We outline a kinetic theory of non-thermal fixed points for the example of a dilute Bose gas, partially reviewing results obtained earlier, thereby extending, complementing, generalizing and straightening them out. We study universal dynamics after a cooling quench, focusing on situations where the time evolution represents a pure rescaling of spatial correlations, with time defining the scale parameter. The non-equilibrium initial condition set by the quench induces a redistribution of particles in momentum space. Depending on conservation laws, this can take the form of a wave-turbulent flux or of a more general self-similar evolution, signaling the critically slowed approach to a non-thermal fixed point. We identify such fixed points using a non-perturbative kinetic theory of collective scattering between highly occupied long-wavelength modes. In contrast, a wave-turbulent flux, possible in the perturbative Boltzmann regime, builds up in a critically accelerated self-similar manner. A key result is the simple analytical universal scaling form of the non-perturbative many-body scattering matrix, for which we lay out the concrete conditions under which it applies. We derive the scaling exponents for the time evolution as well as for the power-law tail of the momentum distribution function, for a general dynamical critical exponent z and an anomalous scaling dimension η. The approach of the non-thermal fixed point is, in particular, found to involve a rescaling of momenta in time t by t β , with β = 1/z, within our kinetic approach independent of η. We confirm our analytical predictions by numerically evaluating the kinetic scattering integral as well as the non-perturbative many-body coupling function. As a side result we obtain a possible finite-size interpretation of wave-turbulent scaling recently measured by Navon et al.

We study the effect of correlation on the direction of particle exchange between local thermal sub-systems where the total system is isolated. Our focus is the situation where both sub-systems have the same temperature but different chemical potentials to eliminate the effect of energy transfer due to the temperature difference. The analysis is done in two limits; in the short time scale where the final state of each sub-system is close to its initial thermal state and in a longer time scale where each sub-system's final state can be arbitrary. The results indicate that the conventional flow of particles from a higher chemical potential to a lower one occurs when the correlation which is quantified by mutual information increases. In contrast, an anomalous flow of particles in the reverse direction has a chance to happen when the correlation goes down. Our findings show that the direction of the particle exchange cannot be predetermined by the chemical potential difference in the presence of correlation.

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