What is the difference among MongoDB's [aggregation $min](http://docs.mongodb.org/manual/reference/operator/aggregation/min/) versus [query modifier $min](http://docs.mongodb.org/manual/reference/operator/meta/min/) versus [find().sort()](http://docs.mongodb.org/manual/reference/method/db.collection.find/#order-documents-in-the-result-set) that returns the minimum of the\_field with: `db.the_collection.find().sort({the_field:1}).limit(1)`?

**Which one should I use if there are about a few hundred calls per minute to retrieve the minimum element in a collection and work with it independently each time?**

Side question: Can someone show me the correct syntax using either $min to give me the minimum of a field in a collection? `db.the_collection.find().min({the_field:10})` doesn't work.