From 58f00e57c0095ad913887acef137fc176d86df79 Mon Sep 17 00:00:00 2001 From: "Rose K. Cersonsky" <47536110+rosecers@users.noreply.github.com> Date: Fri, 15 Sep 2023 12:34:27 -0500 Subject: [PATCH 1/2] Fixing some things for the github pages --- .gitignore | 3 +- _update.sh | 7 + lecture_files/figs/fig_17_1-01.png | Bin 0 -> 109151 bytes lecture_files/figs/fig_17_2-01.png | Bin 0 -> 14301 bytes lecture_files/figs/fig_17_3-01.png | Bin 0 -> 6371 bytes lecture_files/figs/fig_17_4-01.png | Bin 0 -> 6380 bytes lecture_files/figs/fig_17_5-01.png | Bin 0 -> 5762 bytes lecture_files/figs/fig_17_6-01.png | Bin 0 -> 5743 bytes lecture_files/figs/fig_17_7-01.png | Bin 0 -> 6150 bytes lecture_files/figs/fig_17_8-01.png | Bin 0 -> 6152 bytes problems/ps_1/problem_set_1.md | 222 ++++++++++ problems/ps_1/pset_1_mixing_entropy_fig.png | Bin 0 -> 64052 bytes problems/ps_1/pset_1_plot_stirling.png | Bin 0 -> 318221 bytes problems/ps_1/pset_1_plot_stirling_error.png | Bin 0 -> 234119 bytes problems/ps_1/pset_1_random_walk_fig.png | Bin 0 -> 12135 bytes problems/ps_2/problem_set_2.md | 238 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lecture_files/figs/fig_17_8-01.png create mode 100644 problems/ps_1/problem_set_1.md create mode 100644 problems/ps_1/pset_1_mixing_entropy_fig.png create mode 100644 problems/ps_1/pset_1_plot_stirling.png create mode 100644 problems/ps_1/pset_1_plot_stirling_error.png create mode 100644 problems/ps_1/pset_1_random_walk_fig.png create mode 100644 problems/ps_2/problem_set_2.md create mode 100644 problems/ps_2/pset_2_Magnetization_H.PNG create mode 100644 problems/ps_2/pset_2_Magnetization_convergence.PNG create mode 100644 problems/ps_2/pset_2_pulling_polymer_fig.png create mode 100644 problems/ps_3/1D_Ising-01.png create mode 100644 problems/ps_3/problem_set_3.md create mode 100644 problems/ps_3/pset_3_Energy_J.PNG create mode 100644 problems/ps_3/pset_3_Energy_convergence.PNG create mode 100644 problems/simulation_project.md diff --git a/.gitignore b/.gitignore index a4e48a8..da15d92 100644 --- a/.gitignore +++ b/.gitignore @@ -1,4 +1,3 @@ -problems _build -*solutions +*solutions* 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Like ideal gases, the monomers in an ideal polymer chain do +not interact with each other; that is, the segments of the chain do not +exclude volume and can overlap. Consider a **one-dimensional** ideal +polymer chain composed of $N$ independent segments. + +One end of the chain is placed at the origin. A single chain conformation can then be +generated by iteratively placing a single segment of the chain a +distance $b$ in either the positive or negative $x$ dimension from the +current chain end - *i.e.*, the chain elongates by taking "steps" along +the one-dimensional coordinate. The end-to-end distance of the chain, +$x$, is the distance between the origin and the end of the last segment +placed. + +![image](pset_1_random_walk_fig.png) +Figure 1 shows an example for $N = 8$ in which the end-to-end distance is after all 8 steps are taken. + +```{admonition} **(a)** + +For a one-dimensional ideal chain with $N$ segments and segment +size $b$, calculate the probability, $p(x)$, that the end-to-end +distance of the polymer is equal to $x$. Assume that there is an **equal +likelihood** of taking a step in either the positive or negative +direction for each chain segment. + +
+ Hints + First compute the probability that we take n steps to the right. +
+ +``` + + +```{admonition} **(b)** + +For the chain described in part **a**, using Stirling's +approximation, show that in the large +$N$ limit the probability distribution $p(x)$ can be approximated by: + +$$p(x) = C \exp \left ( -\frac{x^2}{2Nb^2}\right )$$ + +where $C=\frac{1}{\sqrt{2\pi Nb^2}}$ is a normalization constant +to enforce that $\int_{-\infty}^{\infty} p(x) dx = 1$. Make sure +to show how you obtain $C$. + +
+ Hints + Define the quantity a = x/Nb, where a << 1 in the large N limit, + and write a Taylor series expansion for ln(1-a) when appropriate. + +
+ +``` + + +```{admonition} **(c)** + +Show that the entropy of an ideal one-dimensional chain in the +large $N$ limit is given by: + +$S(N, x) = -\frac{k_Bx^2}{2Nb^2} + S(N) + k_B \ln{C}$. + +where $C$ is our normalizing constant from above, and S(N) is the x-independent entropy term. + +``` + +## Question 2: Magnetization of a paramagnet + +Consider a system of $N$ distinguishable, non-interacting atoms with +magnetic dipole moments (spins) in a magnetic field $H$ at constant +temperature. Each spin $s_i$ has a magnetic moment $\mu$ and can be in +one of two states: parallel to the field ($s_i = 1$) or anti-parallel to +the field ($s_i = -1$). The energy of each microstate is due only to +interactions between the spins and the magnetic field, and is given by: + +$$E = -\sum_{i=1}^N s_i \mu H$$ + +The magnetization of the material is defined as: + +$$\begin{aligned} +M = \sum_i^N s_i \mu = -\frac{E}{H} +\end{aligned}$$ + +This model is commonly used to describe paramagnetic materials. In this +problem, we will derive an expression for the ensemble-average +magnetization of a paramagnet in a magnetic field. + + +```{admonition} **(a)** + +Assuming that the paramagnet has a fixed energy $E$, write an +expression for the entropy as a function of the number of spins aligned +with the field, $N_+$, and the total number of spins, $N$. + + +
+ Hints + Use the [Boltzmann formulation](https://en.m.wikipedia.org/wiki/Boltzmann's_entropy_formula) for entropy. + +
+``` + +```{admonition} **(b)** + +Show that the expression for $N_+$ as a function of the number of +spins $N$, the magnetic field strength $H$, the magnetic moment $\mu$, +and the temperature $T$ (or as a function of $\beta \equiv 1/k_B T$) is + + +$$ N_+ = \frac{N}{1+exp(-2H\beta\mu)}$$ + + +
+ Hints + Procedurally, this is almost identical to our previous lecture. + +
+``` + + + +```{admonition} **(c)** +Show that the magnetization of a paramagnet is given by: + +$$\begin{aligned} +M = N\mu \tanh (\beta \mu H) +\end{aligned}$$ + +``` + +## Question 3: Mixing entropy + +Consider two different ideal fluids containing $N_1$ and $N_2$ +molecules, respectively, for a total of $N$ molecules. All molecules +exclude the same molar volume and are assumed to interact weakly with +each other and with themselves, such that the potential energy of the +system is negligible. The two fluids are initially completely demixed +due to an impermeable partition; the partition is then removed and the +fluids are allowed to mix at constant volume as illustrated in Figure 2. +Assume also that the molecules of each fluid are **indistinguishable** +from the molecules of the same fluid. + +![image](pset_1_mixing_entropy_fig.png) + +```{admonition} **(a)** + +Assume that the two fluids occupy a fictitious lattice that +spans the available volume. Each molecule occupies a single lattice site +and all lattice sites are occupied; thus, there are are $N_1$ lattice +sites occupied by molecule 1 and $N_2$ lattice sites occupied by +molecule 2. Using this approximation, calculate $\Omega(N_1, N_2)$, +which is defined as the number of microstates for the mixture of fluids. + + +
+ Hints + Identical arrangements of the fluids are indistinguishable. + +
+ +``` + +```{admonition} **(b)** + +Derive an expression for the entropy change associated with +mixing the two ideal fluids in terms of the mole fractions, +$x_1 = N_1/N$ and $x_2 = N_2/N$, of the two components. That is, +determine an expression for: + +$$\Delta S_{\textrm{mix}} = S_{\textrm{mixed}} - S_{\textrm{demixed}}$$ + +
+ Hints + How many distinguishable configurations are there for totally demixed systems? + +
+ +``` + +```{admonition} **(c)** + +Is the lattice model assumption reasonable for molecules +occupying a continuous set of positions (rather than discrete points)? +Why or why not? + +``` + +## Question 4: Stirling's approximation + + +```{admonition} Python Exercise + +Write a Python program that calculates the percent error of Stirling's +approximation as a function of $N$, where Stirling's approximation is +defined as: + +$$\ln N! \approx N \ln N - N$$ + +Include with your solution a copy of your Python code (including +comments as necessary), a plot of $\ln N!$ and Stirling's approximation +as a function of $N$ up to $N=100$, and a plot of the error of +Stirling's approximation as a function of $N$ up to $N=100$. Your grade +for this problem will be based in part on the readability of your code +and plots in addition to the accuracy of the solution. +``` + +**Note: this problem is intended to provide you with practice in Python +programming prior to the assignment of the simulation project. If you +need resources for learning to code in Python, see [here](https://sts.doit.wisc.edu/).** + +The two requested plots are provided below. + +![Comparison of Stirling's approximation vs. $\ln N!$ as a function of +$N$.](pset_1_plot_stirling.png) + +![Relative error of Stirling's approximation vs. $N$, which approaches +zero as $N$ exceeds +$\approx 50$.](pset_1_plot_stirling_error.png) diff --git a/problems/ps_1/pset_1_mixing_entropy_fig.png b/problems/ps_1/pset_1_mixing_entropy_fig.png new file mode 100644 index 0000000000000000000000000000000000000000..b80424b80cd041b1d23c6e8a5bc772dfcc4baed9 GIT binary patch literal 64052 zcmd3NWl&sA_vZjf2m}wVfdmhb1h;_%hv4q+?mD<@65K7g1q*J$-8HzoOMn2|JkR^z z+S=XP{jgtlswi&F%)Pho>F#rWawbGZN(2>|5E%l2po)nK%0VDcenKGdGtc3{JBM`p zW8fc>ji{y^n<7ZqY=A+dOU{iQQ>rMNfBKbf90=+Gl;J66WY$jpEEg?&b}Ol89Qnz`LDVTnkZ zThgA_IDue@^UMvpxq=XcZhxI(4^1DwwTI$`NadY@fj+*DkB3!68!goKlu5- z2hTC#{yiY4`1k)S!6*M7_y_;{-x~w--vbPa$FP9c|8Nl)#R>&4FE2ND!_S{nl6J45 zP%Yvm^NwIlVQo}nC2HHe#iI 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+$$S(N,V,E) = k_B \ln \Omega(N,V,E)$$ + +where $\Omega(N,V,E)$ is the degeneracy (or number of unique +microstates) in the microcanonical ensemble. An alternative, general +definition of the entropy, referred to as the Gibbs entropy, is: + +$$S = - k_B\sum_i p_i \ln p_i$$ + +where $p_i$ is the probability of microstate $i$ and the sum runs over +all microstates. + +```{admonition} **(a)** +Show that the Boltzmann and Gibbs entropy are equivalent for the microcanonical ensemble. +

+ Hints +Recall that according to the second postulate of statistical mechanics, +all microstates in the microcanonical ensemble are equally probable. +
+``` + +```{admonition} **(b)** +Starting from an appropriate partial derivative of the Helmholtz +free energy, show that entropy can be expressed in terms of the +{term}`canonical partition function` and the temperature as +$S = k_B \left[T\left ( \frac{\partial \ln Z}{\partial T} \right )_{N,V} + \ln Z\right] $. +
+ Hints + +$dF = -SdT - PdV + \mu dN$ + +
+ +``` + +```{admonition} **(c)** +Show that the expression derived in part **b** reduces to the +Boltzmann entropy if the canonical ensemble is dominated by a single +energy level (i.e., the contributions of all other energy levels to the +partition function are negligible). +``` + +```{admonition} **(d)** +Calculate the Gibbs entropy of the canonical ensemble, and show +that this expression for the entropy leads to the correct thermodynamic +expression for the Helmholtz free energy ($F=E-TS$). +``` + +## Question 2: Pulling on a protein + +A simplified model for the protein keratin is a linear, long chain +molecule consisting of $N$ units. Each unit can be in either an +unstretched state with energy $\epsilon_\alpha$ and length $a$, or a +stretched state with energy $\epsilon_\beta$ and length $b$. The chain +is stretched at constant tension, $\tau$, which causes it to reach a +total length $L$; $\tau$ is a force that is conjugate to $L$. Assume +that stretching the chain (i.e. changing $L$) does not change the total +chain volume, the chain is held at constant temperature, and each unit +is independent. + +![image](pset_2_pulling_polymer_fig.png) + +
+ Hint + +Hint: for a system of $N$ distinguishable, independent subsystems for +which the energy of the combined system, $E$, is equal to the sum of the +energies of the subsystems, $E = \sum_i \epsilon_i$, the partition +function of the combined system, $Z$, is related to the partition +function of each subsystem, $z$, by: + +$$Z = z^N$$ + +This relationship is true of any generalized partition function $\Phi$, +as well as the canonical partition function, $Z$. + +
+ +```{admonition} **(a)** +Using the first law of thermodynamics, show that the thermodynamic +potential $B$ corresponding to this ensemble is expressed +as $B = - PdV - SdT - L d\tau$. +``` + +```{admonition} **(b)** +Derive an expression for a generalized partition function with +fixed thermodynamic parameters that correspond to the natural variables +of the potential derived in part **a**. Your result should be in terms +of $\epsilon_a$, $\epsilon_b$, $a$, $b$, $\tau$, and $N$. + +
+ Hint + +Since each unit of the molecule is independent and the different units +are distinguishable, we can use the first hint within this problem +to factorize the partition function and write: + +$$\begin{aligned} + \Phi &= \phi^N + +\end{aligned}$$ + +Here, $\phi$ is the single-unit partition function, which we see is a +two-state model with one state characterized by energy $\epsilon_a$ and +length $a$ and the other state characterized by energy $\epsilon_b$ and +length $b$. + +
+``` + +```{admonition} **(c)** + +Calculate the average length, $\langle L \rangle$, of the +molecule as a function of the tension, $\tau$. +``` + + +## Question 3: Magnetization of a paramagnet revisited + +In [Problem Set 1](../ps_1/problem_set_1.md), we used the microcanonical ensemble to show that the +magnetization of a paramagnet is: + +$$\begin{aligned} +M = N\mu \tanh (\beta \mu H) +\end{aligned}$$ + +The system is $N$ distinguishable, non-interacting spins that are fixed +on a lattice and exposed to a magnetic field $H$ at constant +temperature. Each spin $s_i$ has a magnetic moment of $\mu$, and can be +in one of two states: parallel to the field ($s_i = 1$) or anti-parallel +to the field ($s_i = -1$). The energy of a particular microstate is due +only to interactions between the spins and the magnetic field: + +$$E = -\sum_{i=1}^N s_i \mu H$$ + +The magnetization is defined as: + +$$\begin{aligned} +M = \sum_i^N s_i \mu +\end{aligned}$$ + +```{admonition} Derive the magnetization of the system using the canonical ensemble. +
+ Hint + +$\cosh(x) = \frac{e^x + e^{-x}}{2}$ + +$\sinh(x) = \frac{e^x - e^{-x}}{2}$ + +
+``` + +## Question 4: Numerical calculation of the magnetization of a paramagnet + +As an alternative to the analytical derivation of the magnetization of a +paramagnet, the Metropolis Monte Carlo algorithm can be used to +calculate the ensemble-average magnetization. We will derive this +algorithm in a [later lecture](../../lecture_files/Lecture10.md); +in this question, you will apply the algorithm +to compare to the analytical result in Question 3. + +Consider a system of $N=100$ distinguishable, non-interacting spins in a +magnetic field $H$ at constant temperature. Each spin is fixed to a +lattice position such that the relative positions of spins cannot +interchange. The system can assume a set of configurations in which each +spin is either parallel to the field or antiparallel to the field, with +the energy of each configuration defined in Question 3. The Metropolis- +Monte Carlo algorithm for calculating the ensemble-average magnetization +consists of the following steps: + +1. Initialize the system in a random configuration. + +2. Randomly generate a new configuration, in which at least 1 spin has + switched orientation. + +3. Calculate the energy of the new configuration, $E_{\textrm{new}}$, + and old configuration, $E_{\textrm{old}}$. + +4. Calculate the probability of accepting the trial configuration, + $\alpha$, using the following equation: + $$\alpha = \text{min} \left ( 1, \exp \left \{ -\beta \left [ E_{\textrm{new}} - E_{\textrm{old}} \right ] \right \} \right )$$, + where $\beta = (k_B T)^{-1}$. + +5. Generate a uniform random number in the range \[0, 1\]. If the + random number is less than $\alpha$, then retain the new + configuration as the system configuration; the new configuration is + *accepted*. If the random number is greater than $\alpha$, then + retain the old configuration as the system configuration; the new + configuration is *rejected*. Note that all new configurations for + which the system energy is lowered are automatically accepted. + +6. Performing steps 2-5 defines a single Monte Carlo trial. Repeat + steps 2-5 for a large number of trials (discussed below). + +7. Calculate $\langle M \rangle$ by averaging values of $M$ over all + configurations sampled during the trials. Note that configurations + will be repeated when new configurations are rejected in step 5. + +Think carefully about how to efficiently implement this algorithm. +**This problem is intended to provide you with practice in Python +programming prior to the assignment of the simulation project.** Your +grade for this problem will be based in part on the readability of your +code and plots in addition to the accuracy of the solution. + +```{admonition} **(a)** +Write a Python program that computes $\langle M \rangle$ for the +paramagnetic spin lattice described in Question 3 using the Metropolis- +Monte Carlo algorithm described above. Attach a copy of your code with +appropriate comments explaining your implementation. +``` + +```{admonition} **(b)** +Perform independent Monte Carlo simulations for field strengths +of $\mu H =$ -3.0 $k_BT$ to $\mu H =$ 3.0 $k_B T$ in increments of 0.5 +$k_B T$. Each simulation should include 100,000 Monte Carlo trials to +obtain accurate approximations of $\langle M \rangle$. Plot +$\langle M \rangle$ vs. $\mu H$ and compare to the analytical result +from Question 3. +``` + +The Python code for part **a** and **b** is included on the Canvas +website. The plots below show the number of MC iterations necessary to +obtain converged values of the ensemble-average magnetization and the +comparison between the ensemble-average mangetization and the analytical +result from Question 3. + +![Convergence of ensemble-average +magnetization.](pset_2_Magnetization_convergence.png){width="70%"} + +![Comparison of the ensemble-average magnetization obtained from the +Monte Carlo method to the analytical solution. 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Hints +Consider what entities can be distinguished from one another, and therefore +separated in our partition function formulation. + +``` + +**(b)** Using the result from part **a** and assuming that both $N_A$ +and $N_B$ are large, show that the Boltzmann distribution law follows +from the condition that the chemical potential of a molecule in state +$A$ is equal to the chemical potential of a molecule in state $B$ +(*i.e.*, $\mu_A = \mu_B$), for a system at constant volume and constant +temperature. + +In part a, we derived an expression for the canonical partition function +of the entire combined system. The relationship between the canonical +partition function and the Helmholtz free energy is: + +$$F = -k_B T \ln Z$$ + +We can write the differential form of the Helmholtz free energy of a +two-component mixture as: + +$$dF = -SdT - PdV + \mu_A dN_A + \mu_B dN_B$$ + +Therefore, we can write an expression for the chemical potential of each +species as: + +$$\begin{aligned} + \mu_A &= \left ( \frac{\partial F}{\partial N_A} \right )_{N_B, V, T} \\ + &= -k_B T \left ( \frac{\partial \ln Z}{\partial N_A} \right )_{N_B, V, T} \\ + \mu_B &= -k_B T \left ( \frac{\partial \ln Z}{\partial N_B} \right )_{N_A, V, T} + +\end{aligned}$$ + +We can write $\ln Z$ for the partition function from part **a** as: + +$$\begin{aligned} + \ln Z &= \ln \frac{\left (\Omega_A z_{\textrm{trans}}e^{-\epsilon_A/kT} \right )^{N_A}}{N_A!} \frac{\left (\Omega_B z_{\textrm{trans}}e^{-\epsilon_B/kT}\right )^{N_B}}{N_B!} \\ + &= N_A \ln \left ( \Omega_A z_{\textrm{trans}}e^{-\epsilon_A/kT}\right ) - N_A \ln N_A + N_A + N_B \ln \left ( \Omega_B z_{\textrm{trans}}e^{-\epsilon_B/kT}\right ) - N_B \ln N_B + N_B + +\end{aligned}$$ + +Using this expression, we can write the chemical potential for each +species as: + +$$\begin{aligned} + \mu_A &= -k_B T \left ( \frac{\partial\ln Z}{\partial N_A}\right )_{N_B, V, T} \\ + &= -k_B T \ln \left ( \Omega_A z_{\textrm{trans}}e^{-\epsilon_A/kT}\right ) - \ln N_A \\ + &= -k_B T \ln \frac{\Omega_A z_{\textrm{trans}}e^{-\epsilon_A/kT}}{N_A} \\ + \mu_B &=-k_B T \ln \frac{\Omega_B z_{\textrm{trans}}e^{-\epsilon_B/kT}}{N_B} + +\end{aligned}$$ + +Setting $\mu_A = \mu_B$ yields: + +$$\begin{aligned} + \ln \frac{\Omega_A z_{\textrm{trans}}e^{-\epsilon_A/kT}}{N_A} &= \ln \frac{\Omega_B z_{\textrm{trans}}e^{-\epsilon_B/kT}}{N_B} \\ + \frac{N_A}{N_B} &= \frac{\Omega_A z_{\textrm{trans}}e^{-\epsilon_A/kT}}{\Omega_B z_{\textrm{trans}}e^{-\epsilon_B/kT}} \\ + \frac{N_A}{N_B} &= \frac{\Omega_A}{\Omega_B} e^{\beta \Delta \epsilon} + +\end{aligned}$$ + +The last simplification is because the translational partition function +for each state is identical as specified in the problem statement. We +thus recover the Boltzmann distribution law. + +## Question 2: Adsorption of a gas mixture + +Consider a two-component mixture of two ideal gas species, denoted as +$A$ and $B$, that can both adsorb to a surface. Each gas molecule of +type $A$ adsorbs with an energy $\epsilon_i = -\epsilon_A$ while each +gas molecule of type $B$ adsorbs with energy $\epsilon_i = -\epsilon_B$. +Gas molecules of either type that are not adsorbed can be treated as an +ideal gas, and gas molecules adsorbed on the surface can exchange with +particles in the bulk. Gas particles exclude each other on the surface - +*e.g.*, if an $A$ particle adsorbs to a part of the surface, a $B$ +particle cannot adsorb to the same part of the surface. The partial +pressures of each gas can be written as $P_A = P x_A$ and $P_B = P x_B$, +where $P$ is the pressure, $x_A$ is the mole fraction of gas $A$ in the +mixture, $x_B$ is the mole fraction of gas $B$ in the mixture, and +$x_A + x_B = 1$. Both the surface and bulk are at constant temperature +and constant volume. + +**(a)** Derive a grand canonical partition function for the surface. +Define any variables that you introduce. + +This question is an extension of a Langmuir isotherm by allowing for the +adsorption of multiple components on a single lattice site. As in the +Langmuir isotherm derivation, let us assume there are $N_s$ +distinguishable sites on the surface. Each site for a given +configuration can be either unoccupied (0 energy), occupied by species +$A$ ($-\epsilon_A$ energy), or occupied by species $B$ ($-\epsilon_B$ +energy). The partition function for a site in the grand canonical +ensemble is then: + +$$\begin{aligned} + \xi &= \sum_j e^{-E_j/k_BT}e^{\mu_j N_j / k_B T} \\ + &= 1 + e^{\beta(\epsilon_A + \mu_A^s)} + e^{\beta(\epsilon_B + \mu_B^s)} + +\end{aligned}$$ + +Here, $\mu_A^s$ is the chemical potential of molecule type $A$ on the +surface, $\mu_B^s$ is the chemical potential of molecule type $B$ on the +surface. The partition function for the entire system of $N_s$ sites is +then: + +$$\Xi = \xi^{N_s} = \left ( 1 + e^{\beta(\epsilon_A + \mu_A^s)} + e^{\beta(\epsilon_B + \mu_B^s)} \right )^{N_s}$$ + +**(b)** Derive an expression for the fraction of the surface that is +occupied by molecules of type $A$, $f_A$, and the fraction of the +surface that is occupied by molecules of type $B$, $f_B$. Write the +expressions for $f_A$ and $f_B$ in terms of the partial pressures of +each component and any other constants that you choose to define. + +We can now express the fraction of the surface associated with each +species. We define $N_A$ as the number of molecules of type $A$ adsorbed +to the surface and $N_B$ as the number of molecules of type $B$ adsorbed +to the surface. We can also write the thermodynamic potential +corresponding to the grand canonical partition function as: + +$$\Sigma = -k_B T \ln \Xi$$ + +Then: + +$$\begin{aligned} + f_A &= \frac{\langle N_A \rangle}{N_s} = -\frac{1}{N_s}\left ( \frac{\partial \Sigma}{\partial \mu_A^s} \right )_{\mu_B^s,V,T} \\ + &= \frac{ e^{\beta(\epsilon_A + \mu_A^s)}}{ 1 + e^{\beta(\epsilon_A + \mu_A^s)} + e^{\beta(\epsilon_B + \mu_B^s)}} \\ + f_B &= \frac{\langle N_B \rangle}{N_s} = -\frac{1}{N_s}\left ( \frac{\partial \Sigma}{\partial \mu_B^s} \right )_{\mu_A^s,V,T} \\ + &= \frac{ e^{\beta(\epsilon_B + \mu_B^s)}}{ 1 + e^{\beta(\epsilon_A + \mu_A^s)} + e^{\beta(\epsilon_B + \mu_B^s)}} + +\end{aligned}$$ + +In the multicomponent system, chemical equilibrium is satisfied by +$\mu_A^s = \mu_A^g$ and $\mu_B^s = \mu_B^g$ where +$\mu_A^g = \beta^{-1} \ln (\beta P_A\lambda_A^3)$ is the chemical +potential in the gas phase, $P_A$ is the partial pressure of molecule +$A$, and $\lambda_A$ is the thermal de Broglie wavelength of molecule +$A$. Substituting in these expressions, we obtain the following: + +$$\begin{aligned} + f_A &= \frac{ \beta P_A\lambda_A^3 e^{\beta\epsilon_A }}{ 1 + \beta P_A\lambda_A^3 e^{\beta\epsilon_A } + \beta P_B\lambda_B^3 e^{\beta\epsilon_B }} \\ + f_B &= \frac{ \beta P_B\lambda_B^3 e^{\beta\epsilon_B }}{ 1 + \beta P_A\lambda_A^3 e^{\beta\epsilon_A } + \beta P_B\lambda_B^3 e^{\beta\epsilon_B }} + +\end{aligned}$$ + +If we define +$P_A^0 =\left ( \frac{1}{\beta\lambda_A^3}\right )e^{-\beta \epsilon_A}$ +and +$P_B^0 =\left ( \frac{1}{\beta\lambda_B^3}\right )e^{-\beta \epsilon_B}$ +we obtain an expression similar to the Langmuir isotherm: + +$$\begin{aligned} + f_A &= \frac{P_A / P_A^0}{1 + P_A / P_A^0 + P_B / P_B^0} \\ + f_B &= \frac{P_B / P_B^0}{1 + P_A / P_A^0 + P_B / P_B^0} + +\end{aligned}$$ + +This expression shows that the fraction surface coverage of a component +increases if either its partial pressure increases (via $x_A$) or if the +binding energy $\epsilon_A$ becomes larger, leading to an decrease in +$P_A^0$. + +## Question 3: 1D Ising model + +In class, we discussed the **Ising model** for a magnetic system in +which a set of $N$ distinguishable magnetic dipoles are each +characterized by a spin, $s_i$, that has one of two discrete values: +$s_i = \pm 1$. The total magnetization, $M$, can be approximated as the +sum of all spins, $M = \sum_{i=1}^{N} \mu s_i$. Spins are placed on a +fixed lattice and can interact with an external magnetic field, $H$, and +other nearest-neighbor spins. The energy with which spin $i$ interacts +with its neighbors is: + +$$\epsilon_i = - J \sum_j^\prime s_i s_j\$$ + +Here, $J$ is the value of the coupling parameter (with units of energy) +that describes the interaction energy between each *pair* of +nearest-neighbor spins. The sum $\sum_j^\prime$ indicates a sum over all +$n$ nearest-neighbor spins of $i$ on the lattice. The *total* energy of +a single microstate is: + +$$E = -\sum_i^N \mu H s_i - \frac{J}{2} \sum_{i}^N \sum_j^\prime s_i s_j\$$ + +The factor of $1/2$ in front of the interaction energy term accounts for +the double-counting of interactions between pairs of spins when summing +over the entire system. + +In this problem, we will analytically solve the **1D Ising model**, +which refers to a set of spins that are placed on a one-dimensional +lattice. Each spin therefore only has two nearest-neighbors. We will +assume **free boundary conditions**, which means that the spins at the +beginning and end of the chain ($s_1$ and $s_N$) only have a single +neighbor. An example of a single microstate of a 1D Ising model is +schematically drawn below. + +![image](1D_Ising-01.png){width="30%"} + +**(a)** Show that in the zero field limit ($H=0$), the partition +function for the 1D Ising model is: + +$$\begin{aligned} +Z &=2 \left[2 \cosh(\beta J)\right]^{N-1} +\end{aligned}$$ + +Hint: Consider each *pair* of spins when simplifying the partition +function. + +The energy of a configuration in the absence of a field ($H=0$) is: + +$$\begin{aligned} +E &= -\frac{J}{2} \sum_i^N\sum_j^\prime s_i s_j +\end{aligned}$$ + +The full partition function, accounting for all possible states, is +then: + +$$\begin{aligned} +Z &= \sum_{s_1=\pm 1} \sum_{s_2=\pm 1} \dots \sum_{s_N =\pm 1}\exp \left ( \frac{\beta J}{2} \sum_i^N\sum_j^\prime s_i s_j \right ) +\end{aligned}$$ + +We can simplify the expression for the energy by recognizing that spin +$i$ interacts only with spin $i-1$ and spin $i+1$. We can exactly count +all of these interactions *without* overcounting by starting at spin $1$ +and counting interactions between spin $i$ and spin $i+1$ only, while +summing over all spins. We exclude the last spin because there is no +$i+1$ interaction. This gives: + +$$\begin{aligned} +E &= -J \sum_i^{N-1} s_i s_{i+1} \\ +&= -J \sum_i^{N-1}\sigma_i^\prime +\end{aligned}$$ + +Here, $i$ now indicates the index of the pair of spins and +$\sigma_i^\prime = s_i s_{i+1}$. We have eliminated the factor of $1/2$ +because there is no longer any overcounting when we sum over the $N-1$ +interactions as opposed to summing over the $N$ spins. The full +partition function is then: + +$$\begin{aligned} +Z &= \sum_{s_1=\pm 1} \sum_{s_2=\pm 1} \dots \sum_{s_N =\pm 1}\exp \left ( \beta J \sum_i^{N-1}\sigma_i^\prime \right ) +\end{aligned}$$ + +The outer sums of this partition function iterate over all possible +arrangements of spins in the system to enumerate all possible +microstates. However, for a 1D system, we can write each arrangement by +summing over each *pair* of spins using the index as noted above. In +general, each pair of spins can only be in one of two states: the spins +can either be aligned or unaligned. The alignment of one pair of spins +does not affect the next pair of spins - while individual spins are +correlated, pairs of spins are uncorrelated or independent. We can thus +factorize the partition function into $N-1$ independent +"single-pair-of-spins" partition functions in analogy with the +factorization of partition functions performed for the ideal gas: + +$$\begin{aligned} +z &= \exp \left ( \beta J \sum_{ \sigma_i=\pm 1} \sigma_i^\prime \right )\\ +&= \exp \left (\beta J \right ) + \exp \left (-\beta J \right ) +\end{aligned}$$ + +There are two terms in the single-pair-of-spins partition function +accounting for the two spins either being aligned or unaligned. Pairs of +spins are distinguishable because they are positioned on a lattice, +allowing us to write the full partition function in terms of the +single-pair-of-spins partition function without dividing by $(N-1)!$. +Finally, we recognize that we have to add a prefactor of 2 to the full +partition function because we can generate a unique microstate by +inverting the sign of all spins, doubling the number of possible states. +The full partition function is then: + +$$\begin{aligned} +Z &= 2z^{N-1} \\ +&= 2\left[ \exp \left (\beta J \right ) + \exp \left (-\beta J \right ) \right]^{N-1} \\ +&= 2\left[2 \cosh(\beta J)\right]^{N-1} +\end{aligned}$$ + +**(b)** Calculate the ensemble-average energy per spin in the limit that +$N \rightarrow \infty$. + +We can obtain the average energy from an appropriate derivative of the +partition function. We can relate the partition function to the +ensemble-average energy using the expression from Lecture 4: + +$$\begin{aligned} +\langle E \rangle &= -\left ( \frac{\partial \ln Z }{\partial \beta}\right )_{N,V} +\end{aligned}$$ + +We can first simplify $\ln Z$: + +$$\begin{aligned} +Z &= 2\left[2 \cosh(\beta J)\right]^{N-1} \\ +\ln Z &= \ln \left ( 2\left[2 \cosh(\beta J)\right]^{N-1} \right ) \\ +&= \ln 2 + (N-1) \ln 2 + (N-1) \ln \cosh(\beta J) +\end{aligned}$$ + +The first two terms have no $\beta$ dependence, so when we evaluate the +derivative only the third term survives. The average energy is then: + +$$\begin{aligned} +\langle E \rangle &= -(N-1) J \frac{\sinh(\beta J)}{\cosh(\beta J)} \\ +&= -(N-1) J \tanh(\beta J) +\end{aligned}$$ + +We assume that as $N\rightarrow \infty$, $N-1 \approx N$. The +ensemble-average energy per spin is then: + +$$\begin{aligned} +\frac{\langle E \rangle}{N} &= -\frac{N-1}{N} J \tanh(\beta J) \\ +&\approx -J \tanh(\beta J) +\end{aligned}$$ + +## Question 4: Monte Carlo simulation of the Ising model + +In Problem Set 2, we introduced the Metropolis Monte Carlo algorithm as +a means of computing ensembe-average quantities. In Problem Set 2, we +studied the magnetization of non-interacting spins. We will now use the +same method to study the 1D Ising model which we analytically studied in +##Question 3.-- + +Consider a 1D Ising model in which $N = 100$ spins are placed on a 1D +lattice, such that each lattice point has $n=2$ nearest-neighbors. +Assume that the system has free boundaries so that spin $1$ and spin +$100$ only have 1 nearest-neighbor (this directly mimics the problem +from ##Question 3).-- + +**(a)** Write a Python program that computes $\langle E \rangle / N$ for +the 1D Ising model using the Metropolis Monte Carlo algorithm described +above. Attach a copy of your code with appropriate comments explaining +your implementation. + +**(b)** Perform independent Monte Carlo simulations for $J =$ -3, -2, +-1, 0, 1, 2, and 3 $k_B T$. Each simulation should include 100,000 +trials to obtain accurate approximations of $\langle E \rangle / N$. +Plot $\langle E \rangle / N$ vs. $J$ and compare to the analytical +result from ##Question 3.-- + +The Python code for part **a** and **b** is included on the Canvas +website. The plots below show the number of MC iterations necessary to +obtain converged values of the ensemble-average energy and the +comparison between the ensemble-average energy and the analytical result +from ##Question 3.-- + +![Convergence of ensemble-average +energy.](pset_3_Energy_convergence.png){width="70%"} + +![Comparison of the ensemble-average energy obtained from the Monte +Carlo method to the analytical solution. Points indicate the model +calculations and the line indicates the analytical +solution.](pset_3_Energy_J.png){width="70%"} diff --git a/problems/ps_3/pset_3_Energy_J.PNG b/problems/ps_3/pset_3_Energy_J.PNG new file mode 100644 index 0000000000000000000000000000000000000000..24e69d85c090651e14173bc139a1c253f2d045d8 GIT binary patch literal 300828 zcmeFZcRZJU-#@N8+e1<+8AY~82w7z&qGZqP6|y%i8HI{Uh^%aqnVFTAtc(bu?7g?| z^EkWC^SbWy{{0@`zrT;i^|>GSbxQg;j`#6;zh2Mv{`g&zm)^aDat8?s$!?kR5{e`w zgwf!|S{iBXXwsvOLR!2{9;Q!cwl(3r z8})T33CU3s8HsaB&JiOWZV#Rqb!?sJ8hf3;(^ipvZ?KW@UOJ!CydUNErICHerXr2q z^2Po|{bzak!;I#(2f0 zdGow)C?b7KR7IkmEtAru4;PfSWYreCjc1KKjbZ=v8h7!}*NqNl4*P$8llZgimd%d; zAKyQI`RdmH{q3{+{`aC9L@dd)>KeXAXdGZc-%LR?-=(|U|Z1<+7rVa@?U*C23)bYD}s40D@ z9o6nr^0G<=(NNP%2j1^2c01^5@#lNKYB_V}jDAfRZ>H<)kXo^;!`H80D_>p=@2TPI zx_|fX-PT-7&1+pH!foyCv2k(7FUZRF-q^IqcWrgK`t8k426ZpaaoG>l9XfqYZP$_0 z?1B6}zWhrAu?CgCWSz3Vslks;GB>GzU%&g?k?(J~lT%U*CVQ$TN3v?)P1lOjo;Y#h 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20 points available, click for grading breakdown + +- Magnetization plot: 5 pts +- $C_{v}$ plot: 5 pts +- Correct plot shape 4 pts +- Correct smoothness (sampled a few times or reached good + convergence): 2 pts +- Explain how each curve was calculated: 4 pts + +
+``` + +```{admonition} **(b)** + +A [first-order phase transition](https://en.wikipedia.org/wiki/Phase_transition#Ehrenfest_classification) +is characterized by a discontinuity in a first derivative of the free energy with respect to +the temperature when the system undergoes a phase transition, while a +[second-order phase transition](https://en.wikipedia.org/wiki/Phase_transition#Ehrenfest_classification) is characterized by the divergence of +a second derivative of the free energy with respect to temperature when +the system undergoes a phase transition. Based on your results in part +**a**, is the ferromagnetic order-disorder transition classified as a +first or second order phase transition? Explain your rationale. + +
10 points available, click for grading breakdown + +- Correct order identified: 5 pts + +- Rationale: 5 pts +
+``` + +```{admonition} **(c)** +Determine an approximate relationship between $T_C$ and $J$ +(assuming $J>0$) for the 2D Ising model using your simulations. You may +assume that $T_C$ and $J$ are linearly related. Compare your expression +to the mean-field result and the exact Onsager solution for the 2D Ising +model. + +
15 points available, click for grading breakdown + +- Comparison: 5 pts + +- Show that Ising model is better: 5 pts + +- Explicitly write Onsager's solution: 3 pts + +- Multiple J values tested: 2 pts +
+``` + +```{admonition} **(d)** +Explain the competition between driving forces in the Ising +model and why the values of the magnetization identified at temperatures +above and below the Curie temperature are obtained. Assume that $J>0$ in +your explanation. + +5 points available, no grading breakdown +``` + +## Question 2: Molecular dynamics simulations of a Lennard-Jones fluid + +Consider a set of particles that interact via a {term}`Lennard-Jones potential`. +This system, referred to as a *Lennard-Jones fluid*, captures the +behavior of simple solvents. Develop a simulation model for a +Lennard-Jones fluid in which $N = 125$ particles are placed in a 3D, +periodic, cubic simulation box. Assume particles only interact via a +Lennard-Jones potential with characteristic energy $\epsilon$ and length +$\sigma$ which set the energy and length scales of the simulations, +respectively. + +
Recommendations + +- Use a dimensionless timestep of $\Delta t = 0.001$. + +- Assume the mass of each particle is $m=1.0$. + +- Set the Boltzmann constant to $k_B = 1.0$. + +- Sample and save system properties every 100 timesteps. + +- For your final simulations, run for at least 100,000 timesteps (this + should take $< 15$ mins). + +- Reduce the number of particles and +simulation timesteps while you are testing code. +- Save your trajectories (positions and velocities), to prevent running the +code an excessive number of times. + +
+ +```{admonition} **(a)** +Simulate a Lennard-Jones fluid initialized at a temperature +$T=1.2$ $\epsilon/k_B$ and a density $\rho = 0.85$ particles/$\sigma^3$ +using Molecular Dynamics with a velocity Verlet integrator and no +thermostat. Plot the kinetic energy, potential energy, and total energy +per particle as a function of the simulation time and comment on their +relative values. Indicate the approximate number of timesteps necessary +for the system to reach equilibrium. + +
15 points available, click for grading breakdown + +- Plot Kinetic, Potential and Total Energy: 5 pts + +- Constant energy demonstrated: 5 pts + +- Indicated approx number of steps: 5 pts +
+ +``` + +```{admonition} **(b)** +Using an Andersen thermostat, rerun the molecular dynamics +simulation of the same system with the temperature maintained at +$T = 1.2$ $\epsilon/k_B$ using a coupling constant $\eta = 1$. Plot the +kinetic energy, potential energy, and total energy per particle as a +function of the simulation time, and indicate the number of timesteps +necessary for the system to reach thermal equilibrium. Compare features +of these plots to the results from part **a**. + +
15 points available, click for grading breakdown + +- Plot Kinetic, Potential and Total Energy: 5 pts + +- Energy fluctuations, but convergence demonstrated: 5 pts + +- Compare results: 3 pts + +- Correct comparison: 2pts +
+``` + +```{admonition} **(c)** +Plot the equilibrium distribution of particle **speeds** for the +simulations from part **a** and part **b**. Specify the time interval +used for the calculation. Compare these results to the Maxwell-Boltzmann +distribution and calculate the ensemble-average temperature for the +thermostatted simulation. Explain if your results match your +expectations. I recommend using a histogram size of +$0.2 \sigma/\text{unit time}$. + +
10 points available, click for grading breakdown + +- Plot speeds correctly: 5 pts + +- Compare results: 3 pts + +- Correct comparison: 2pts +
+``` + +```{admonition} **(d)** +Perform MD simulations of Lennard-Jones fluids at $T=1.5$ +$\epsilon/k_B$ and at densities of $\rho = 0.01$, $\rho = 0.10$, and +$\rho=0.85$ particles/$\sigma^3$ and compare the corresponding radial +distribution functions. Specify the time interval used for the +calculation and report the length scale in units of $\sigma$. Comment on +key features of the plot. I recommend using a histogram size of 0.02 +$\sigma$. Explain system phase behavior based on your results. +Simulation visualization is also recommended. + +
5 points available, click for grading breakdown + +- Correct RDFs (zero at large r, not showing past L/2): 3 pts + +- Compare results for three different densities: 2 pts +
+``` + +```{admonition} **(e)** +Calculate the chemical potential of a Lennard-Jones particle for +each of the systems considered in part **d**. Recall that the chemical +potential is the change in the free energy of the system upon addition +or subtraction of a single particle. Explain the method that you use for +this calculation. Hint: you do not need to perform any additional +simulations if you have saved your trajectories. + +
5 points available, click for grading breakdown + +- Explain method: 2 pts + +- Attempted:1 pts + +- Correct results: 2 pts +
+``` From f9dcfaa8ecc8ae232ed8d3d4bc9fcca8918bba86 Mon Sep 17 00:00:00 2001 From: cbefan <145715919+cbefan@users.noreply.github.com> Date: Thu, 21 Sep 2023 15:04:59 -0500 Subject: [PATCH 2/2] Update Lecture5.md --- lecture_files/Lecture5.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/lecture_files/Lecture5.md b/lecture_files/Lecture5.md index dbdfd69..41a3b48 100644 --- a/lecture_files/Lecture5.md +++ b/lecture_files/Lecture5.md @@ -229,7 +229,7 @@ referred to as the [**Gibbs entropy**](https://en.wikipedia.org/wiki/Entropy_(st $$S = -k_B \sum_j p_j \ln p_j$$ -You will shows this expression's equivalence to the Boltzmann definition +You will show this expression's equivalence to the Boltzmann definition of the entropy on [Problem Set 2](../../problems/ps_2/problem_set_2). We now have the expressions: $$\begin{aligned}

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point deductions** +For each plot, if you are missing the title, axis label(s), or have not labeled +your units, you will incur a 1 point deduction per missing item per plot. + +## Resources for learning Python +1. The links for CS 301: Intro to Data Programming are useful for setting up Python on multiple platforms. The class website is [here](http://pages.cs.wisc.edu/~cs301-1/). +2. DoIT manages subscriptions to Lynda.com, which contains ~2 hour long training videos on Python. See the links [here](https://sts.doit.wisc.edu/) and [here](https://www.lynda.com/search?q=Python). +3. [Google’s Python class](https://developers.google.com/edu/python/?csw=1) +4. [Python tutorial](https://docs.python.org/3/tutorial/) + + +## Question 1: Ferromagnetic phase transition + +Ferromagnetic materials undergo an order-disorder transition as a +function of temperature. The 2D Ising model is able to reproduce this +transition. Recall that the [Ising model](../lecture_files/Lecture8A) consists of $N$ distinguishable +spins on a lattice. Each spin has one of two discrete values: +$s_i = \pm 1$. The magnetization, $M$, is the sum of all spins, +$M = \sum_{i=1}^{N} \mu s_i$. Spins can interact with an external +magnetic field, $H$, and other nearest-neighbor spins. For this problem +we will assume $H=0$. The energy with which spin $i$ interacts with its +neighbors is: + +```{math} +\epsilon_i = - J \sum_j^\prime s_i s_j +``` + +$J$ is the coupling parameter (in the same units as energy) that determines the +interaction energy between each *pair* of nearest-neighbor spins. The +sum $\sum_j^\prime$ indicates a sum over all $n$ nearest-neighbor spins +of $i$ on the lattice. The *total* energy of the system is: + +$$E_\nu = - \frac{J}{2} \sum_{i}^N \sum_j^\prime s_i s_j$$ + +The factor of $1/2$ accounts for the double-counting of interactions +between pairs of spins when summing over the entire system. In class, we +used a mean-field approximation to show that + +$$T_C = \frac{nJ}{k_B}$$ + +$T_C$ is the temperature at which the system transitions from non-zero +magnetization to zero magnetization. However, the mean-field +approximation introduces significant errors. In this problem, we will +show that Monte Carlo simulations can identify a more accurate +expression for $T_C$. + + +```{admonition} **(a)** + +Develop a simulation representation of an Ising model in which +$N = 400$ spins are placed on a 2D, periodic, square lattice, such that +each lattice point has $n=4$ nearest-neighbors. Plot the absolute value +of the ensemble-average magnetization per spin, +$| \langle M \rangle / N |$, and the heat capacity, $C_V$, as a function +of temperature for a coupling parameter $J = 1.0$. + +Choose a temperature range of $T = 0.1$ $J/k_B$ to $T = 6.0$ $J/k_B$ in increments of $0.1$ +$J/k_B$. The coupling parameter is expressed in dimensionless units; +accordingly, $J$ is the unit of energy and $J/k_B$ is the unit of +temperature. + +I suggest running for at least 1,000,000 trials, saving the +magnetization every 500 steps, and averaging the results from multiple +independent simulations. Explain any other choices you make for +simulation parameters. Consider which pairwise interactions actually +contribute during evaluation of the MC acceptance condition; it is not +necessary to compute the energy of the entire lattice. + +

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