Exam 3, 03/07/03

Exam 3 CHEM 430b/530b
Statistical Methods and Thermodynamics
03/07/03

 Exercise 1

(10 points) Item (1.1): Explain the underlying approximation of Mean Field theory and illustrate it with a specific Hamiltonian.
(20 points) Item (1.2): Show that Mean Field theory predicts spontaneous magnetization for the 2-dimensional Ising model when $T < 4 J/k$, where J is the coupling constant between spins.
(20 points) Item (1.3): Derive the Gibbs-Bogoliubov-Feynman inequality.
(20 points) Item (1.4): Derive the renormalization group equations for the 1-dimensional Ising model.

Exercise 2

(30 points) Compute the grand canonical partition function of a 1-dimensional lattice gas by implementing the transfer matrix approach. Hint: Assume that the total energy for a given set of occupation numbers $\{n_j\}$ is

$$E=-\mu \sum_{j=1}^N n_j -\epsilon \sum_{j=1}^N \sum_k n_j n_k,$$ (363)

where $\mu$ is the chemical potential of the particles, the occupation numbers $n_j=0, 1$. The indices $k$, in Eq. (eq363) label the cells that are next to cell $j$ and N is the total number of cells.
 

Solution

Item (1.1):
The underlying approximation of Mean Field theory is to assume that the most important contribution to the interaction between each particle and its neighboring particles is determined by the mean field due to the neighboring particles. Section Mean Field Theory on page 65 of your lecture notes illustrates this concept as applied to the description of the 1-dimensional Ising model.

Item (1.2):
In the 2-dimensional Ising model, the average force $\overline{F_k}$ exerted on spin $S_k$ is

$$\overline{F_k} \equiv -\overline{\frac{\partial H}{\partial S_k}} = \bar{\mu} B + J \sum_{j} \overline{S_j},$$ (364)

where the index $j$ includes all the nearest neighbors of spin $S_k$. Therefore, the average magnetic field $\overline{B}$ acting on spin $S_k$ is

$$\overline{B} \equiv \frac{\overline{F_k}}{\bar{\mu}} = B + \Delta B,$$ (365)

where

$$\Delta B = J 4 \overline{S_k}/ \bar{\mu},$$ (366)

is the contribution to the mean field due to the nearest neighbors. Note that $\overline{S_k}=\overline{S_j}$ when all spins are identical. The partition function, under the mean field approximation, is

$$Z \approx \sum_{S_1} \sum_{S_2} ... \sum_{S_N} e^{\beta \sum_j S_j (B+\Delta B) \bar{\mu}} = 2^N ^N (\beta \bar{\mu} \overline{B}),$$ (367)

and the average value of $S_k$ is

$$\overline{S_k} = \frac{1}{N} \frac{\partial \text{ln} Z}{\partial......\overline{B})} = \text{tanh}(\beta \bar{\mu} (B+4 J \overline{S_k}/\bar{\mu})).$$ (368)

Note that Eq. (368) involves a transcendental equation. Its solution corresponds to the value of $\overline{S_k}=m$ for which the function on the left hand side of Eq.(368) (i.e., $\overline{S_k}$) equals the function on the right hand side of Eq. (368) (i.e., tanh$(\beta \bar{\mu} (B+4 J \overline{S_k}/\bar{\mu})$). In the absence of an external magnetic field (i.e., when $B=0$), Eq. (368) always has the trivial solution $S_k=0$ and a non-trivial solution $S_k=m$ only when $\beta 4 J > 1$.
Item (1.3):
See derivation of Eq. (312) on page 69 of your lecture notes.

Item (1.4):
See derivation of Eqs. (326) and (327) on page 72 of your lecture notes.

Exercise 2:

The Hamiltonian of the system is

$$H = -\mu \sum_j n_j - \epsilon \sum_{jk} n_j n_k,$$ (369)

where the sum of products $n_j n_k$ defines the interaction between cells that are nearest neighbors. The grand canonical partition function of the system is

$$\Xi = \sum_{n_1=0,1} \sum_{n_2=0,1} ... \sum_{n_N=0,1} e^{\beta (......n_2)/2 + \epsilon n_2 n_3)} ... e^{\beta (\mu (n_N+n_1)/2 + \epsilon n_N n_1)}.$$ (370)

In order to perform a rigorous calculation of the grand canonical partition function introduced by Eq.(370), we define the transfer function as follows,

$$T(n_i, n_{i+1}) \equiv (\beta (\mu (n_i+n_{i+1})/2 + \epsilon n_i n_{i+1})).$$ (371)

Substituting Eq.(371) into Eq.(370) we obtain

$$\Xi = \sum_{n_1=0,1} \sum_{n_2=0,1} ... \sum_{n_N=0,1} T(n_1, n_2) T(n_2, n_3) ... T(n_N, n_1).$$ (372)

This expression corresponds to the trace of a product of N identical $2 \times 2$ matrices. Thus the calculation of the grand canonical partition function is reduced to that of computing the trace of the $N$th power of the transfer matrix. Now, the trace of a matrix is the sum of its eigenvalues and the eigenvalues of ${\bf T}^N$ are $\lambda_{\pm}^N$, where $\lambda_{\pm}$ are the eigenvalues of ${\bf T}$ determined by the equation

$$\Bigg \vert \begin{matrix}e^{\beta (\mu + \epsilon)}-\lambda & e^{\beta \mu/2} \\ e^{\beta \mu/2} & 1-\lambda \\ \end{matrix} \Bigg \vert = 0,$$ (373)

with solutions

$$\lambda_{\pm} = \frac{1}{2} [e^{\beta (\mu+\epsilon)} + 1] \pm \sqrt{[e^{\beta (\mu+\epsilon)}-1]^2/4+e^{\beta \mu}}.$$ (374)

Hence, the partition function is simply,

$$Z = \lambda_+^N +\lambda_-^N,$$ (375)

where $\lambda_{\pm}$ are defined by Eq. (374).