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Manuscript Title: Symbolic test of the Jacobi identity for given generalized 'Poisson' bracket.
Authors: M. Kroger, M. Hutter, H.C. Ottinger
Program title: jacobi2.0
Catalogue identifier: ADOE_v1_0
Distribution format: tar.gz
Journal reference: Comput. Phys. Commun. 137(2001)325
Programming language: Mathematica.
Computer: Alpha-Workstation, Silicon Graphics, Sun, Linux-PC, Windows-PC, MacIntosh.
Operating system: DEC-Unix, Irix, Solaris, Linux, Windows 98/NT.
RAM: 2M words
Word size: 16
Keywords: Computer algebra, Jacobi identity, Poisson brackets, GENERIC, Nonequilibrium thermodynamics, Reversible motion, Symbolic programming, Statistical physics, General purpose, Algebras.
Classification: 4.2, 5, 23.

Nature of problem:
The problem is to evaluate single and nested arbitrary generalized Poisson brackets and the cyclic sum of these in order to test the Jacobi identity on a given state space for systems described in terms of discrete or of continuous variables. The Jacobi identity has to be fulfilled for Poisson brackets consistently describing the reversible dynamics of physical systems as desired, e.g., within the framework of nonequilibrium thermodynamics [1-3].

Solution method:
By symbolic programming the algorithm inserts linear combinations of discrete state variables or functionals of field variables into the relevant terms of the Jacobi identity. Subsequent transformations such as partial integrations, functional derivatives, and recognition rules are used to perform the operation.

Restrictions:
The machine must provide the main memory needed (see Long Write-up Sec. 3.4). There is no restrictions concerning discrete problems. The distributed version handles fixed and variable bounded and unbounded integrals for continuous problems (concerning checkjacobi, see Tab.1). Other methods, in particular directF and REDUCE (see Tab.1) disregard any surface contributions, and apply partial integrations by neglecting surface terms.

Additional comments:
A more powerful solution for problems involving continuous fields is available as AEGW_v1_0

Running time:
The typical running time increases with the number of ingredients for the Poisson bracket. For short brackets, e.g. with 5 terms, and one-dimensional integration, the running time is of the order of seconds to minutes on any modern computer.

References:
[1] A.N. Beris and B.J. Edwards, Thermodynamics of flowing systems with internal microstructure (Oxford University Press, Oxford, U.K., 1994).
[2] M. Grmela and H.C. Ottinger, Phys. Rev. E 56 (1997) 6620-32.
[3] H.C. Ottinger and M. Grmela, Phys. Rev. E 56 (1997) 6633-35.