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While physical layer capture has been observed in real implementations of wireless devices which randomly accessing shared channels, fair rate control algorithms based on accurate channel models describing the phenomenon have not been developed. In this paper, using our general physical channel model, we formally present the characteristics of the feasible domain of a general fairness problem and utility fairness under physical layer capture. We show that the allocation domain is not convex and the previous optimization schemes cannot be applied. We further show the objective function for utility fairness is concave on the domain of channel access attempt probability. We clarify that the optimal attempt probability of the node in the log utility fairness is proportional to its interference effect on its neighbors. Based on our analysis, we propose a rate determination algorithm to achieve log-utility fairness for random access networks. Running in a fully distributed manner, our rate control algorithm can reach an assignment close to the optimum, in contrast of the previous algorithms causing node starvation in presence of physical layer capture. The accuracy of our algorithm is verified through simulations and our results indicate the accuracy error for log utility fairness is less than 5% for 97% of the nodes.