#### quasar987

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I find the mathematical manipulation of thermodynamics are often dubious and this one is doubly so. It is the derivation of the Boltzman probability distribution made by Reif (Section 6.2).

He considers an isolated system [itex]A^{(0)}[/itex] of energy [itex]E^{(0)}[/itex] in thermodynamical equilibrium composed of two system A and A' in thermal contact, where A' has much more degrees of freedom that A (A'>>A). He also supposes that the energy of interaction of A and A' are negligible so that [itex]E^{(0)}=E+E'[/itex].

These initial conditions set, he tries to find the probability of finding system A in state

[tex]P_r=\frac{\Omega'(E^{(0)}-E_r)}{\Omega^{(0)}}[/tex]

He then says, let us expand [itex]\Omega'(E^{(0)}-E_r)[/itex] in a Taylor series about [itex]E^{(0)}[/itex]. But he actually expands the log of that in order to get a faster convergence:

[tex]\ln(\Omega'(E^{(0)}-E_r))\approx \ln(\Omega'(E^{(0)}))-\left[\frac{\partial \Omega'}{\partial E'} \right]_{E^{(0)}}E_r[/tex]

And he justifies the dropping of the higher order terms by the fact that [itex]E_r <<E^{(0)}[/itex].

But I have three problems with this justification. (problem #3 is closely linked to #2)

1) [itex]E_r=E^{(0)}[/itex], E'=0

2) It seems to me that [itex]E_r <<E^{(0)}[/itex] justifies nothing! We want [itex]E_r<<1[/itex] so that the higher the power n, the smaller [itex](E_r)^n[/itex] is.

3) And wheter [itex]E_r[/itex] is <1 actually depends on the units of energy used! In, say, a relativistic taylor expansion, the argument is v/c which is dimensionless, so the problem of dimensionality does not arise. But in this case?!

I would really appreciate it if someone could nullify these two objections for me.

Many thanks!

*Reif uses [itex]\Omega'(E')[/itex] to denote the cardinality of the set of all states for wich the energy of A' is E' and [itex]\Omega^{(0)}[/itex] to denote the cardinality of the set of all possible states of the system [itex]A^{(0)}[/itex] as a whole.

He considers an isolated system [itex]A^{(0)}[/itex] of energy [itex]E^{(0)}[/itex] in thermodynamical equilibrium composed of two system A and A' in thermal contact, where A' has much more degrees of freedom that A (A'>>A). He also supposes that the energy of interaction of A and A' are negligible so that [itex]E^{(0)}=E+E'[/itex].

These initial conditions set, he tries to find the probability of finding system A in state

*r*of corresponding energy [itex]E_r[/itex]. It follows from the postulate of equiprobability that the probability of finding A in state*r*is just the ratio of all the possible states of [itex]A^{(0)}[/itex] given that A is in state*r*to the ratio of all possible states of [itex]A^{(0)}[/itex]*:[tex]P_r=\frac{\Omega'(E^{(0)}-E_r)}{\Omega^{(0)}}[/tex]

He then says, let us expand [itex]\Omega'(E^{(0)}-E_r)[/itex] in a Taylor series about [itex]E^{(0)}[/itex]. But he actually expands the log of that in order to get a faster convergence:

[tex]\ln(\Omega'(E^{(0)}-E_r))\approx \ln(\Omega'(E^{(0)}))-\left[\frac{\partial \Omega'}{\partial E'} \right]_{E^{(0)}}E_r[/tex]

And he justifies the dropping of the higher order terms by the fact that [itex]E_r <<E^{(0)}[/itex].

But I have three problems with this justification. (problem #3 is closely linked to #2)

1) [itex]E_r=E^{(0)}[/itex], E'=0

*is*a possible state! And in principle, we do not know just*how*unlikely it is until we've effectively found the formula for [itex]P_r[/itex]!2) It seems to me that [itex]E_r <<E^{(0)}[/itex] justifies nothing! We want [itex]E_r<<1[/itex] so that the higher the power n, the smaller [itex](E_r)^n[/itex] is.

3) And wheter [itex]E_r[/itex] is <1 actually depends on the units of energy used! In, say, a relativistic taylor expansion, the argument is v/c which is dimensionless, so the problem of dimensionality does not arise. But in this case?!

I would really appreciate it if someone could nullify these two objections for me.

Many thanks!

*Reif uses [itex]\Omega'(E')[/itex] to denote the cardinality of the set of all states for wich the energy of A' is E' and [itex]\Omega^{(0)}[/itex] to denote the cardinality of the set of all possible states of the system [itex]A^{(0)}[/itex] as a whole.

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