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\line{Physics 352 \hfill \today \hfill Handout E}
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\beginsection Mean field approximation
 
In class we studied a limit where the mean-field approximation becomes valid. 
But the approximation derived there didn't look exactly like the one studied in 
earlier lectures.  In this handout I show that the two mean field approximations
are the same [cf. Landau and Liftshitz, 3rd edition, Section 147].  We first
considered a lattice gas of $V$ sites with occupation numbers
$n_i$ and interaction energy $\half \frac v \zeta
\sum_{\expectation{i,j}}^\zeta n_i(1-n_j)$. The sum $\expectation{i, j}$ denotes
a sum over all sites $i$ and those sites $j$ that interact with it directly.  The
number of such interacting sites for each site $i$ is denoted $\zeta$.  In the
simplest case, the sum
$j$ is over the nearest neighbors of each site $i$.  We noted that the
average
$\expectation{n}$ was simply the volume fraction---denoted $\phi$---of occupied
sites.  We then argued that the thermodyanmic potential $A(\phi)$ could be
approximated by replacing $n$ by its average $\phi$:
$$\eqalign{
A(\phi) \definedas& \expectation{U} - T S\cr \aboutequal & 
\left . \expectation{U} \right |_{n\goesto \phi} - TS(\phi)\cr
&= v V\phi(1-\phi) + T V\phi \log \phi + T V(1-\phi) \log (1-\phi)
\definedas A_1(\phi)
}\eqno(1)$$
In the last lecture we showed that if the number of interacting sites $\zeta$
was sufficiently large, the grand potential $\Omega(\mu)\definedas -T \log Q$ 
could be written as 
$$\eqalign{
\Omega(\mu) &\definedas -T \log \sum_c \exp(-\beta U + \beta \mu \sum_i n_i)\cr
&\aboutequal  -T \log \sum_c \exp[-\beta \sum_i n_i( v(1-\phi) - \mu)]
}\eqno(2)$$
In both of these approximations $n$ has been replaced by its average, $\phi$, to
make a simpler thermodynamic function.  But it is not clear whether the two
approximations (1) and (2) are equivalent.  We now show that they are.

In general the interaction energy $U$ depends on several degrees of freedom. 
If the degrees of freedom are denoted as $n_i$ (borrowing the notation of the
lattice gas), then $U$ can very often be written in the form 
$U\{n\} = v\sum_i n_i M_i\{n\}$, where the quantity $M$ depends on degrees of
freedom other than $n_i$.  The mean-field approximation of Eq. 2 amounts to
$$
\Omega(\mu, v) \aboutequal \Omega(\mu - vM(\phi), 0)
.$$
It relates the grand potential of the real system to that of a
{\it noninteracting} system with a shifted chemical potential: $\mu\goesto \mu -
v M(\phi)$.  To find the thermodynamic potential $A(\phi)$ for $\phi$, we have to
use the Legendre-transform relation between $\Omega$ and $A$:
$$
\Omega(\mu, v) = A(\phi, v) - \mu N
, \quad {\rm or} \quad A(\phi, v) = \Omega(\mu, v) + \mu N
= \Omega(\mu - vM(\phi), 0) + \mu N
.$$
We denote the shifted $\mu$ as $\tilde \mu$, so that $\mu = \tilde \mu + v
M(\phi)$.  Then
$$\eqalign{
A(\phi) = \quad\Omega(\tilde\mu, 0) +\quad [\tilde\mu  + v& M(\phi)] N\cr
= [\Omega(\tilde\mu, 0) + \tilde\mu N]\quad\hfill  + v& M(\phi) N \cr
= \quad A(\phi, 0) \quad\quad\quad  +  v& M(\phi) N
}$$
The $v M N$ on the right is simply $\left . \expectation{U}\right |_{n\goesto
\phi}$, as in Eq. 1.  Thus the mean-field approximation of Eq. 2 is equivalent
to that of Eq. 1, as claimed.

\beginsection Correlations at long distances: Ornstein-Zernike Approximation

[ref: Landau and Liftshitz 3rd edition, section 146]
\head{Susceptibility and curvature}  The susceptibility $\chi$ of a system with a
phase transition is defined as the derivitive of its order parameter with
respect to its conjugate field. For a lattice gas, the order parameter can be
taken as the volume fraction $n - \half$.  Thus the
susceptibility $\chi = \partial \expectation{n}/\partial \mu = 
\partial^2
\frac \Omega V/\partial \mu^2 = \beta \expectation{(n -
\expectation{n})^2}$.  This susceptibility has a simple relationship to
derivitives of the thermodynamic potential for $\phi\definedas \expectation{n}$,
namely $A(\phi)$:  
$$
\mu = \partial A/\partial N =  \partial \frac A V/\partial \phi
$$  so
$$
\partial \mu /\partial \phi = \partial^2 \frac A V/\partial \phi^2
$$
But since\footnote*
{No other variables were Legendre-transformed, so the same quantities are held
fixed for $\partial/\partial \mu$ and for its inverse, $\partial/\partial \phi$.
}
 $\partial \mu /\partial \phi = (\partial \phi /\partial \mu)^{-1}$,
the susceptibility $\chi$ is evidently an {\it inverse} second derivitive of
$A$:
$$
\chi = {1\over \partial^2 \frac A V/\partial \phi^2}
.$$
This relationship is general; it is true for any susceptibility.  Thus whenever
the curvature of the thermodynamic potential with respect to the order
parameter vanishes, the susceptibility diverges.

\head{Free energy of nonuniform states} {\multiply \tolerance by 10 
If the
average of
$n$ is forced to have some nonuniform value $\phi(r)$, the free energy
$A[\phi(r)]$ is influenced by the nonuniformity.  We can find this free energy
in the mean-field approximation.  
$$
A[\phi(r)] = \sum_i A_0(\phi(r_i))/V + \left .\expectation{U}\right|_{n\goesto
\phi(r)}
.$$
} %tolerance
The $A_0/V$ is the free energy per site of the noninteracting lattice. Each
site's free energy depends only on its own volume fraction $\phi(r_i)$.  But
the potential energy $U = \frac v \zeta \sum_{\expectation{i,j}}^\zeta n_i
(1-n_j)$ depends nonlocally on $\phi$.  This dependence has a simple form if
the imposed $\phi(r)$ is gently varying on the scale $\xi_0$ of the interaction
range.  In that case, $\phi$ can be well approximated by its Taylor expansion
in $U$: $\phi(r_j) \aboutequal \phi(r_i) + \grad \phi \cdot (r_j - r_i)
+ \half \grad^2 \phi~~ (r_j - r_i)^2 + ...
$.  Thus 
$$
\expectation{U} \aboutequal \sum_i v\phi(r_i) (1 - \phi(r_i))
- v\sum_i \phi(r_i) \grad^2 \phi(r_i) \left (\frac 1 \zeta \sum_j^\zeta
(r_j-r_i)^2\right ) .$$
The linear gradient term has vanished since the $\sum_j$ is symmetric about the
site $i$.  Other quadratic gradients also vanish by symmetry.  The $\sum_i
\phi \grad^2 \phi$ can be re-expressed using Green's theorem as $-\sum_i (\grad
\phi)^2$.  The term in $(...)$ is the mean-square distance between interacting
sites.  We call it $\xi_0^2$.  The expression for the free energy $A$ can now
be written as 
$$
A[\phi] = \sum_i {A(\phi(r_i)\over V} + \half v ~\xi_0^2 ~ (\grad \phi(r_i))^2 +
...  .$$
The first term is local.  For each site $i$ we take the free energy per site of a
uniform system whose density is $\phi(r_i)$.  This is not the whole free energy
because the interaction energy is nonlocal.  There is an additional
contribution involving gradients of the density.  We have chosen our
interaction energy to resist nonuniform states; thus naturally the gradient
energy is positive.  This is true in general.  If it were not, the system would
spontaneously lower its free energy by becoming nonuniform.  The $...$ denotes 
higher-order gradients, assumed negligible for our gently-varying density.

Even where the mean-field approximation is not valid, we expect the free energy 
for nonuniform densities to have this form, with $v$ replaced by some other
coefficient $\hat v$.  We shall assume below that $A[\phi]$ has a ``gradient
expansion" of this form.

\head{Spatial fluctuations} Knowing the free energy for a nonuniform density, we
may calculate the nonuniform fluctuations in density.  It is convenient to think
of the density as being composed of plane waves.  Accordingly, we define 
$$
\tilde n_q \definedas \sum_j n_j e^{i q \cdot r_j}
.$$
Evidently, we can express the local density in terms of the $\tilde n_q$:
$$
n_j = \sum'_q \tilde n_q e^{-i q \cdot r_j}.
$$
The $\sum'_q$ is over all plane-wave modes defined on the lattice, and it
includes any needed factors of $2\pi$.  

To find the magnitude of these fluctuations, we consider the free energy for a
nonuniform $\phi(r)$ of sinusoidal form: $\phi(r) = \tilde \phi_q e^{i q
r}$., where the amplitude is assumed very small.  Then from the gradient
expansion for
$A[\phi]$ we obtain
$$
A[\phi] = \frac 1 V\sum_i A(\phi(r)) + \half \hat v \xi_0^2 q^2 \sum_i |\tilde
\phi_q|^2 .$$
Now, the local $A$ has the form $A_0 + \half A'' \phi^2$, where the $A''$ is
taken at the unperturbed density.  The second term has a nonzero average of
$\half A'' |\tilde\phi_q|^2$.  Combining, 
$$
A[\phi] = A_0 + \half  A'' ~ |\tilde \phi_q|^2 + \half \hat v
\xi_0^2 q^2 |\tilde\phi_q|^2
,$$
where $A_0$ is the free energy without the perturbation.  We may find the
fluctuations of $\tilde n_q$ as we can for any other system
variable:
$$
\expectation{|n_q - \expectation{n_q}|^2} = {T \over \partial^2 \frac A V /
\partial\tilde\phi_q^2} = {T \over A''/V + \hat v ~\xi_0^2 q^2}
= {\chi T \over 1 + q^2 \xi^2}
,\eqno(10)$$
where $\xi^2 \definedas \hat v ~\xi_0^2~ \chi$.
These are equilibium fluctuations, so $\expectation{n_q}=0$.  This expression
gives the amplitude of spontaneous plane-wave fluctuations in the system for
arbitrary small $q$.  Shorter-wavelength fluctuations have higher $q$, the
denominator of (10) is larger, and thus the fluctuations are weaker.  The
wavevector appears in combination with a characteristic length $\xi$ which is
proportional to the square root of $\chi$.  This expression is the
{\it Ornstein-Zernike} formula for the q-dependent susceptibility.  It is valid
for small-enough $q$ for almost every system.  

From the Ornstein-Zernike susceptibility we can infer the spatial density
correlations in our system.  Substituting the expression for $n_j$, we find
$$
\expectation{|n_q|^2} = 
\expectation{\sum_j n_j \sum_k n_k e^{i q (r_j - r_k)}}
= \sum_{\vector r_j} \sum_{\vector r_j - \vector r_k}\expectation{n_j n_k}e^{i q
(r_j - r_k)}  .$$
We note that the $\expectation{n_j n_k}$ depends only on the relative
separation of sites $j$ and $k$, denoted $\vector r$, so that the site $j$ might
as well be the origin: $\expectation{n_j n_k} = \expectation{n(0) n
(\vector r)}$. Using these facts we may simplify
$$
\expectation{|n_q|^2} = \sum_{\vector r_j} \sum_{\vector r} \expectation{n(0)
n(r)} e^{i q r} = V \sum_r \expectation{n(0) n(r)} e^{i q r}
.$$
Thus the spatial correlation function $\expectation{n(0) n(r)}$ is related to the
Ornstein-Zernike susceptibility by a simple Fourier transform:
$$
\expectation{n(0) n(r)} = {\chi T \over V}\sum'_q {1 \over 1 + q^2
\xi^2} e^{-iq \cdot r}
.$$
This $q$ dependence arises often in physics.  It describes the electrostatic
potential of a screened point charge or the propagator of a massive
scalar field.  We know from these other contexts that $\expectation{n(0) n(r)}$
must be proportional to
$e^{-r/\xi}/r$ in three dimensions.  In two dimensions, $\expectation{n(0)
n(r)}$ is a modified Bessel function, which also dies exponentially at
infinity.  

As our system approaches a critical point the susceptibility $\chi$ diverges. 
but the coefficient $\hat v \xi_0$ of the gradient term does not.  This term
reflects the microscopic features of the interaction potential: its strength
and its range.  Since $\xi$ is proportional to $\chi^{1/2}$, it must diverge as
the critical point is approached.   The divergence for real critical point
involves a  power of $\chi$ different from 1/2.  This difference arises because
the mean-field approximation becomes invalid as the critical point is
approached.

\beginsection Yang-Lee theorem: no phase transitions in a finite system.

[ref: Ma, Chapter 9]

The abrupt, discontinuous behavior seen at a phase transition is hard to
rationalize intuitively.  Indeed, one can prove that such discontinuities 
are impossible whenever the system has a finite number of degrees of freedom. 
To illustrate, we consider a lattice gas of $V$ sites.  
In order for a system variable to change discontinuously as a function of a
field such as $\mu$, the grand potential $\Omega(\mu)$ must be discontinuous. 
This in turn means that the grand partition function $Q = \sum_c f_c$  is
discontinuous.

But in a finite system such discontinuities can't happen.  To see this we write
out $Q$ by grouping all the configurations having a given number of particles
$N$.  
$$
Q = \sum_{N=1}^V Q_N e^{\beta \mu N}
.$$
Defining the {\it fugacity} or {\it activity} $z \definedas e^{\beta \mu}$, we
can express $Q$ as
$$
Q = \sum_{N=1}^V Q_N z^N
.$$

Evidently, $Q$ is a polynomial in $z$ whenever $V$ is finite.  The coefficients
$Q_N$ are the sums of positive Boltzmann weights and are therefore positive. 
Thus $Q(z)$ cannot vanish for positive $z$.  Since this includes all possible
values of $z$, the $Q(z)$ cannot vanish at all.  

We now find the grand potential $\Omega$ by taking the log of $Q(z)$.  Since
$Q$ is never zero for physical $z$, the grand potential is never singular. 
If we consider $z$ to be a complex variable $\Omega(z)$ must be an {\it
analytic}  function near the positive real axis. The function $\Omega(z)$ and
all its derivitives have to be completely smooth.  There can be no
discontinuities as
$z$ or $\mu$ are varied.  Thus there can be no phase transition.  This
analyticity is the content of the {\it Yang-Lee theorem}.

How does this picture break down as the size $V$ of the system grows larger? 
Yang and Lee showed that in the limit of large size, the zeros of the partition
function converge towards the real axis at one point.  This point then becomes
the $z$ value at which the phase transition occurs.
\end