Liquid-State Physical Chemistry: Fundamentals, Modeling, and Applications (2013)
5. The Transition from Microscopic to Macroscopic: Statistical Thermodynamics
5.4. A Few General Aspects
In this section, a few general aspects are dealt with that could have been made at various points in the foregoing discussion. Since the line-of-thought has been interrupted several times before, and these remarks would have interrupted this even more, we present them here together.
First, the presented line-of-thought takes the mode of the distribution as representing the average behavior, while one might think that one should calculate the average behavior directly. Apart from the fact that phenomenological thermodynamics in a way also calculates the mode (i.e., maximizes S), one can show that the distribution is extremely sharp and that mode and average are virtually the same. Essentially, one writes Σiexp(−βεi) ≅ ∫ exp(−βε)dn(ε) = ∫ exp(−βε)g(ε)dε with g(ε) = dn(ε)/dε. The sharpness is due to the compensating effects of the rapidly increasing value of the density of states g(ε) and the rapidly decreasing value of the Boltzmann factor exp(−βε) (see Problem 5.16). In the so-called thermodynamic limit (N → ∞ and V → ∞ but N/V remains finite) mode and average are identical. The fact that mode and mean are virtually the same leads also to the (surprising) maximum-term method (see Justification 5.4) which allows one to calculate the value of a logarithmic sum by taking just the largest term. A completely rigorous calculation of average values is via the Darwin–Fowler method (see, e.g., Ref. ).
Second, note the difference in the behavior of conjugated variables in statistical thermodynamics as compared to phenomenological thermodynamics. Whilst in the latter case both members of a pair of conjugated variables can be fixed, in the former case they behave complementarily. For example, if the temperature is prescribed via a temperature bath (closed system), the energy of the system will fluctuate, whereas if the energy is fixed (isolated system) the temperature will fluctuate. The same applies to other pairs of conjugated variables, although for macroscopic systems these fluctuations are normally small. Fluctuations resulting from our statistical considerations must not be confused with the intrinsic uncertainty of quantum mechanics, as described by the Heisenberg uncertainty relations (Eq. 2.61), in which both members of a conjugated pair of variables can have an uncertainty. Our statistical fluctuations obviously add to the quantum uncertainty, but usually they are much larger.
Third and finally, we note that other ensembles can be defined by using other constraints to the energy expression given by the Hamilton operator . For example, using constant pressure P instead of constant volume V, we can calculate directly G (P) = −kTlnΔ, where Δ = Σi exp[−β(Ei + PV)] using as energy expression  instead of calculating G(P) = F(P) + PV(P) after solving V from P = −∂F(V)/∂V. Using an obvious indication of the independent variables, the partition function Δ refers to the N-P-T or pressure ensemble, while Z refers to the N-V-T or canonical ensemble. Although, strictly speaking, and similar expressions do not represent internal energy, in statistical thermodynamics they are still often referred to as Hamiltonians.
Justification 5.4: The maximum-term method
Surprisingly, in statistical thermodynamics the logarithm of a sum often can be approximated by the logarithm of its maximum term. Consider a typical statistical thermodynamics expression such as
and and , where indicates the order of magnitude of x. The sum S can be evaluated exactly as
The largest term for S with index N* is also the largest term for lnS. Since N* and M − N* are both large, we can use Stirling's approximation for the factorials and write
The maximum term evaluates to , identical to the sum itself! To better understand this behavior, we expand ln tN about N* to obtain
where the linear term is missing because . From Eq. (5.40)
This is a very sharp Gaussian distribution centered about N*, since the relative standard deviation . Higher-order terms would yield a completely negligible contribution, and we have 〈N 〉 = N*.
Problem 5.16: The sharpness of the microstate distribution
Consider as system X a particle in a 3D box with density of states (DoS) . Take as reservoir R also such a system, so that the total energy becomes ε = εX + εR. Maximize the total DoS g = gXgR using ∂g/∂εX = 0, and show that εX = ½ε. Is this the expected result, and is the associated energy distribution sharp? Next, consider a many-particle system with NX particles with DoS with α = (3NX/2) − 1. Couple this system to a many-particle reservoir with NR particles with DoS with β = (3NX/2) − 1. Show that in this case that g is extremely sharply peaked for large NX and NR with mode EX = αE/(α + β), where E = EX + ER.