Extended phase space 179∂P
∂ρ=
∂P
∂V
∂V
∂ρ∂^2 P
∂ρ^2=
[
∂^2 P
∂V^2
(
∂V
∂ρ) 2
+
∂P
∂V
∂^2 V
∂ρ^2]
. (4.7.49)
Both derivatives vanish at the critical point because of the conditions in eqn. (4.7.36).
It can be easily verified, however, that the third derivative is not zero, so that the first
nonvanishing term in eqn. (4.7.48) (apart from the constant term)is
P−Pc∼(ρ−ρc)^3 , (4.7.50)which leads to the prediction thatδ = 3. The calculation ofβis somewhat more
involved, so for now, we simply quote the result, namely, that the van der Waals
theory predictsβ= 1/2. We will discuss this exponent in more detail in Chapter 16.
In summary, the van der Waals theory predicts the four principal exponents to be
α= 0,β= 1/2,γ= 1, andδ= 3. Experimental determination of these exponents
givesα= 0.1,β= 0.34,γ= 1.35, andδ= 4.2, and we can conclude that the van
der Waals theory is only a qualitative theory: It predicts the existence of a gas-liquid
phase transition but cannot yield quantitative agreement with experiment for the
critical exponents of the transition.
4.8 Molecular dynamics in the canonical ensemble: Hamiltonian
formulation in an extended phase space
Our treatment of the canonical ensemble naturally raises the question of how molecular
dynamics simulations can be performed under the external conditions of this ensemble.
After all, as noted in the previous chapter, simply integrating Hamilton’s equations of
motion generates a microcanonical ensemble as a consequence of the conservation of the
total Hamiltonian. By contrast, in a canonical ensemble, energy is not conserved but
fluctuates so as to generate the Boltzmann distribution exp[−βH(x)] due to exchange
of energy between the system and the thermal reservoir to whichit is coupled. Although
we argued that these energy fluctuations vanish in the thermodynamic limit, most
simulations are performed far enough from this limit that the fluctuations cannot be
neglected.
In order to generate these fluctuations in a molecular dynamics simulation, we need
to mimic the effect of the thermal reservoir. Various methods to achieve this have been
proposed (Andersen, 1980; Nos ́e and Klein, 1983; Berendsenet al., 1984; Nos ́e, 1984;
Evans and Morriss, 1984; Hoover, 1985; Bulgac and Kusnezov, 1990; Martynaet al.,
1992; Liu and Tuckerman, 2000). We will discuss several of these approaches in the
remainder of this chapter. It must be mentioned at the outset, however, that most
canonical “dynamics” methods do not actually yield any kind of realistic dynamics
for a system coupled to a thermal bath. Rather, the trajectories generated by these
schemes comprise a set of microstates consistent with the canonical distribution. In
other words, they produce asamplingof the canonical phase space distribution from