You need to add an opacityPP attribute to the particleShape and map it with a ramp.

This is done in the attribute editor by clicking the Opacity button under Add Dynamic Attributes.
From the dialog that pops up choose Add Per Particle Attribute.
Now you should see an opacityPP under the Per Particle (Array) Attributes section.
Right-click on the field next to it and choose Create Ramp. A text "<- arrayMapper1.outValuePP" appears in the field.
Right click on the text and choose "<- arrayMapper1.outValuePP -> Edit Ramp".
By default the ramp is a white to black (bottom to top) gradient and is driven by the particle's age. This means that at birth the particle is completely opaque and at death it's completely transparent. Edit the ramp and see what happens...
To add more randomness you might want to change the particles lifespan to Random Range and play with the settings.

I know this wasn't exactly what you asked but if you really want the opacityPP be driven by the distance travelled you'd need to write a per particle runtime expression which could get a bit involved (but not impossible)...

Hope this helps...