Existence proof for orthogonal dynamics and the Mori-Zwanzig formalism
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We study the existence of solutions to the orthogonal dynamics equation, which arises in the Mori-Zwanzig formalism in irreversible statistical mechanics. This equation generates the random noise associated with a reduction in the number of variables. IfL is the Liouvillian, or Lie derivative associated with a Hamiltonian system, andP an orthogonal projection onto a closed subspace ofL 2, then the orthogonal dynamics is generated by the operator (I −P)L. We prove the existence of classical solutions for the case whereP has finite-dimensional range. In the general case, we prove the existence of weak solutions.
KeywordsWeak Solution Conditional Expectation Volterra Equation Energy Inequality Hamiltonian Vector Field
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