1549380323-Statistical Mechanics Theory and Molecular Simulation

Harmonic systems 93

Let us now repeat the analysis using eqn. (3.5.29). Introducing Stirling’s approx-
imation as a logarithm of eqn. (3.5.19), lnN!≈NlnN−N, eqn. (3.5.29) can be
rewritten as

S=Nkln

[

V

Nh^3

(

2 πm

β

) 3 / 2 ]

+

5

2

Nk, (3.5.35)

which is known as theSackur–Tetrodeequation. Using eqn. (3.5.35), the entropy
difference ∆Sbecomes

∆S= (N 1 +N 2 )kln

(

V 1 +V 2

N 1 +N 2

)

−N 1 kln(V 1 /N 1 )−N 2 kln(V 2 /N 2 )

=N 1 kln(V/V 1 ) +N 2 kln(V/V 2 )−N 1 kln(N/N 1 )−N 2 kln(N/N 2 )

=N 1 kln

(

V

N

N 1

V 1

)

+N 2 kln

(

V

N

N 2

V 2

)

. (3.5.36)

However, since the densityρ=N 1 /V 1 =N 2 /V 2 =N/Vis constant, the logarithms all
vanish, which leads to the expected results ∆S= 0. A purely classical treatment of
the particles is, therefore, unable to resolve the paradox. Only byaccounting for the
identical nature of the particlesa posteriorior via a proper quantum treatment of the
ideal gases (see Chapter 11) can a consistent thermodynamic picture be obtained.

3.6 The harmonic oscillator and harmonic baths

The second example we will study is a single harmonic oscillator in one dimension
and its extention to a system ofNoscillators in three dimensions (also known as a
“harmonic bath”). We are returning to this problem because harmonic oscillators lie
at the heart of a wide variety of important problems. They are often used to describe
intramolecular bond and bend vibrations in biological force fields, they are used to
describe ideal solids, they form the basis of normal mode analysis (see Section 1.7),
and they turn up repeatedly in quantum mechanics.
Consider first a single particle in one dimension with coordinatexand momentum
pmoving in a harmonic potential

U(x) =

1

2

kx^2 , (3.6.1)

wherekis the force constant. The Hamiltonian is given by

H=

p^2

2 m

+

1

2

kx^2. (3.6.2)

In Section 1.3, we saw that the harmonic oscillator is an example of a bound phase
space. We shall consider that the one-dimensional “container” is larger than the maxi-
mum value ofx(as determined by the energyE), so that the integration can be taken
over all space.