Chapter 5 - WebRing

CHAPTER 5. MAGNETIC SYSTEMS 271
This change is an example of phase separation, and the simultaneous existence of both a gas and
liquid is referred to as gas-liquid coexistence. 13 The order parameter of the lattice gas is taken to
be ρ ∗ ≡ (ρL −ρG)/ρL, where ρL is the particle density of the liquid region and ρG is the density
of the gas region. Above Tc there is no phase separation (there are no separate liquid and gas
regions) and thus ρ ∗ = 0. At T = 0, ρL = 1 and ρG = 0. 14 The power law behavior of ρ ∗ as
T → Tc from below Tc is described by the critical exponent β, which has the same value as that
for the magnetization in the Ising model. The equality of the critical exponents for the Ising and
lattice gas models as well as more realistic models of liquids and gases is an example of universality
of critical phenomena for qualitatively different systems (see Sections 9.5 and 9.6).
If the number of occupied and unoccupied sites is unequal, then the transition from a fluid
to two-phase coexistence as the temperature is lowered is discontinuous. For particle systems it is
easier to analyze the transition by varying the pressure rather than the temperature.
As we will discuss in Chapter 7, there is a jump in the density as the pressure is changed along
an isotherm in a real fluid. This density change can occur either for a fixed number of particles
with a change in the volume or as a change in the number of particles for a fixed volume. We will
consider the latter by discussing the lattice gas in the grand canonical ensemble. From (2.168)
we know that the thermodynamic potential Ω associated with the grand canonical ensemble is
given by Ω = −PV. For the lattice gas the volume V is equal to the number of sites N, a fixed
quantity. We know from (4.144) that Ω = −kT lnZG, where ZG is the grand partition function.
The grand partition function for the lattice gas with the energy given by (5.137) is identical to the
partition function for the Ising model in a magnetic field with the energy given by (5.35), because
the difference is only a change of variables. The free energy for the Ising model in a magnetic field
is F = −kT lnZ. Because Z = ZG, we have that F = Ω, and thus we conclude that −PV for
the lattice gas equals F for the Ising model. This identification will allow us to understand what
happens to the density as we change the pressure along an isotherm. In a lattice gas the density
is the number of particles divided by the number of sites or ρ = ni. In Ising language ρ is
ρ = 1
2 (si +1) = (m+1)/2, (5.138)
where m is the magnetization. Because −PV = F, changing the pressure at fixed temperature
and volume in the lattice gas corresponds to changing the free energy F(T,V,H) by changing the
field H in the Ising model. We know that when H changes from 0 + to 0 − for T < Tc there is a
jump in the magnetization from a positive to a negative value. From (5.138) we see that this jump
corresponds to a jump in the density in the lattice gas model, corresponding to a change in the
density from a liquid to a gas.
Problem 5.23. Simulation of the two-dimensional lattice gas
(a) What is the value of the critical temperature Tc for the lattice gas in two dimensions (in units
of u0/k)?
(b) Program LatticeGas simulates the lattice gas on a square lattice of linear dimension L. The
initial state has all the particles at the bottom of the simulation cell. Choose L = 32 and set
13If the system were subject to gravity, the liquid region would be at the bottom of a container and the gas would
be at the top.
14For T > 0 the cluster representing the liquid region would have some unoccupied sites and thus ρL < 1, and
the gas region would have some particles so that ρG > 0.