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My question is are there any systems in physics that can only be formulated as an integral equation? Or do all integral equations have an equivalent differential equation?

Anode
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1 Answers1

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My first response would be, that all systems with memory require some form of integral representation to "store" the past states. Even the description of systems with a simple delay requires either an integral equation, or a partial differential equation, which can be approximated by an infinite number of ordinary differential equations. While the theory of equivalence between classes of both types of equations is probably going to be intriguing, I have a feeling, that for physical considerations it is of limited use. The generalization of both, integral and differential equations, to operators and their spectra plays a much, much larger role in modern physics.

CuriousOne
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  • The interaction between two electrostatic charges which are relatively far away has 'memory' in that the force on either points towards some past position of the other, and yet the whole system can be phrased as coupled ODEs and PDEs. There is therefore a nontrivial class of systems for which 'auxiliary' degrees of freedom, and their dynamics, can be used to store the system's memory. – Emilio Pisanty Aug 11 '14 at 17:54
  • Conventionally I have heard the term "memory" only being used for systems with an infinite number of degrees of freedom, and not for Hamiltonian systems with a finite number of particles. You are, however, pointing out correctly, that one can approximate both types of systems by the other type, and in practice both physicists and engineers are using delay lines (e.g. coaxial cable) as approximations of lumped element filters and lumped element chains as approximations for delay lines. – CuriousOne Aug 11 '14 at 18:09
  • The point is that the fact that a system has some sort of memory is not enough for it to require some form of integral equation to represent its dynamics. – Emilio Pisanty Aug 11 '14 at 18:13