Re: [Phys-l] Microcanonical ensemble

[John Denker <jsd@av8n.com>]

On 08/20/2007 02:19 PM, D.V.N. Sarma wrote:
> Some books say that N,E,V are constant for microcanonical ensemble. 
> Some others say that only E and N are constant and V is not. They say
>  that it can under go free expansion without changing either E or N.

Here's my take on this:

Suppose our system consists of a gas in some kind of container.

1a)  When I hear the term "microcanonical ensemble", it tells me 
 that
 -- the energy of the system is uniform across the ensemble, 
  i.e. all members of the ensemble have the same energy.
 -- the number of particles in the system is uniform across
  the ensemble, i.e. all members of the ensemble have the
  same number of particles.

1b) When I hear the term "canonical ensemble", it tells me
 -- the temperature of the system is uniform across the ensemble, 
  i.e. all members of the ensemble have the same temperature.
 -- the number of particles in the system is uniform across
  the ensemble, i.e. all members of the ensemble have the
  same number of particles.

1c) When I hear the term "grand canonical ensemble", it tells me
 -- the temperature of the system is uniform across the ensemble, 
  i.e. all members of the ensemble have the same temperature
 -- the chemical potential of the system is uniform across
  the ensemble, i.e. all members of the ensemble have the
  same chemical potential.

These are not the most-general definitions, because we want the 
terms to apply to systems other than gases in containers, but this 
is a way of introducing the main ideas.

2) Implementation:

2a) The most-common way of making a microcanonical system is 
 to have all the containers have the same volume, and to have
 the walls be thermally insulated, and to have the walls be
 impervious to the gas particles.

 I'm not saying it is /necessary/ to have the volume be 
 uniform across the ensemble, but it is common and convenient.

2b) The most-common way of making a canonical ensemble is to
 tie the system to a humongous heat sink, and to have the walls
 be impervious to the gas particles.

2c) The most-common way of making a grand canonical ensemble is
 to tie the system to a humongous heat sink and also to tie it
 to a humongous reservoir of particles.


>        Some books say that N,E,V are constant for microcanonical ensemble.
> Some others say that only E and N are constant and V is not. They say
> that it can under go free expansion without changing either E or N.

That is alarmingly open to misinterpretation.  There is potential
for confusion about the meaning of "constant".  I was careful to
not use the word "constant" until now.

In particular, suppose that our container is the usual setup with
a piston in a cylinder.  The volume will be not "constant" if we
move the piston, but still the volume will be /uniform/ across
all members of the ensemble.

So it is easily possible to have a microcanonical system where
the system undergoes expansion ... including free expansion, or
even non-free expansion against a piston.  The key idea is that 
at each moment during the expansion, the energy is uniform across 
the ensemble.

 -- Having the energy be uniform across the ensemble is important
  because it greatly simplifies the analysis (and is a big part
  of the definition of microcanonical).
 -- Having the volume be adjustable (in a way that keeps the
  energy uniform across the ensemble) is important because 
  otherwise it would be hard to define pressure.  Pressure is
  normally defined as ∂E/∂V at constant S.

There is a huge conceptual difference between 
 ++ "constant" meaning uniform across the ensemble, and
 ++ "constant" meaning immutable for all time.

Professionals are commonly confused about this;  it's little
wonder that students are confused.