Abstract
We investigate non-equilibrium relativistic ϕ4 theory in open systems (the relevant system and the two reservoirs) as a toy model of quantum field theory of the brain, Quantum Brain Dynamics (QBD). We give time evolution equations of quantum fields, that is the Klein–Gordon (KG) equations for background coherent scalar fields and the Kadanoff–Baym (KB) equations for quantum fluctuations. Next we introduce a relativistic kinetic entropy current with the gradient expansion technique and show the H-theorem in the presence of Next-to-Leading-Order self-energy of the coupling expansion and tunneling between systems in d+1 dimensions. Finally we demonstrate numerical simulations with KG and KB equations in 1+1 dimensions. We find that the decoherence (field–particle conversion), the entropy production and the chemical equilibration occur in time evolution of the relevant system and the two reservoirs. We also demonstrate how the background coherent fields transfer between systems. The presented formalism is proposed as a mathematical representation of QBD.
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