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Starting about half an hour or an hour after waking up and about three to five hours after, gradually subsiding in the late afternoon. All this depends on various factors, including how well you slept, how you had breakfast, and whether you did exercise. But in general and on average, something like this.
It seems to me that there are general patterns about the fact that the brain works better in the morning, but here it is better to conduct experiments on yourself and, for example, work (if the work is mental)/study for 5-6 days in strictly allotted time.�
For example, first from 8: 00 to 12: 00, then from 10: 00 to 14: 00, and so on.
You will also be able to notice your overall effectiveness at all times, as you have now somehow limited yourself and worked/studied in a more organized and focused way.
Among other things, the brain works perfectly during a session, deadlines and other stressful situations, when “thinking” is equal to “surviving in this world” and there is proper motivation for mental load.
From 9 am to 2 pm is the best time for solving important questions and cognitive activities. After a lunch nap before 9 pm, it is best to engage in physical activity. At 21: 00, the brain begins to generate original ideas.
I am fond of music, and I want to say this for myself:
In the interval from 1 o'clock in the morning to 2-3 o'clock, the muse seems to find some inspiration. I'm absolutely serious, this is the time when you feel different, and this state lasts for about 2 hours
In my opinion, the answer depends on what kind of productivity is meant. I remember that while preparing for exams at school, we were told that the most productive time for mental activity is from 14 to 16 hours (probably referring to memorization and the ability to analyze operations), but I have not yet seen confirmation/justification of this information in the scientific literature.
At the same time, I agree with other answers, original, creative ideas most often arise at night (for me, between midnight and 2-3 hours). At this time, I can generate the structure of the course paper, add a few lines if it's time, edit the video better than usual, and so on. However, then I regret that I did not sleep, it affects productivity for the rest of the day/week/month, depending on how much I work in this mode.
In general, most likely, the time of greatest productivity is directly related to the well-known types of “owl” and “lark”. But in any case, in order for the brain to work as productively as possible, you need to rest properly (you don't have to sleep 8 hours, someone can and even needs less).