1Revision of equilibrium statistical physics
III Theoretical Physics of Soft Condensed Matter
1.2 Coarse Graining
Usually, in statistical mechanics, we distinguish between two types of objects
— microstates, namely the exact configuration of the system, and macrostates,
which are variables that describe the overall behaviour of the system, such that
pressure and temperature. Here we would like to consider something in between.
For example, if we have a system of magnets as in the Ising model, we the
microstate would be the magnetization at each site, and the macrostate would
be the overall magnetization. A coarse-graining of this would be a function
m
(
r
)
of space that describes the “average magnetization around
r
”. There is no fixed
prescription on how large an area we average over, and usually it does not matter
much.
In general, the coarse-grained variable would be called
ψ
. We can define a
coarse-grained partition function
Z[ψ(r)] =
X
i∈ψ
e
−βE
i
,
where we sum over all states that coarse-grain to
ψ
. We can similarly define the
energy and entropy by restricting to all such ψ, and get
F [ψ] = E[ψ] − T S[ψ].
The probability of being in a state ψ is then
P[ψ] =
e
−βF [ψ]
Z
TOT
, Z
TOT
=
Z
e
−βF [ψ]
D[ψ].
What we have on the end is a functional integral, where we integrate over all
possible values of ψ. We shall go into details later. We then have
F
TOT
= −k
B
T log Z
TOT
.
In theory, one can obtain
F
[
ψ
] by explicitly doing a coarse graining of the
macroscopic laws.
Example.
Consider an interacting gas with
N
particles. We can think of
the energy as a sum of two components, the ideal gas part (
d
2
NkT
), and an
interaction part, given by
E
int
=
1
2
X
i6j
U(r
i
− r
j
),
where
i, j
range over all particles with positions
r
i
, r
j
respectively, and
U
is
some potential function. When we do coarse-graining, we introduce a function
ρ
that describes the local density of particles. The interaction energy can then be
written as
E
int
=
1
2
ZZ
U(r −r
0
)ρ(r)ρ(r
0
) dr dr
0
.
Similarly, up to a constant, we can write the entropy as
S[ρ] = −k
B
Z
ρ(r) log ρ(r) dr.
In practice, since the microscopic laws aren’t always accessible anyway, what
is more common is to take a phenomenological approach, namely we write down
a Taylor expansion of
F
[
ψ
], and then empirically figure out what the coefficients
should be, as a function of temperature and other parameters. In many cases, the
signs of the first few coefficients dictate the overall behaviour of the system, and
phase transition occurs when the change in temperature causes the coefficients
to switch signs.