Callen H.B - Thermodynamics And An Introduction To Thermostatistics · Problem 15.5-2

THE CANONICAL FORMALISM; STATISTICAL MECHANICS IN HELMHOLTZ REPRESENTATION

Physics — Statistical Mechanics · high-dimensionality-energy-shell

Problem

Recalling that

\lim_{x \to 0}(1+x)^{1/x} = e \quad (\equiv 2.718\ldots)

show that

\left(1 - \frac{\Delta}{U}\right) \approx e^{-\Delta/U} \quad \text{for } \frac{\Delta}{U} \ll 1

With this approximation discuss the accuracy of equation 15.31 for a range of reasonable values of

\Delta/U

(ranging perhaps from

10^{-3}

10^{-10}

). With what precision

\Delta/U

would the energy have to be known in order that corrections to equation 15.31 might become significant? Assume a system with

\tilde{N} = 10^{23}