When analyzing Bitcoin users' balance distribution, we observed that it follows a log-normal pattern. Drawing parallels from the successful application of Gibrat's law of proportional growth in explaining city size and word frequency distributions, we tested whether the same principle could account for the log-normal distribution in Bitcoin balances. However, our calculations revealed that the exponent parameters in both the drift and variance terms deviate slightly from one. This suggests that Gibrat's proportional growth rule alone does not fully explain the log-normal distribution observed in Bitcoin users' balances. During our exploration, we discovered an intriguing phenomenon: Bitcoin users tend to fall into two distinct categories based on their behavior, which we refer to as ``poor" and ``wealthy" users. Poor users, who initially purchase only a small amount of Bitcoin, tend to buy more bitcoins first and then sell out all their holdings gradually over time. The certainty of selling all their coins is higher and higher with time. In contrast, wealthy users, who acquire a large amount of Bitcoin from the start, tend to sell off their holdings over time. The speed at which they sell their bitcoins is lower and lower over time and they will hold at least a small part of their initial holdings at last. Interestingly, the wealthier the user, the larger the proportion of their balance and the higher the certainty they tend to sell. This research provided an interesting perspective to explore bitcoin users' behaviors which may apply to other finance markets.
