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The average channel capacity of a free space optical (FSO) communication system running an intensity modulated Gaussian beam is examined in anisotropic non-Kolmogorov atmospheric weak turbulence based on Rytov variance. Results are obtained by employing the log-normal distribution of irradiance fluctuations corresponding to weak turbulence regime. Our results show that average channel capacity increases together with the increase in anisotropy factor in x and y direction, non-Kolmogorov power law exponent, quantum efficiency of photo detector, Gaussian beam source size and the inner scale length. However, the average channel capacity is found to decrease when turbulence strength, link length and noise variance increase.