Hm. Ok. Firstly, im going to assume you're up to date on firmware, drivers, bios, etc. I'm also assuming you're running with the latest version of Crysis (1.21), and that you're not running any mods, tweaks, configs, etc.
1. Try setting everything to low, but keep your resolution at native. Then, run the game. At those settings, there SHOULDN'T be any FPS issues.
2. Run HW monitor
in the background. It will record temperatures of each part. The CPU shouldn't go over 50c, and the GPU shouldn't go over 80c.
3. If the FPS issue starts happening - STOP. Simply stop moving for a few minutes. If the issue is caused by HDD reads, the issue will go away after a bit of time (a few seconds to a minute). However, if it stays the same (i.e. the FPS doesn't return to normal, we have a bit more complex issue. Firstly, try turning around. If its an issue with poly count, or textures, or VRAM, the FPS will only tank when facing certain directions.
4. When you say "once the sun came up", I'm guessing you're talking about the first level - specifically the area called "First Light". This is the first area of the game with massive view distances, and a ton of objects on screen. This could be what is causing it.
Again, this is simply going on what I'm getting from your posts. A more detailed description of what is happening would be great.