Statistical Physics of Particles

Mehran Kardar

Chapter 5 Interacting particles - all with Video Answers

Educators

Chapter Questions

Problem 1

Debye-Hückel theory and ring diagrams: the virial expansion gives the gas pressure as an analytic expansion in the density $n=N / V$. Long-range interactions can result in non-analytic corrections to the ideal gas equation of state. A classic example is the Coulomb interaction in plasmas, whose treatment by Debye-Hückel theory is equivalent to summing all the ring diagrams in a cumulant expansion.

For simplicity consider a gas of $N$ electrons moving in a uniform background of positive charge density $\mathrm{Ne} / \mathrm{V}$ to ensure overall charge neutrality. The Coulomb interaction takes the form
$$
\mathcal{U}_Q=\sum_{i<j} \mathcal{V}\left(\vec{q}_i-\vec{q}_j\right), \quad \text { with } \quad \mathcal{V}(\vec{q})=\frac{e^2}{4 \pi|\vec{q}|}-c
$$
The constant $c$ results from the background and ensures that the first-order correction vanishes, that is, $\int \mathrm{d}^3 \vec{q} \mathcal{V}(\vec{q})=0$.
(a) Show that the Fourier transform of $\mathcal{V}(\vec{q})$ takes the form
$$
\tilde{\mathcal{v}}(\vec{\omega})=\left\{\begin{array}{ll}
e^2 / \omega^2 & \text { for } \vec{\omega} \neq 0 \\
0 & \text { for } \vec{\omega}=0
\end{array}\right. \text {. }
$$
(b) In the cumulant expansion for $\left\langle\mathcal{U}_Q^l\right\rangle_c^0$, we shall retain only the diagrams forming a ring, which are proportional to
$$
R_{\ell}=\int \frac{\mathrm{d}^3 \vec{q}_1}{V} \cdots \frac{\mathrm{d}^3 \vec{q}_{\ell}}{V} \mathcal{V}\left(\vec{q}_1-\vec{q}_2\right) \mathcal{V}\left(\vec{q}_2-\vec{q}_3\right) \cdots \mathcal{V}\left(\vec{q}_{\ell}-\vec{q}_1\right)
$$
Use properties of Fourier transforms to show that
$$
R_{\ell}=\frac{1}{V^{\ell-1}} \int \frac{\mathrm{d}^3 \vec{\omega}}{(2 \pi)^3} \tilde{\mathcal{v}}(\vec{\omega})^{\ell}
$$
(c) Show that the number of ring graphs generated in $\left\langle u_Q^{\ell}\right\rangle_c^0$ is
$$
S_{\ell}=\frac{N !}{(N-\ell) !} \times \frac{(\ell-1) !}{2} \approx \frac{(\ell-1) !}{2} N^{\ell} .
$$
(d) Show that the contribution of the ring diagrams can be summed as
$$
\begin{aligned}
\ln Z_{\text {riags }} & =\ln Z_0+\sum_{\ell=2}^{\infty} \frac{(-\beta)^{\ell}}{\ell !} S_{\ell} R_{\ell} \\
& \approx \ln Z_0+\frac{V}{2} \int_0^{\infty} \frac{4 \pi \omega^2 \mathrm{~d} \omega}{(2 \pi)^3}\left[\left(\frac{\kappa}{\omega}\right)^2-\ln \left(1+\frac{\kappa^2}{\omega^2}\right)\right],
\end{aligned}
$$
where $\kappa=\sqrt{\beta e^2 N / V}$ is the inverse Debye screening length.
(Hint. Use $\ln (1+x)=-\sum_{\ell=1}^{\infty}(-x)^{\ell} / \ell$.)
(e) The integral in the previous part can be simplified by changing variables to $x=\kappa / \omega$, and performing integration by parts. Show that the final result is
$$
\ln Z_{\text {rings }}=\ln Z_0+\frac{V}{12 \pi} \kappa^3 \text {. }
$$
(f) Calculate the correction to pressure from the above ring diagrams.
(g) We can introduce an effective potential $\bar{V}\left(\vec{q}-\vec{q}^{\prime}\right)$ between two particles by integrating over the coordinates of all the other particles. This is equivalent to an expectation value that can be calculated perturbatively in a cumulant expansion. If we include only the loopless diagrams (the analog of the rings) between the particles, we have
$$
\begin{gathered}
\bar{V}\left(\vec{q}-\vec{q}^{\prime}\right)=V\left(\vec{q}-\vec{q}^{\prime}\right)+\sum_{\ell=1}^{\infty}(-\beta N)^{\ell} \int \frac{\mathrm{d}^3 \vec{q}_1}{V} \cdots \frac{\mathrm{d}^3 \vec{q}_{\ell}}{V} \mathcal{V}\left(\vec{q}-\vec{q}_1\right) \mathcal{V}\left(\vec{q}_1-\vec{q}_2\right) \cdots \\
\mathcal{V}\left(\vec{q}_{\ell}-\vec{q}^{\prime}\right) .
\end{gathered}
$$
Show that this sum leads to the screened Coulomb interaction $\bar{V}(\vec{q})=$ $e^2 \exp (-\kappa|\vec{q}|) /(4 \pi|\vec{q}|)$

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03:15

Problem 2

Virial coefficients: consider a gas of particles in $d$-dimensional space interacting through a pairwise central potential, $\mathcal{V}(r)$, where
$$
\mathcal{V}(r)= \begin{cases}+\infty & \text { for } 0<r<a, \\ -\varepsilon & \text { for } a<r<b, \\ 0 & \text { for } b<r<\infty\end{cases}
$$
(a) Calculate the second virial coefficient $B_2(T)$, and comment on its high- and lowtemperature behaviors.
(b) Calculate the first correction to isothermal compressibility
$$
\kappa_T=-\left.\frac{1}{V} \frac{\partial V}{\partial P}\right|_{T, N} .
$$
(c) In the high-temperature limit, reorganize the equation of state into the van der Waals form, and identify the van der Waals parameters.
(d) For $b=a$ (a hard sphere), and $d=1$, calculate the third virial coefficient $B_3(T)$.

Farhana Sharmin

Numerade Educator

01:54

Problem 3

Dieterici's equation: a gas obeys Dieterici's equation of state:
$$
P(v-b)=k_B T \exp \left(-\frac{a}{k_B T v}\right),
$$
where $v=V / N$.
(a) Find the ratio $P v / k_B T$ at the critical point.
(b) Calculate the isothermal compressibility $\kappa_T$ for $v=v_c$ as a function of $T-T_c$.
(c) On the critical isotherm expand the pressure to the lowest non-zero order in $\left(v-v_c\right)$.

Manik Pulyani

Numerade Educator

Problem 4

Two-dimensional Coulomb gas: consider a classical mixture of $N$ positive and $N$ negative charged particles in a two-dimensional box of area $A=L \times L$. The Hamiltonian is
$$
\mathcal{H}=\sum_{i=1}^{2 N} \frac{\vec{p}_i^2}{2 m}-\sum_{i<j}^{2 N} c_i c_j \ln \left|\vec{q}_i-\vec{q}_j\right|,
$$
where $c_i=+c_0$ for $i=1, \cdots, N$, and $c_i=-c_0$ for $i=N+1, \cdots, 2 N$, denote the charges of the particles; $\left\{\vec{q}_i\right\}$ and $\left\{\vec{p}_i\right\}$ their coordinates and momenta, respectively.
(a) Note that in the interaction term each pair appears only once, and there is no selfinteraction $i=j$. How many pairs have repulsive interactions, and how many have attractive interactions?
(b) Write down the expression for the partition function $Z(N, T, A)$ in terms of integrals over $\left\{\vec{q}_i\right\}$ and $\left\{\vec{p}_i\right\}$. Perform the integrals over the momenta, and rewrite the contribution of the coordinates as a product involving powers of $\left\{\vec{q}_i\right\}$, using the identity $\mathrm{e}^{\ln x}=x$
(c) Although it is not possible to perform the integrals over $\left\{\vec{q}_i\right\}$ exactly, the dependence of $Z$ on $A$ can be obtained by the simple rescaling of coordinates, $\vec{q}_i{ }^{\prime}=\vec{q}_i / L$. Use the results in parts (a) and (b) to show that $Z \propto A^{2 N-\beta c_0^2 N / 2}$.
(d) Calculate the two-dimensional pressure of this gas, and comment on its behavior at high and low temperatures.
(e) The unphysical behavior at low temperatures is avoided by adding a hard core that prevents the coordinates of any two particles from coming closer than a distance $a$. The appearance of two length scales, $a$ and $L$, makes the scaling analysis of part (c) questionable. By examining the partition function for $N=1$, obtain an estimate for the temperature $T_c$ at which the short distance scale $a$ becomes important in calculating the partition function, invalidating the result of the previous part. What are the phases of this system at low and high temperatures?

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Problem 5

Exact solutions for a one-dimensional gas: in statistical mechanics, there are very few systems of interacting particles that can be solved exactly. Such exact solutions are very important as they provide a check for the reliability of various approximations. A one-dimensional gas with short-range interactions is one such solvable case.
(a) Show that for a potential with a hard core that screens the interactions from further neighbors, the Hamiltonian for $N$ particles can be written as
$$
\mathcal{H}=\sum_{i=1}^N \frac{p_i^2}{2 m}+\sum_{i=2}^N \mathcal{V}\left(x_i-x_{i-1}\right) .
$$
The (indistinguishable) particles are labeled with coordinates $\left\{x_i\right\}$ such that
$$
0 \leq x_1 \leq x_2 \leq \cdots \leq x_N \leq L
$$
where $L$ is the length of the box confining the particles.
(b) Write the expression for the partition function $Z(T, N, L)$. Change variables to $\delta_1=$ $x_1, \delta_2=x_2-x_1, \cdots, \delta_N=x_N-x_{N-1}$, and carefully indicate the allowed ranges of integration and the constraints.
(c) Consider the Gibbs partition function obtained from the Laplace transformation
$$
z(T, N, P)=\int_0^{\infty} \mathrm{d} L \exp (-\beta P L) Z(T, N, L),
$$
and by extremizing the integrand find the standard formula for $P$ in the canonical ensemble.
(d) Change variables from $L$ to $\delta_{N+1}=L-\sum_{i=1}^N \delta_i$, and find the expression for $Z(T, N, P)$ as a product over one-dimensional integrals over each $\delta_i$.
(e) At a fixed pressure $P$, find expressions for the mean length $L(T, N, P)$, and the density $n=N / L(T, N, P)$ (involving ratios of integrals that should be easy to interpret).
Since the expression for $n(T, P)$ in part (e) is continuous and non-singular for any choice of potential, there is in fact no condensation transition for the onedimensional gas. By contrast, the approximate van der Waals equation (or the meanfield treatment) incorrectly predicts such a transition.
(f) For a hard-sphere gas, with minimum separation $a$ between particles, calculate the equation of state $P(T, n)$. Compare the excluded volume factor with the approximate result obtained in earlier problems, and also obtain the general virial coefficient $B_{\ell}(T)$

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Problem 6

The Manning transition: when ionic polymer (polelectrolytes) such as DNA are immersed in water, the negatively charged counter-ions go into solution, leaving behind a positively charged polymer. Because of the electrostatic repulsion of the charges left behind, the polymer stretches out into a cylinder of radius $a$, as illustrated in the figure. While thermal fluctuations tend to make the ions wander about in the solvent, electrostatic attractions favor their return and condensation on the polymer. If the number of counter-ions is $N$, they interact with the $N$ positive charges left behind on the rod through the potential $U(r)=-2(\mathrm{Ne} / L) \ln (r / L)$, where $r$ is the radial coordinate in a cylindrical geometry. If we ignore the Coulomb repulsion between counter-ions, they can be described by the classical Hamiltonian
$$
\mathcal{H}=\sum_{i=1}^N\left[\frac{p_i^2}{2 m}+2 e^2 n \ln \left(\frac{r}{L}\right)\right],
$$
where $n=N / L$.
(a) For a cylindrical container of radius $R$, calculate the canonical partition function $Z$ in terms of temperature $T$, density $n$, and radii $R$ and $a$.
(b) Calculate the probability distribution function $p(r)$ for the radial position of a counter-ion, and its first moment $\langle r\rangle$, the average radial position of a counter-ion.
(c) The behavior of the results calculated above in the limit $R \gg a$ is very different at high and low temperatures. Identify the transition temperature, and characterize the nature of the two phases. In particular, how does $\langle r\rangle$ depend on $R$ and $a$ in each case?
(d) Calculate the pressure exerted by the counter-ions on the wall of the container, at $r=R$, in the limit $R \gg a$, at all temperatures.
(e) The character of the transition examined in part (d) is modified if the Coulomb interactions between counter-ions are taken into account. An approximate approach to the interacting problem is to allow a fraction $N_1$ of counter-ions to condense along the polymer rod, while the remaining $N_2=N-N_1$ fluctuate in the solvent. The free counter-ions are again treated as non-interacting particles, governed by the Hamiltonian
$$
\mathcal{H}=\sum_{i=1}^N\left[\frac{p_i^2}{2 m}+2 e^2 n_2 \ln \left(\frac{r}{L}\right)\right],
$$
where $n_2=N_2 / L$. Guess the equilibrium number of non-interacting ions, $N_2^*$, and justify your guess by discussing the response of the system to slight deviations from $N_2^*$. (This is a qualitative question for which no new calculations are needed.)

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02:47

Problem 7

Hard rods: a collection of $N$ asymmetric molecules in two dimensions may be modeled as a gas of rods, each of length $2 l$ and lying in a plane. A rod can move by translation of its center of mass and rotation about the latter, as long as it does not encounter another rod. Without treating the hard-core interaction exactly, we can incorporate it approximately by assuming that the rotational motion of each rod is restricted (by the other rods) to an angle $\theta$, which in turn introduces an excluded volume $\Omega(\theta)$ (associated with each rod). The value of $\theta$ is then calculated self-consistently by maximizing the entropy at a given density $n=N / V$, where $V$ is the total accessible area.
(a) Write down the entropy of such a collection of rods in terms of $N, n, \Omega$, and $A(\theta)$, the phase space volume associated with the rotational freedom of a single rod. (You may ignore the momentum contributions throughout, and consider the large $N$ limit.)
(b) Extremizing the entropy as a function of $\theta$, relate the density to $\Omega, A$, and their derivatives $\Omega^{\prime}, A^{\prime}$; express your result in the form $n=f\left(\Omega, A, \Omega^{\prime}, A^{\prime}\right)$.
(c) Express the excluded volume $\Omega$ in terms of $\theta$ and sketch $f$ as a function of $\theta \in[0, \pi]$, assuming $A \propto \theta$.
(d) Describe the equilibrium state at high densities. Can you identify a phase transition as the density is decreased? Draw the corresponding critical density $n_c$ on your sketch. What is the critical angle $\theta_c$ at the transition? You don't need to calculate $\theta_c$ explicitly, but give an (implicit) relation defining it. What value does $\theta$ adopt at $n<n_{\mathrm{c}}$ ?

Narayan Hari

Numerade Educator

Problem 8

Surfactant condensation: $N$ surfactant molecules are added to the surface of water over an area $A$. They are subject to a Hamiltonian
$$
\mathcal{H}=\sum_{i=1}^N \frac{\vec{p}_i^2}{2 m}+\frac{1}{2} \sum_{i, j} \mathcal{V}\left(\vec{r}_i-\vec{r}_j\right),
$$
where $\vec{r}_i$ and $\vec{p}_i$ are two-dimensional vectors indicating the position and momentum of particle $i$. (This simple form ignores the couplings to the fluid itself. The actual kinetic and potential energies are more complicated.)
(a) Write down the expression for the partition function $Z(N, T, A)$ in terms of integrals over $\vec{r}_i$ and $\vec{p}_i$, and perform the integrals over the momenta.

The interparticle potential $\mathcal{V}(\vec{r})$ is infinite for separations $|\vec{r}|<a$, and attractive for $|\vec{r}|>a$ such that $\int_a^{\infty} 2 \pi r \mathrm{~d} r \mathcal{V}(r)=-u_0$.
(b) Estimate the total non-excluded area available in the positional phase space of the system of $N$ particles.
(c) Estimate the total potential energy of the system, within a uniform density approximation $n=N / A$. Using this potential energy for all configurations allowed in the previous part, write down an approximation for $Z$.
(d) The surface tension of water without surfactants is $\sigma_0$, approximately independent of temperature. Calculate the surface tension $\sigma(n, T)$ in the presence of surfactants.
(e) Show that below a certain temperature, $T_c$, the expression for $\sigma$ is manifestly incorrect. What do you think happens at low temperatures?
(f) Compute the heat capacities, $C_A$, and write down an expression for $C_\sigma$ without explicit evaluation, due to the surfactants.

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08:54

Problem 9

Critical point behavior: the pressure $P$ of a gas is related to its density $n=N / V$, and temperature $T$ by the truncated expansion
$$
P=k_B T n-\frac{b}{2} n^2+\frac{c}{6} n^3,
$$
where $b$ and $c$ are assumed to be positive temperature-independent constants.
(a) Locate the critical temperature $T_c$ below which this equation must be invalid, and the corresponding density $n_c$ and pressure $P_c$ of the critical point. Hence find the ratio $k_B T_c n_c / P_c$.
(b) Calculate the isothermal compressibility $\kappa_T=-\left.\frac{1}{V} \frac{\partial V}{\partial P}\right|_T$, and sketch its behavior as a function of $T$ for $n=n_c$.
(c) On the critical isotherm give an expression for $\left(P-P_c\right)$ as a function of $\left(n-n_c\right)$.
(d) The instability in the isotherms for $T<T_c$ is avoided by phase separation into a liquid of density $n_{+}$and a gas of density $n_{-}$. For temperatures close to $T_c$, these densities behave as $n_{ \pm} \approx n_c(1 \pm \delta)$. Using a Maxwell construction, or otherwise, find an implicit equation for $\delta(T)$, and indicate its behavior for $\left(T_c-T\right) \rightarrow 0$. (Hint. Along an isotherm, variations of chemical potential obey $\mathrm{d} \mu=\mathrm{d} P / n$.)

Yaqub Khan

Numerade Educator

02:49

Problem 10

The binary alloy: a binary alloy (as in $\beta$ brass) consists of $N_A$ atoms of type $A$, and $N_B$ atoms of type $B$. The atoms form a simple cubic lattice, each interacting only with its six nearest neighbors. Assume an attractive energy of $-J(J>0)$ between like neighbors $A-A$ and $B-B$, but a repulsive energy of $+J$ for an $A-B$ pair.
(a) What is the minimum energy configuration, or the state of the system at zero temperature?
(b) Estimate the total interaction energy assuming that the atoms are randomly distributed among the $N$ sites; that is, each site is occupied independently with probabilities $p_A=N_A / N$ and $p_B=N_B / N$.
(c) Estimate the mixing entropy of the alloy with the same approximation. Assume $N_A, N_B \gg 1$
(d) Using the above, obtain a free energy function $F(x)$, where $x=\left(N_A-N_B\right) / N$. Expand $F(x)$ to the fourth order in $x$, and show that the requirement of convexity of $F$ breaks down below a critical temperature $T_c$. For the remainder of this problem use the expansion obtained in (d) in place of the full function $F(x)$.
(e) Sketch $F(x)$ for $T>T_c, T=T_c$, and $T<T_c$. For $T<T_c$ there is a range of compositions $x<\left|x_{s p}(T)\right|$, where $F(x)$ is not convex and hence the composition is locally unstable. Find $x_{s p}(T)$.
(f) The alloy globally minimizes its free energy by separating into $A$-rich and $B$-rich phases of compositions $\pm x_{e q}(T)$, where $x_{e q}(T)$ minimizes the function $F(x)$. Find $x_{e q}(T)$.
(g) In the $(T, x)$ plane sketch the phase separation boundary $\pm x_{e q}(T)$, and the socalled spinodal line $\pm x_{s p}(T)$. (The spinodal line indicates onset of metastability and hysteresis effects.)

Matthew Hurlock

Numerade Educator