Statistical Physics and Condensed Matter

# Flow Equations for Hamiltonians

## Contributions by Mathematicians

Meanwhile (2002) we have learned from
Volker Bach, Mainz
that flow equations were also developed by the mathematicians
Roger W. Brockett, Harvard
and by
Moody T. Chu, NCSU, and
Kenneth R. Driessel, Univ. of
Wyoming.
They call the method double bracket flow and isospectral flow, resp.
We mention three papers:
- M. T. Chu and K. R. Driessel:

The Projected Gradient Method for Least Square Matrix Approximations with
Spectral Constraints.

SIAM J. Numer. Anal. 27 (1990) 1050-1060
Abstract
- R. W. Brockett:

Dynamical Systems That Sort Lists, Diagonalize Matrices, and Solve Linear
Programming Problems.

Lin. Alg. and its Appl. 146 (1991) 79-91
Abstract
- M. T. Chu:

A List of Matrix Flows with Applications.

Fields Institute Communications 3 (1994) 87-97
Paper
Abstract

Meanwhile this concept has returned to physics:
- D.Z. Anderson, R.W. Brockett, N. Nuttall:

Information Dynamics of Photorefractive Two-Beam Coupling.

Phys. Rev. Lett. 82 (1999) 1418-1421
- N. Khaneja, R. Brockett, S. Glaser:

Time Optimal Control in Spin Systems.

quant-ph/0006114
Phys. Rev. A 63 (2001) 032308
- N. Khaneja, S.J. Glaser, R. Brockett:

Sub-Riemannian geometry and time optimal control of three spin systems:
Quantum gates and coherence transfer.

Phys. Rev. A 65 (2002) 032301

The problems of computing least squares approximations for various types of
real and symmetric matrices subject to spectral constraints share a common
structure. This paper describes a general procedure in using the projected
gradient method. It is shown that the projected gradient of the objective
function on the manifold of constraints usually can be formulated explicitely.
This gives rise to the construction of a descent flow that can be followed
numerically. The explicit form also facilitates the computation of the
second-order optimality conditions. Examples of applications are discussed.
With slight modifications, the procedure can be extended to solve least
squares problems for general matrices subject to singular-value constraints.

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We establish a number of properties associated with the dynamical system
H^{·}=[H,[H,N]] whre H and N are symmetric n by n matrices and
[A,B]=AB-BA. The most important of these come from the fact that this equation
is equivalent to a certain gradient flow on the space of orthogonal matrices.
We are especially interested in the role of this equation as an analog
computer. For example, we show how to map the data associated with a linear
programming problem into H(0) and N in such a way as to have
H^{·}=[H,[H,N]] evolve to a solution of the linear programming
problem. This result can be applied to find systems which solve a variety of
generic combinatorial optimization problems, and it even provides an algorithm
for diagonalizing symmetric matrices.

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Many mathematical problems, such as existence questions, are studied by
using an appropriate realization process, either iteratively or
continuously. This article is a collection of differential equations that
have been proposed as special continuous realization processes. In some
cases, there are remarkable connections between smooth flows and discrete
numerical algorithms. In other cases, the flow approach seems advantageous
in tackling very difficult problems. The flow approach has potential
applications ranging from new development of numerical algorithms to the
theoretical solution of open problems. Various aspects of the flow approach
are reviewed in this article.

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Back to the main page for flow equations
July 2003