Execute system cron manually How to execute system cron manually via drush or drupal UI?
To execute just the system_cron() hook, the easiest way is with drush

$ drush php-eval 'system_cron();'


This will start a PHP process, boostrap Drupal, and run the input.  Note that this will run everything that this hook does, in addition to the file cleanup, including garbabe collection on the various database tables with ephemeral data, like {cache}, {batch}, {flood}, etc.  There isn't an individual function in Drupal 7 that just handles the file cleanup.  

Also keep in mind that when you run drush like this, you are running as the user who you logged in as.  This user may or may not have persmissions on the files and tmp directories in order to delete files.

You could copy this code into your own script and then do

$ drush scr cleanup-the-files.php