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 ...
and
physics
Physics is the scientific study of matter, its Elementary particle, fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge whi ...
, the Magnus expansion, named after
Wilhelm Magnus (1907–1990), provides an exponential representation of the
product integral solution of a first-order homogeneous
linear differential equation
In mathematics, a linear differential equation is a differential equation that is linear equation, linear in the unknown function and its derivatives, so it can be written in the form
a_0(x)y + a_1(x)y' + a_2(x)y'' \cdots + a_n(x)y^ = b(x)
wher ...
for a
linear operator
In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that pr ...
. In particular, it furnishes the
fundamental matrix of a system of linear
ordinary differential equations
In mathematics, an ordinary differential equation (ODE) is a differential equation (DE) dependent on only a single independent variable. As with any other DE, its unknown(s) consists of one (or more) function(s) and involves the derivatives ...
of order with varying coefficients. The exponent is aggregated as an infinite series, whose terms involve multiple integrals and nested commutators.
The deterministic case
Magnus approach and its interpretation
Given the coefficient matrix , one wishes to solve the
initial-value problem associated with the linear ordinary differential equation
:
for the unknown -dimensional vector function .
When ''n'' = 1, the solution is given as a
product integral
:
This is still valid for ''n'' > 1 if the matrix satisfies for any pair of values of ''t'', ''t''
1 and ''t''
2. In particular, this is the case if the matrix is independent of . In the general case, however, the expression above is no longer the solution of the problem.
The approach introduced by Magnus to solve the matrix initial-value problem is to express the solution by means of the exponential of a certain matrix function
:
:
which is subsequently constructed as a
series
Series may refer to:
People with the name
* Caroline Series (born 1951), English mathematician, daughter of George Series
* George Series (1920–1995), English physicist
Arts, entertainment, and media
Music
* Series, the ordered sets used i ...
expansion:
:
where, for simplicity, it is customary to write for and to take ''t''
0 = 0.
Magnus appreciated that, since , using a
Poincaré−Hausdorff matrix identity, he could relate the time derivative of to the generating function of
Bernoulli numbers and
the
adjoint endomorphism of ,
:
to solve for recursively in terms of "in a continuous analog of the
BCH expansion", as outlined in a subsequent section.
The equation above constitutes the Magnus expansion, or Magnus series, for the solution of matrix linear initial-value problem. The first four terms of this series read
:
where is the matrix commutator of ''A'' and ''B''.
These equations may be interpreted as follows: coincides exactly with the exponent in the scalar ( = 1) case, but this equation cannot give the whole solution. If one insists in having an exponential representation (
Lie group
In mathematics, a Lie group (pronounced ) is a group (mathematics), group that is also a differentiable manifold, such that group multiplication and taking inverses are both differentiable.
A manifold is a space that locally resembles Eucli ...
), the exponent needs to be corrected. The rest of the Magnus series provides that correction systematically: or parts of it are in the
Lie algebra
In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an operation called the Lie bracket, an alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi ident ...
of the
Lie group
In mathematics, a Lie group (pronounced ) is a group (mathematics), group that is also a differentiable manifold, such that group multiplication and taking inverses are both differentiable.
A manifold is a space that locally resembles Eucli ...
on the solution.
In applications, one can rarely sum exactly the Magnus series, and one has to truncate it to get approximate solutions. The main advantage of the Magnus proposal is that the truncated series very often shares important qualitative properties with the exact solution, at variance with other conventional
perturbation theories. For instance, in
classical mechanics
Classical mechanics is a Theoretical physics, physical theory describing the motion of objects such as projectiles, parts of Machine (mechanical), machinery, spacecraft, planets, stars, and galaxies. The development of classical mechanics inv ...
the
symplectic character of the
time evolution is preserved at every order of approximation. Similarly, the
unitary character of the time evolution operator in
quantum mechanics
Quantum mechanics is the fundamental physical Scientific theory, theory that describes the behavior of matter and of light; its unusual characteristics typically occur at and below the scale of atoms. Reprinted, Addison-Wesley, 1989, It is ...
is also preserved (in contrast, e.g., to the
Dyson series solving the same problem).
Convergence of the expansion
From a mathematical point of view, the convergence problem is the following: given a certain matrix , when can the exponent be obtained as the sum of the Magnus series?
A sufficient condition for this series to
converge for is
:
where
denotes a
matrix norm
In the field of mathematics, norms are defined for elements within a vector space. Specifically, when the vector space comprises matrices, such norms are referred to as matrix norms. Matrix norms differ from vector norms in that they must also ...
. This result is generic in the sense that one may construct specific matrices for which the series diverges for any .
Magnus generator
A recursive procedure to generate all the terms in the Magnus expansion utilizes the matrices defined recursively through
:
:
which then furnish
:
:
Here ad
''k''Ω is a shorthand for an iterated commutator (see
adjoint endomorphism):
:
while are the
Bernoulli numbers with .
Finally, when this recursion is worked out explicitly, it is possible to express as a linear combination of ''n''-fold integrals of ''n'' − 1 nested commutators involving matrices :
:
which becomes increasingly intricate with .
The stochastic case
Extension to stochastic ordinary differential equations
For the extension to the stochastic case let
be a
-dimensional
Brownian motion
Brownian motion is the random motion of particles suspended in a medium (a liquid or a gas). The traditional mathematical formulation of Brownian motion is that of the Wiener process, which is often called Brownian motion, even in mathematical ...
,
, on the
probability space
In probability theory, a probability space or a probability triple (\Omega, \mathcal, P) is a mathematical construct that provides a formal model of a random process or "experiment". For example, one can define a probability space which models ...
with finite time horizon
and natural filtration. Now, consider the linear matrix-valued stochastic Itô differential equation (with Einstein's summation convention over the index )
:
where
are progressively measurable
-valued bounded
stochastic processes
In probability theory and related fields, a stochastic () or random process is a mathematical object usually defined as a family of random variables in a probability space, where the index of the family often has the interpretation of time. Stoc ...
and
is the
identity matrix
In linear algebra, the identity matrix of size n is the n\times n square matrix with ones on the main diagonal and zeros elsewhere. It has unique properties, for example when the identity matrix represents a geometric transformation, the obje ...
. Following the same approach as in the deterministic case with alterations due to the stochastic setting the corresponding matrix logarithm will turn out as an Itô-process, whose first two expansion orders are given by
and
, where
with Einstein's summation convention over and
:
Convergence of the expansion
In the stochastic setting the convergence will now be subject to a
stopping time
In probability theory, in particular in the study of stochastic processes, a stopping time (also Markov time, Markov moment, optional stopping time or optional time ) is a specific type of "random time": a random variable whose value is interpre ...
and a first convergence result is given by:
Under the previous assumption on the coefficients there exists a strong solution
, as well as a strictly positive
stopping time
such that:
#
has a real logarithm
up to time
, i.e.
#:
# the following representation holds
-almost surely:
#:
#:where
is the -th term in the stochastic Magnus expansion as defined below in the subsection Magnus expansion formula;
# there exists a positive constant , only dependent on
, with
, such that
#:
Magnus expansion formula
The general expansion formula for the stochastic Magnus expansion is given by:
:
where the general term
is an Itô-process of the form:
:
The terms
are defined recursively as
:
with
:
and with the operators being defined as
:
Applications
Since the 1960s, the Magnus expansion has been successfully applied as a perturbative tool in numerous areas of physics and chemistry, from
atomic and
molecular physics
Molecular physics is the study of the physical properties of molecules and molecular dynamics. The field overlaps significantly with physical chemistry, chemical physics, and quantum chemistry. It is often considered as a sub-field of atomic, mo ...
to
nuclear magnetic resonance
Nuclear magnetic resonance (NMR) is a physical phenomenon in which nuclei in a strong constant magnetic field are disturbed by a weak oscillating magnetic field (in the near field) and respond by producing an electromagnetic signal with a ...
and
quantum electrodynamics
In particle physics, quantum electrodynamics (QED) is the Theory of relativity, relativistic quantum field theory of electrodynamics. In essence, it describes how light and matter interact and is the first theory where full agreement between quant ...
.
[ (See Rabi model.)] It has been also used since 1998 as a tool to construct practical algorithms for the numerical integration of matrix linear differential equations. As they inherit from the Magnus expansion the
preservation of qualitative traits of the problem, the corresponding schemes are prototypical examples of
geometric numerical integrators.
See also
*
Baker–Campbell–Hausdorff formula
*
Derivative of the exponential map
Notes
References
*
*
*
*
*
External links
*
UCSD
The University of California, San Diego (UC San Diego in communications material, formerly and colloquially UCSD) is a public land-grant research university in San Diego, California, United States. Established in 1960 near the pre-existing ...
*
* {{cite web, url=https://www.youtube.com/watch?v=PujxIFxk3qs , title=Motivation for Magnus and Fer expansions , date=December 25, 2020 , website=YouTube
Ordinary differential equations
Stochastic differential equations
Lie algebras
Mathematical physics