A question on basics of statistical mechanics

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We don't have rigorous approach to statistical mechanics but have an intuitive one.We follow Concepts of Modern Physics by Sir Arthur Beiser.
It is given that
n(ε)=g(ε)f(ε) Here ε is the energy state.
where according to my understanding
n(ε)=number of particles which are present in an energy state ε.
g(ε)=number of states corresponding to energy state ε.
f(ε)=average number of particles in each state.
Is my understanding right? I can't figure this out.
In the next two lines he states that if the energy distribution is continous rather than discrete, then..
1.The number of energy states between ε and ε+dε(dεis small change in energy) is g(ε)dε.My question is how is this possible? If there are two states, ε and ε+dε. then the number of states between them is g(ε+dε)-g(ε), is it not?
2. The same is said for number number of particles between ε and ε+dε. He says that the number is equal to n(ε)dε,How? Should it not be like g(ε+dε)-g(ε)??

He uses these two assumptions to derive the Maxwell-Boltzmann distribution(We don't derive partition function). Please help me out as I need atleast some basic level of understanding(if not rigorous)..
 

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  • #2
atyy
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For a normal density ρ(x), the amount of matter in a small volume Δχ is ρ(x)Δχ , which one integrates over to find the amount of matter in larger volumes.

Similarly, for the density of states g(ε), the number of states in small energy range Δε is g(ε)Δε.
 
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  • #3
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For a normal density ρ(x), the amount of matter in a small volume Δχ is ρ(x)Δχ , which one integrates over to find the amount of matter in larger volumes.

Similarly, for the density of states g(ε), the number of states in small energy range Δε is g(ε)Δε.
I think I got it. Well, my understanding was wrong.Is g(ε) measured over a small interval of energy Δε.
Is this mathematical formulation right?

g(ε)=dn(ε)/dε..here n(e) is the number of states which correspond to a particular energy ε.
 
  • #4
atyy
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I think I got it. Well, my understanding was wrong.Is g(ε) measured over a small interval of energy Δε.
Is this mathematical formulation right?

g(ε)=dn(ε)/dε..here n(e) is the number of states which correspond to a particular energy ε.
Yes, that's right.
 
  • #5
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For a normal density ρ(x), the amount of matter in a small volume Δχ is ρ(x)Δχ , which one integrates over to find the amount of matter in larger volumes.

Similarly, for the density of states g(ε), the number of states in small energy range Δε is g(ε)Δε.
also what is the precise meaning of f(ε)?
 
  • #6
atyy
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I don't know the precise definition. It's easiest for me to think of f(ε) before we take energy to be continuous. In that case f(ε) is the probability that a particle occupies a particular state of energy ε. Still keeping energy discrete, g(ε) is the number of states with energy ε. Then the probability that a particle has energy ε is n(ε)=g(ε)f(ε).

After this we make energy continuous so that the probability that a particle has energy between ε and ε + dε is n(ε)dε=g(ε)f(ε)dε. So now g(ε) has become the density of states per energy, just as n(ε) has slightly changed its meaning to the density of particles per energy.
 

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