Inner product structure for classical diff equations

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SUMMARY

The discussion focuses on the application of inner product structures, typically associated with quantum mechanics, to differential equations in classical physics, specifically regarding the non-Hermitian differential operator of the Fokker-Planck equation. Participants clarify the concept of an "inner product structure" as a vector space of functions equipped with an inner product. A reference to a specific resource, a lecture on the Fokker-Planck equation, is provided, highlighting its limitations to scenarios involving gradient flows and Hermitian operators.

PREREQUISITES
  • Understanding of inner product spaces in functional analysis
  • Familiarity with differential equations, particularly the Fokker-Planck equation
  • Knowledge of Hermitian and non-Hermitian operators
  • Basic concepts of classical physics related to diffusion processes
NEXT STEPS
  • Research the mathematical foundations of inner product spaces in functional analysis
  • Study the Fokker-Planck equation in detail, focusing on its applications in classical physics
  • Explore the implications of Hermitian vs. non-Hermitian operators in differential equations
  • Investigate gradient flows and their relationship to diffusion processes in physics
USEFUL FOR

Physicists, mathematicians, and researchers interested in the intersection of quantum mechanics and classical physics, particularly those exploring advanced topics in differential equations and diffusion processes.

julian
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I'm interested in what people know about the application of inner product structures (usually reserved for QM) to diff equations describing classical physics, in particular non- hermitician diff operator of the Fokker-Plank equation. Thanks.
 
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Before your thread gets the automated courtesy bump, let me suggest you explain what you mean by an "inner product structure". Do you mean a vector space of functions with an inner product?
 
Stephen Tashi said:
Before your thread gets the automated courtesy bump, let me suggest you explain what you mean by an "inner product structure". Do you mean a vector space of functions with an inner product?

Yes. I've found this:

http://wwwf.imperial.ac.uk/~pavl/lec_fokker_planck.pdf

and I'm looking at it now, but it only applies to where you have gradient flows (that's where the diffusing particle is also subject to steady state flow given by the grad of a potential) and the operator is hermitician.
 
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