Add tutorial on GCMC #4670
Add tutorial on GCMC #4670
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| "source": [ | ||
| "## Grand-Canonical Ensemble\n", | ||
| "The case of changing particle numbers is best described by the grand canonical ensemble <a href='#[1]'>[1]</a>.\n", | ||
| "In the grand canonical ensemble, the volume $V$, the temperature $T$ and the chemical potentials $\\mu_i$ of the exchangeable species $i$ rather than the particle numbers $N_i$ are fixed.The chemical potentials $\\mu_i$correspond to the free energy cost of adding a particle of type $i$ while keeping everything else fixed:\n", |
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missing space in last sentence
| "The grand canonical ensemble is thus also referred to as the $(\\mu, V, T)$-ensemble.\n", | ||
| "In the grand canonical ensemble, the probability for the system to contain $N_i$ particles of type $i$, i.e. a total of $N=\\sum_{i=1}^{N_\\mathrm{s}}N_i$ and to be in a specific microstate $\\xi\\in\\Gamma_{N}$ is given by the probability distribution\n", | ||
| "\\begin{align}\n", | ||
| "p^\\mathrm{G}\\left(\\left\\{N_i\\right\\},\\xi\\right) = \\frac{\\exp\\left(-\\beta \\left(H(\\xi)-\\sum_{i=1}^{N_\\mathrm{s}}\\mu_iN_i\\right)\\right)}{Z^\\mathrm{G}h^{3N}\\prod_{i=1}^{N_\\mathrm{s}} N_i!}. \\quad \\tag{1}\n", |
| "\\end{align}\n", | ||
| "Here, $H$ denotes the Hamiltonian of the system with $N_i$ particles of type $i$ and $Z^\\mathrm{G}$ is the grand-canonical partition function which is given by\n", | ||
| "\\begin{align}\n", | ||
| "Z^\\mathrm{G}\\left(\\left\\{\\mu_i\\right\\},V,T\\right) = \\sum_{N_1=0}^{\\infty}...\\sum_{N_\\mathrm{s}=0}^{\\infty}Z\\left(\\left\\{N_i\\right\\},V,T\\right)\\exp\\left(\\beta \\sum_{i=1}^{N_\\mathrm{s}}\\mu_iN_i\\right).\n", |
| "\\mu_i^{\\mathrm{sys}} = \\mu_i^{\\mathrm{res}}.\n", | ||
| "\\end{equation}\n", | ||
| "This condition is referred to as a chemical equilibrium between the two phases and is equivalent to a minimization of the free energy of the total system (system + reservoir).\n", | ||
| "In the case where one requires electroneutrality of the phases, due to the additional constraint, the electrochemical potential $\\overline{\\mu}_i=\\mu_i+z_ie\\psi$ rather than the chemical potential has to be equal." |
| "\\frac{P_{i\\rightarrow j}^\\mathrm{acc}}{P_{j\\rightarrow i}^\\mathrm{acc}} = \\frac{p_j^\\mathrm{eq}}{p_i^\\mathrm{eq}}. \n", | ||
| "\\quad \\tag{6}\n", | ||
| "\\end{align}\n", | ||
| "It is easy to see that the acceptance probability \n", |
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avoid such phrases to not discourage beginners
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| "100%|██████████████████████████████████████████████████████████████████████████████████████████████████████████████████████████████████████████████████████████████████████| 50/50 [00:04<00:00, 11.72it/s]" |
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remove output from tutorial before commiting/pushing
| " K_XX =salt_concentration ** 2 * np.exp(excess_chemical_potential_monovalent_pairs_in_bulk(salt_concentration))\n", | ||
| " RE.change_reaction_constant(gamma= K_XX,reaction_id = 0)\n", | ||
| " for i in tqdm(range(warmup_loops),bar_format='{l_bar}{bar:60}{r_bar}{bar:-60b}'):\n", | ||
| " system.integrator.run(1000)\n", |
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maybe give these numbers as predefined variables
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| "**Exercise**\n", | ||
| "* Plot the universal partition coeffiecients obtained by measuring the partition coefficient $\\xi_-$ and $\\xi_+$ in the simulation against $c_{\\mathrm{M}^-}/(2c_{\\mathrm{salt}}^{\\mathrm{res}})$\n", |
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not sure plotting is a meaningful exercise
| "source": [ | ||
| "**Exercise**\n", | ||
| "* Plot the universal partition coeffiecients obtained by measuring the partition coefficient $\\xi_-$ and $\\xi_+$ in the simulation against $c_{\\mathrm{M}^-}/(2c_{\\mathrm{salt}}^{\\mathrm{res}})$\n", | ||
| "* Interpret the deviation of the simulation results from the ideal prediction." |
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there should definitely be a solution to this question
| "<figcaption>Fig. 1: Schematic representation of the considered two phase setup.</figcaption>\n", | ||
| "</center>\n", | ||
| "</figure>\n", | ||
| "Here we want to investigate a polyelectrolyte solution that is coupled to a reservoir containing a monovalent salt, as shown schematically in figure 1. The small salt ions (X$^+$ and X$^-$) can be exchanged between the two phases while the polyelectrolyte is confined to the system. Experimentally such a setup could be realized by seperating the two phases with a semi-permeable membrane. Similiar setups also occur in the study of polyelectrolyte hydrogels.\n", |
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maybe add a sentence on the relevance/usefulness of this system
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Thank you for your review and your annotations. I have now worked in your annotations and added an interpretation of the final plot. In addition I have changed the sample sizes a bit to speed up the runtimes. |
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Looks really nice now, thanks for the work you put in. |
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Reduce sampling times. Improve testing. Redesign plots.
Fixes #4575
Description of changes: