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:
  1. 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
  2. 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
  3. 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:
  1. D.Z. Anderson, R.W. Brockett, N. Nuttall:
    Information Dynamics of Photorefractive Two-Beam Coupling.
    Phys. Rev. Lett. 82 (1999) 1418-1421
  2. N. Khaneja, R. Brockett, S. Glaser:
    Time Optimal Control in Spin Systems.
    quant-ph/0006114 Phys. Rev. A 63 (2001) 032308
  3. 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

    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

    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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    R. W. Brockett:
    Dynamical Systems That Sort Lists, Diagonalize Matrices, and Solve Linear Programming Problems.
    Lin. Alg. and its Appl. 146 (1991) 79-91

    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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    M. T. Chu:
    A List of Matrix Flows with Applications.
    Fields Institute Communications 3 (1994) 87-97

    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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    July 2003