Computational Physics

7.1 Basic theory 175

canonical ensemble is given by

〈E〉NVT=

∑

Xe

−βH(X)H(X)

∑

Xe−βH(X)

(7.25)

and from this it is readily seen that

〈E〉NVT=−

∂lnZ

∂β

. (7.26)

The specifc heat at constant volumeCVis defined as

CV=

(

∂E

∂T

)

N,V

(7.27)

and it can therefore be related to the root mean square (rms) fluctuation of the energy:

CV=

1

kBT^2

∂^2 lnZ

∂β^2

=

1

kBT^2

[∑

Xe

−βH(X)H (^2) (X)
∑
Xe−βH(X)

−

(∑

Xe

−βH(X)H(X)

∑

Xe−βH(X)

) 2 



=

1

kBT^2

(〈E^2 〉NVT−〈E〉^2 NVT). (7.28)

Information about the microscopic properties of the system is given by correlation
functions, which can sometimes be measured experimentally, for example through
neutron scattering experiments [9]. In the next section we shall encounter several
examples of correlation functions.
In later chapters, we shall describe the molecular dynamics and Monte Carlo
simulation methods, which enable us to evaluate ensemble averages of different
physical quantities expressed in terms of the system coordinates. Such ensemble
averages are calledmechanical averages. Free energies and chemical potentials are
not directly given as mechanical averages but as phase space integrals. Integrals
over phase space cannot be estimated directly in simulations, but fortunately dif-
ferences between free energies at two different temperatures can be formulated as
ensemble averages. Suppose, for example, that we know the free energy of system
at a temperatureT, and we would like to know it at a different temperatureT′. The
differenceβF(T)−β′F(T′)is then found as

exp[βF(β)−β′F(β′)]=

Z(β′)

Z(β)

=

∑

Xexp[−β

′H(X)]

∑

Xexp[−βH(X)]

=〈exp[(−β′+β)H]〉β (7.29)

where〈···〉βdenotes a canonical ensemble average evaluated at inverse temperature
β. Determination of this expectation value in a simulation suffers from bad statistics.