Re: macroscopic vs microscopic degrees of freedom

[John Denker <jsd@MONMOUTH.COM>]

At 05:30 PM 10/29/99 -0700, Leigh Palmer wrote:
> In what follows I ask only rhetorical questions, so I have supplied
> my answers in parentheses
>
> I paddle my canoe. The water
> is churned on a scale comparable to the size of the paddle, and the
> turbulent motion then evolves to smaller scales over an interval of
> time measured in seconds. What part of the work which was done on the
> water (and I will argue that work was done on the water) goes into
> mechanical energy and what part goes into internal energy? Does that
> fraction vary with time? (yes)  Does the amount of work which *was*
> done on the water also vary with time? (obviously not)

These are intelligent (indeed profound) rhetorical questions.

Alas I disagree with the "obvious" answer to the final question.


The notion of "paddling" a fluid cuts to the core of the distinction
between "work" and "heat".  As I recall, James Prescott Joule literally
used a paddle-wheel to quantify for the first time the mechanical
equivalent of heat.

If the problem is mis-formulated, there will be innumerable paradoxes.  The
only way out of these paradoxes is to consider *timescales*.  Feynman makes
this point in the opening section of his _Statistical Mechanics_ book.
Here is an approximate quote from memory (alas my copy of the book is not
easily accessible):  "Equilibrium is when the fast things have happened and
the slow things have not.  A gas confined to a cylinder will eventually
erode the walls of the container, but before that it will have come to a
state of definite temperature, pressure, ...."

So, in this spirit, consider the question

> Does the amount of work which *was*
> done on the water also vary with time?

If I may answer a slightly different question, namely

%Does the amount of work which *was*
%done on the water depend on the timescale?

then the answer is absolutely _yes_.  If you paddle and then look very
quickly, you will see more work than heat, and we can reasonably say that
the paddle did work on the fluid.  OTOH if you paddle and look somewhat
later, you will see essentially all heat and no work, and we can reasonably
summarize events by saying that the paddle heated the fluid.

Each of us is free to choose our preferred timescale, but we have no right
to complain if somebody else chooses some other _reasonable_ timescale.
(If you choose a timescale that is not one of the "natural" or
"conventional" timescales of the problem, it would be good to emphasize
this fact.  Otherwise people will misunderstand you and think you are
crazier than you actually are.)


______________________________________________________________
copyright (C) 1999 John S. Denker             jsd@monmouth.com