisions of the particles (see Fick's l ...
,
:
But this is a
deterministic
Determinism is the metaphysical view that all events within the universe (or multiverse) can occur only in one possible way. Deterministic theories throughout the history of philosophy have developed from diverse and sometimes overlapping mo ...
equation, implying the surface has no random fluctuations. The simplest way to include fluctuations is to add a
noise
Noise is sound, chiefly unwanted, unintentional, or harmful sound considered unpleasant, loud, or disruptive to mental or hearing faculties. From a physics standpoint, there is no distinction between noise and desired sound, as both are vibrat ...
term. Then we may employ the equation
:
with
taken to be the
Gaussian
Carl Friedrich Gauss (1777–1855) is the eponym of all of the topics listed below.
There are over 100 topics all named after this German mathematician and scientist, all in the fields of mathematics, physics, and astronomy. The English eponymo ...
white noise
In signal processing, white noise is a random signal having equal intensity at different frequencies, giving it a constant power spectral density. The term is used with this or similar meanings in many scientific and technical disciplines, i ...
with
mean
A mean is a quantity representing the "center" of a collection of numbers and is intermediate to the extreme values of the set of numbers. There are several kinds of means (or "measures of central tendency") in mathematics, especially in statist ...
zero and
covariance
In probability theory and statistics, covariance is a measure of the joint variability of two random variables.
The sign of the covariance, therefore, shows the tendency in the linear relationship between the variables. If greater values of one ...
. This is known as the Edwards–Wilkinson (EW) equation or
stochastic heat equation with
additive noise (SHE). Since this is a
linear equation
In mathematics, a linear equation is an equation that may be put in the form
a_1x_1+\ldots+a_nx_n+b=0, where x_1,\ldots,x_n are the variables (or unknowns), and b,a_1,\ldots,a_n are the coefficients, which are often real numbers. The coeffici ...
, it can be solved exactly by using
Fourier analysis
In mathematics, Fourier analysis () is the study of the way general functions may be represented or approximated by sums of simpler trigonometric functions. Fourier analysis grew from the study of Fourier series, and is named after Joseph Fo ...
. But since the noise is Gaussian and the equation is linear, the fluctuations seen for this equation are still Gaussian. This means the EW equation is not enough to describe the surface growth of interest, so we need to add a nonlinear function for the growth. Therefore, surface growth change in time has three contributions. The first models lateral growth as a nonlinear function of the form
. The second is a
relaxation, or
regularization
Regularization may refer to:
* Regularization (linguistics)
* Regularization (mathematics)
* Regularization (physics)
* Regularization (solid modeling)
* Regularization Law, an Israeli law intended to retroactively legalize settlements
See also ...
, through the diffusion term
, and the third is the white noise forcing
. Therefore,
:
The key term
, the deterministic part of the growth, is assumed to be a function only of the slope, and to be a
symmetric function
In mathematics, a function of n variables is symmetric if its value is the same no matter the order of its arguments. For example, a function f\left(x_1,x_2\right) of two arguments is a symmetric function if and only if f\left(x_1,x_2\right) = f\ ...
. A great observation of Kardar, Parisi, and Zhang (KPZ)
was that while a surface grows in a
normal direction
In geometry, a normal is an object (e.g. a line, ray, or vector) that is perpendicular to a given object. For example, the normal line to a plane curve at a given point is the infinite straight line perpendicular to the tangent line to the cur ...
(to the surface), we are measuring the height on the height axis, which is perpendicular to the space axis, and hence there should appear a nonlinearity coming from this simple geometric effect. When the surface slope
is small, the effect takes the form
, but this leads to a seemingly intractable equation. To circumvent this difficulty, one can take a general
and expand it as a
Taylor series
In mathematics, the Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor ser ...
,
:
The first term can be removed from the equation by a time shift, since if
solves the KPZ equation, then
solves
:
The second should vanish because of the symmetry of
, but could anyway have been removed from the equation by a constant velocity shift of coordinates, since if
solves the KPZ equation, then
solves
:
Thus the quadratic term is the first nontrivial contribution, and it is the only one kept. We arrive at the KPZ equation
:
See also
*
Fokker–Planck equation
In statistical mechanics and information theory, the Fokker–Planck equation is a partial differential equation that describes the time evolution of the probability density function of the velocity of a particle under the influence of drag (physi ...
*
Fractal
In mathematics, a fractal is a Shape, geometric shape containing detailed structure at arbitrarily small scales, usually having a fractal dimension strictly exceeding the topological dimension. Many fractals appear similar at various scale ...
*
Quantum field theory
In theoretical physics, quantum field theory (QFT) is a theoretical framework that combines Field theory (physics), field theory and the principle of relativity with ideas behind quantum mechanics. QFT is used in particle physics to construct phy ...
*
Renormalization group
In theoretical physics, the renormalization group (RG) is a formal apparatus that allows systematic investigation of the changes of a physical system as viewed at different scales. In particle physics, it reflects the changes in the underlying p ...
*
Rough path
In stochastic analysis, a rough path is a generalization of the classical notion of a smooth path. It extends calculus and differential equation theory to handle irregular signals—paths that are too rough for traditional analysis, such as a Wi ...
*
Stochastic partial differential equation
Stochastic partial differential equations (SPDEs) generalize partial differential equations via random force terms and coefficients, in the same way ordinary stochastic differential equations generalize ordinary differential equations.
They hav ...
*
Surface growth
*
Tracy–Widom distribution
*
Universality (dynamical systems)
In statistical mechanics, universality is the observation that there are properties for a large class of systems that are independent of the Dynamics (mechanics), dynamical details of the system. Systems display universality in a scaling limit, whe ...
Sources
Further reading
*
*
*
*
{{DEFAULTSORT:Kardar-Parisi-Zhang equation
Statistical mechanics
Stochastic differential equations
Partial differential equations
Functions of space and time