Number of ways to distribute particles

[Kashmir]

Pathria, Statistical Mechanics

"
... classical system composed of noninteracting particles... .Now, if there do not exist any spatial correlations among the particles, that is, if the probability of anyone of them being found in a particular region of the available space is completely independent of the location of the other particles, then the total number of ways in which the $N$ particles can be spatially distributed in the system will be simply equal to the product of the numbers of ways in which the individual particles can be accommodated in the same space independently of one another"

Why is that "the total number of ways in which the $N$ particles can be spatially distributed in the system will be simply equal to the product of the numbers of ways in which the individual particles can be accommodated in the same space independently of one another"

Suppose I've two particles in some volume which has 2 compartments which can hold only one particle.

Now there are only 2 ways to distribute the particles in the box but according to the author the total number of ways to distribute them should be product of the number of ways to distribute each of them independently which gives me $2*2=4$Can anyone please help me

 

[Orodruin]

Your example is not applicable because if the boxes can hold only one particle then the distributions of the particles are not independent.

 

[Kashmir]

Your example is not applicable because if the boxes can hold only one particle then the distributions of the particles are not independent.

Can you give me an example which correspondens to what author says?

 

[Orodruin]

Can you give me an example which correspondens to what author says?

Two (distinguishable) particles, two boxes, each box can hold arbitrarily many particles.

 

[Kashmir]

Two (distinguishable) particles, two boxes, each box can hold arbitrarily many particles.

The author uses the word " same space" but you've used two different volumes.

then the total number of ways in which the N particles can be spatially distributed in the system will be simply equal to the product of the numbers of ways in which the individual particles can be accommodated in the same space

 

[DrClaude]

The author uses the word " same space" but you've used two different volumes.

That's the same thing.

It reminds me of the discussion we had in https://www.physicsforums.com/threa...the-barrier-allows-particle-exchange.1013460/. "Volume" can mean an enclosed area, but more generally any extent of space, whether actually defined by a physical boundary or not.