Existence proof for orthogonal dynamics and the Mori-Zwanzig formalism. (English) Zbl 1126.82018
Summary: 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. If \(L\) is the Liouvillian, or Lie derivative associated with a Hamiltonian system, and \(P\) an orthogonal projection onto a closed subspace of \(L^2\), then the orthogonal dynamics is generated by the operator \((I -P)L\). We prove the existence of classical solutions for the case where \(P\) has finite-dimensional range. In the general case, we prove the existence of weak solutions.
MSC:
| 82C10 | Quantum dynamics and nonequilibrium statistical mechanics (general) |
| 37N20 | Dynamical systems in other branches of physics (quantum mechanics, general relativity, laser physics) |
| 42C05 | Orthogonal functions and polynomials, general theory of nontrigonometric harmonic analysis |
Keywords:
irreversible statistical mechanics; existence of classical solutions; existence of weak solutions; LiouvillianReferences:
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