Diagrammatic theory of time correlation functions of facilitated kinetic Ising models

Journal of Chemical Physics

Volumn 114, Issue 3, 2001, Pages 1101-1114

  Pitts, Steven J ^a,b   Andersen, Hans C ^a  

^a STANFORD UNIVERSITY   (United States)

^b Department of Life Sciences   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

COMPUTER SIMULATION; GLASS TRANSITION; MARKOV PROCESSES; MATHEMATICAL MODELS;

KINETIC ISING MODELS; TIME CORRELATION FUNCTIONS;

KINETIC THEORY;

EID: 0035124945     PISSN: 00219606     EISSN: None     Source Type: Journal    
DOI: 10.1063/1.1330578     Document Type: Article

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51. 1(i-t′)C(t′). Note the minus signs on the right.
52. The details of the summation are given elsewhere (Ref. 33). An important consideration is to sum diagrams with all relative horizontal placements of the interactions in the fragments of the diagrams associated with each of the neighbors of spin i. The important lemma that makes the summation possible is closely related to a theorem of Hugenhoitz [N. M. Hugenholtz, Physica (Utrecht) 22, 343 (1956); reprinted in Problems in Quantum Theory of Many-Particle Systems, edited by L. Van Hove, N. M. Hugenhoitz, and L. P. Howland (W A. Benjamin, New York, 1961)] used in quantum many body theory. A standard detinition of convolution (in Laplace variable space) is used. See either of the references above.
53. The details of the summation are given elsewhere (Ref. 33). An important consideration is to sum diagrams with all relative horizontal placements of the interactions in the fragments of the diagrams associated with each of the neighbors of spin i. The important lemma that makes the summation possible is closely related to a theorem of Hugenhoitz [N. M. Hugenholtz, Physica (Utrecht) 22, 343 (1956); reprinted in Problems in Quantum Theory of Many-Particle Systems, edited by L. Van Hove, N. M. Hugenhoitz, and L. P. Howland (W A. Benjamin, New York, 1961)] used in quantum many body theory. A standard detinition of convolution (in Laplace variable space) is used. See either of the references above.
54. This was conjectured by Mauch and Jackle (Ref. 29) and confirmed by P. Diaconis and D. Aldous (private communication).
55. Another interesting feature of this work is the fact that this graphical theory is the first one, to our knowledge, that ascribes a simple diagrammatic meaning to the irreducible memory function (i.e., as the sum of diagrams in the memory function that have no nodal points). The name "irreducible memory function" was coined by Cichocki and Hess (Ref. 49) by analogy to the graphical concept of "one-particle-irreducibility" in field theory, and some of the discussion of this function by them and by Kawasaki (Ref. 31) appears motivated by diagrammatic ideas, but none of their work was based on a diagrammatic formulation of the problems under consideration.