Carnot’s theorem as Noether’s theorem for thermoacoustic engines

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Abstract
Onset in thermoacoustic engines, the transition to spontaneous self-generation of oscillations, is studied here as both a dynamical critical transition and a limiting heat engine behavior. The critical transition is interesting because it occurs for both dissipative and conservative systems, with common scaling properties. When conservative, the stable oscillations above the critical point also implement a reversible engine cycle satisfying Carnot’s theorem, a universal conservation law for entropy flux. While criticality in equilibrium systems is naturally associated with symmetries and universal conservation laws, these are usually exploited with global minimization principles, which dynamical critical systems may not have if dissipation is essential to their criticality. Acoustic heat engines furnish an example connecting equilibrium methods with dynamical and possibly even dissipative critical transitions: A reversible engine is shown to map, by a change of variables, to an equivalent system in apparent thermal equilibrium; a Noether symmetry in the equilibrium field theory implies Carnot’s theorem for the engine. Under the same association, onset is shown to be a process of spontaneous symmetry breaking and the scaling of the quality factor predicted for both the reversible and irreversible engines is shown to arise from the Ginzburg-Landau description of the broken phase.

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