In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
and
physics
Physics is the natural science that studies matter, its 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 which r ...
, the Magnus expansion, named after
Wilhelm Magnus
Hans Heinrich Wilhelm Magnus known as Wilhelm Magnus (February 5, 1907 in Berlin, Germany – October 15, 1990 in New Rochelle, New York) was a German-American mathematician. He made important contributions in combinatorial group theory, Lie alge ...
(1907–1990), provides an exponential representation of the solution of a first-order homogeneous
linear differential equation
In mathematics, a linear differential equation is a differential equation that is defined by a linear polynomial in the unknown function and its derivatives, that is an equation of the form
:a_0(x)y + a_1(x)y' + a_2(x)y'' \cdots + a_n(x)y^ = b ...
for a
linear operator. In particular, it furnishes the
fundamental matrix of a system of linear
ordinary differential equations 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
In multivariable calculus, an initial value problem (IVP) is an ordinary differential equation together with an initial condition which specifies the value of the unknown function at a given point in the domain. Modeling a system in physics or o ...
associated with the linear ordinary differential equation
:
for the unknown -dimensional vector function .
When ''n'' = 1, the solution simply reads
:
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 in ...
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
In mathematics, the Bernoulli numbers are a sequence of rational numbers which occur frequently in analysis. The Bernoulli numbers appear in (and can be defined by) the Taylor series expansions of the tangent and hyperbolic tangent functions, ...
and
the
adjoint endomorphism
In mathematics, the adjoint representation (or adjoint action) of a Lie group ''G'' is a way of representing the elements of the group as linear transformations of the group's Lie algebra, considered as a vector space. For example, if ''G'' is G ...
of ,
:
to solve for recursively in terms of "in a continuous analog of the
CBH 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 that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the additio ...
), 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 Binary operation, operation called the Lie bracket, an Alternating multilinear map, alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow ...
of the
Lie group
In mathematics, a Lie group (pronounced ) is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the additio ...
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
Perturbation or perturb may refer to:
* Perturbation theory, mathematical methods that give approximate solutions to problems that cannot be solved exactly
* Perturbation (geology), changes in the nature of alluvial deposits over time
* Perturbat ...
theories. For instance, in
classical mechanics
Classical mechanics is a physical theory describing the motion of macroscopic objects, from projectiles to parts of machinery, and astronomical objects, such as spacecraft, planets, stars, and galaxies. For objects governed by classical ...
the
symplectic character of the
time evolution
Time evolution is the change of state brought about by the passage of time, applicable to systems with internal state (also called ''stateful systems''). In this formulation, ''time'' is not required to be a continuous parameter, but may be disc ...
is preserved at every order of approximation. Similarly, the
unitary
Unitary may refer to:
Mathematics
* Unitary divisor
* Unitary element
* Unitary group
* Unitary matrix
* Unitary morphism
* Unitary operator
* Unitary transformation
* Unitary representation
* Unitarity (physics)
* ''E''-unitary inverse semigrou ...
character of the time evolution operator in
quantum mechanics
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, ...
is also preserved (in contrast, e.g., to the
Dyson series
In scattering theory, a part of mathematical physics, the Dyson series, formulated by Freeman Dyson, is a perturbative expansion of the time evolution operator in the interaction picture. Each term can be represented by a sum of Feynman diagra ...
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
Converge may refer to:
* Converge (band), American hardcore punk band
* Converge (Baptist denomination), American national evangelical Baptist body
* Limit (mathematics)
* Converge ICT, internet service provider in the Philippines
*CONVERGE CFD s ...
for is
:
where
denotes a
matrix norm
In mathematics, a matrix norm is a vector norm in a vector space whose elements (vectors) are matrices (of given dimensions).
Preliminaries
Given a field K of either real or complex numbers, let K^ be the -vector space of matrices with m ro ...
. 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
In mathematics, the adjoint representation (or adjoint action) of a Lie group ''G'' is a way of representing the elements of the group as linear transformations of the group's Lie algebra, considered as a vector space. For example, if ''G'' is G ...
):
:
while are the
Bernoulli numbers
In mathematics, the Bernoulli numbers are a sequence of rational numbers which occur frequently in analysis. The Bernoulli numbers appear in (and can be defined by) the Taylor series expansions of the tangent and hyperbolic tangent functions, ...
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, 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 ...
,
, 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 t ...
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. Stochastic processes are widely used as mathematical models of systems and phenomena that appe ...
and
is the
identity matrix. 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 inter ...
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, m ...
to
nuclear magnetic resonance
Nuclear magnetic resonance (NMR) is a physical phenomenon in which nuclei in a strong constant magnetic field are perturbed 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 relativistic quantum field theory of electrodynamics. In essence, it describes how light and matter interact and is the first theory where full agreement between quantum mechanics and spec ...
. 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
In the theory of Lie groups, the exponential map is a map from the Lie algebra of a Lie group into . In case is a matrix Lie group, the exponential map reduces to the matrix exponential. The exponential map, denoted , is analytic and has as su ...
Notes
References
*
*
*
*
*
{{refend
Ordinary differential equations
Stochastic differential equations
Lie algebras
Mathematical physics