Math

What is Entropy?

There are many mathematical definitions of entropy.  The mental picture I find most useful is to imagine the following:  

  • you are put in a room and your job is to label everything in the room with a sharpy indelible marker and masking tape.  
  • You are asked to label everything in the room using the binary numbering system.  This binary number will be that particular objects I.D.

As you go about this you may want to number the objects you most commonly refer to with the lower digits that have less length.  That way since you mention "FORK" much more often than "NUMBER 6 SCREW" you will end up having to say less digits.

The measure of entropy in this room is the number of binary digits required to number all the objects.  This is entropy.  The formula for this sentence that I just said is:

             Entropy ~= log2N    where N is the number of different types of objects in the room

Now in a probabalistic situation with outcomes  x1 , x2 ….  xn    with P(xi) = probability of xi

…..more

By Fudgy McFarlen, ago

Use of Maximum Entropy to explain the form of Energy States of an Electron in a Potential Well

The base state of an electron in an infinite potential well has the most "space" for the electron state.  Thus it has the maximum entropy. Take that same state and imagine pinching the electrons existence to nil in the middle of the trough.  Now you have state-2.  The electron now exists in a smaller entropic state and guess what?  It contains exploitable energy now. This is like a spring compressed.  The electron can decompress and exert force / expend energy.  For example in an interaction with another atom possibly a recoil could occur.   In a crystal lattice an electron can transfer its energy to the atom next door and in effect yield conduction.  All these are preliminary suppositions subject to more scrutiny. As mentioned before since the electron exists in this potential well in the form of free fall it can not have any acceleration.  Thus its distribution must thoroughly avoid the edges of the well were it would indeed experience accelerations by bouncing and recoiling off of the walls.

electron-in-infinite-well.bmp

By Fudgy McFarlen, ago