Probability distributions for carbon emissions and atmospheric response

Probability distributions for carbon burning, atmospheric CO₂, and global average temperature are produced by time series calibration of models of utility optimization and carbon and heat balance using log-linear production functions. Population growth is used to calibrate a logistically evolving index of development that influences production efficiency. Energy production efficiency also includes a coefficient that decreases linearly with decreasing carbon intensity of energy production. This carbon intensity is a piecewise linear function of fossil carbon depletion. That function is calibrated against historical data and extrapolated by sampling a set of hypotheses about the impact on the carbon intensity of energy production of depleting fluid fossil fuel resources and increasing cumulative carbon emissions. Atmospheric carbon balance is determined by a first order differential equation with carbon use rates and cumulative carbon use as drivers. Atmospheric CO₂ is a driver in a similar heat balance. Periodic corrections are included where required to make residuals between data and model results indistinguishable from independently and identically distributed normal distributions according to statistical tests on finite Fourier power spectrum amplitudes and nearest neighbor correlations. Asymptotic approach to a sustainable non-fossil energy production is followed for a global disaggregation into a tropical/developing and temperate/more-developed region. The increase in the uncertainty of global average temperature increases nearly quadratically with the increase in the temperature from the present through the next two centuries.