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Is there any unwritten rule in the academia regarding the rate of paper publication by a PhD student?
I.e. how many paper publication per year is considered as a good performance for a PhD student?
How does that vary from 1st to last year?
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The answer to this is dependent on too many things for any reasonable answer. In some fields it is unusual for a doctoral student to publish anything before finishing the dissertation and even that might be published after the degree is awarded, if at all.
In other fields it is common for the dissertation itself to be a summarization of some set of already published papers, but even then, the expectations will vary with the field and with the individual university. They might even vary with the advisor.
Of course, there are papers and there are papers. When publications are required a larger number of less significant papers may not outweigh even a single significant contribution.
The situation is complicated by normal practice in some lab sciences in which papers have a large number of authors and each contributes differently. In such fields, precisely where you are listed in the list of authors can be significant and can indicate the "amount" of contribution. There are many questions on this site that come from fierce disagreements over author placement. Again, however, having lots of minor placements might not get you very close to completion.
Number of papers produced is a terrible measure of productivity. Citations of those papers by others is a better measure.
But to get an answer that applies to yourself, ask your advisor or some other professor(s) in your field and at your university. That will yield the only meaningful expectation of your progress.
