Thermodynamic incompleteness in non-Markovian Majorana transport I: Island dynamics and missing transport statistics

Abstract: We show that the complete knowledge of the non-Markovian island-state dynamics of a floating Majorana island does not, in general, determine the thermodynamic transport statistics measured in the leads. We demonstrate this statement in a Coulomb-blockaded island with $M$ Majorana zero modes coupled to structured reservoirs. In the cotunneling regime, a Schrieffer-Wolff transformation gives reservoir-assisted transitions generated by Majorana bilinears. After the reservoirs are traced out, the island state determines the memory kernel associated with each bilinear, and this is enough to predict all island-state observables within the cotunneling approximation. It is not enough to determine which lead or detector channel supplied the electron, absorbed the electron, or carried the corresponding energy exchange. This is a genuine loss of thermodynamic information, not an error in the island equation. We formulate the result as a thermodynamic completeness criterion: an island memory equation determines a transport observable only when that observable is constant over all assignments of reservoir channels that give the same island memory kernel. The criterion gives a measurable prediction. Two structured-reservoir Majorana devices can have identical island-state tomography and relaxation, but different charge noise measured separately in the leads, heat noise, and mixed charge-energy correlations. The geometry of the projection from reservoir records to island kernels and the topology of the network of tunnel contacts identify which transport information is absent from island-state dynamics.