{"id":446,"date":"2020-05-11T19:53:47","date_gmt":"2020-05-11T19:53:47","guid":{"rendered":"http:\/\/statisticalbiophysicsblog.org\/?p=446"},"modified":"2020-05-11T19:53:51","modified_gmt":"2020-05-11T19:53:51","slug":"theory-stronger-incrementally-exercise-8","status":"publish","type":"post","link":"https:\/\/statisticalbiophysicsblog.org\/?p=446","title":{"rendered":"Theory stronger, incrementally!! (Exercise 8)"},"content":{"rendered":"

The key lesson of all these exercises is that you can push yourself to be better and more confident in theory by tackling simple, paradigmatic problems in an incremental way. You must put pencil to paper! You must do it regularly. But once you do, the benefits come quickly. Each mini-realization builds into knowledge. Each solved simple problem builds your intuition for understanding complex systems.<\/p>\n

<\/p>\n

Below are solutions to the problems from last time. Looking for more? Scroll down for some additional problems.<\/p>\n

Here are the answers to the questions from last time, which explain the connection between continuous and discrete-time kinetic descriptions.<\/p>\n

    \n
  1. \n Give the full definition of \"T_{ji}(\Delta t)\" and explain why it\u2019s dimensionless.<\/p>\n
      \n
    1. \n This is the conditional to probability to be in state \"j\" at time \"t+\Delta t\" if a system was in state \"i\" at time \"t\". Importantly, it\u2019s completely irrelevant to \"T_{ji}\" where the system may have been in between \"t\" and \"t+\Delta t\". It only matters where the system is at the times we\u2019re \u201clooking.\u201d The probability \"T_{ji}\" is dimensionless because it\u2019s simply a probability \u2013 the fractional chance to be in state \"j\" – and not a probability per unit time like a rate constant.<\/li>\n<\/ol>\n<\/li>\n
    2. \n If \"P_i(t)\" is the time-dependent probability of state \"i\", show that \"P_i(t+\Delta t) = \sum_j T_{ij} P_j(t)\".<\/p>\n
        \n
      1. \n At time \"t\", some fraction of the probability is in each state \"j\", including \"j=i\". Then the total probability in state \"i\" at the later time \"t+\Delta t\" is the sum of the \u2018contributions\u2019 to state \"i\" from every state indexed by \"j\" (including \"j=i\") based on the \"T_{ij}\". More precisely, \"T_{ij}\" is the fraction of the state \"j\" probability which ends up in \"i\", implying \"T_{ij} P_j\" is the fraction \u2018contributed\u2019.<\/li>\n<\/ol>\n<\/li>\n
      2. \n Using the exact solutions to the continuous-time<\/em> two-state probabilities, \"P_A(t), P_B(t)\", which we derived previously<\/a>, calculate the discrete-time<\/em> transition probabilities \"T_{AB}(\Delta t)\" and \"T_{AA}(\Delta t)\" for transitions into<\/em> state A. This problem is trickier than it sounds (though the math is easy), but don\u2019t worry the derivation is there for you in the notes<\/a>. The key trick is to write \"P_A(t+\Delta t)\" in terms of \"P_A(t)\" and \"P_B(t)\" using the fact that \"P_A + P_B = 1\" for any\/all \"t\".<\/p>\n
          \n
        1. \n For this, please just look at the notes<\/a>. It\u2019s written out carefully there.<\/li>\n<\/ol>\n<\/li>\n<\/ol>\n

          Going forward, here are some key problems you should solve.<\/p>\n

            \n
          1. \n Demonstrate the correctness of the (differential) continuity equation in one dimension by integrating the probability over a small increment.<\/li>\n
          2. \n Derive the diffusion equation via a Taylor expansion of the time-varying probability distribution. See my online notes<\/a>.<\/li>\n
          3. \n Derive the Smoluchowski equation via a Taylor expansion of the time-varying probability distribution. See my online notes<\/a>.<\/li>\n
          4. \n Show that the stationary distribution of the Smoluchowski equation is the Boltzmann distribution.<\/li>\n
          5. \n Show that overdamped dynamics lead to the Boltzmann distribution. For hints, see Sec. IIB of this paper<\/a> by Chandler\u2019s group.<\/li>\n
          6. \n Derive the replica-exchange Metropolis acceptance criterion. For hints, see Ch. 12 of my textbook.<\/li>\n
          7. \n Explain why one takes a linear average of MD or MC snapshots (equally spaced in time) to obtain a Boltzmann-weighted average. See Ch. 2 of my textbook.<\/li>\n
          8. \n Derive the ideal gas partition function from scratch. See Ch. 5 of my textbook.<\/li>\n<\/ol>\n

            Good luck! Work hard, but work slow \u2026<\/p>\n","protected":false},"excerpt":{"rendered":"

            The key lesson of all these exercises is that you can push yourself to be better and more confident in theory by tackling simple, paradigmatic problems in an incremental way. You must put pencil to paper! You must do it regularly. But once you do, the benefits come quickly. Each mini-realization builds into knowledge. Each […]<\/p>\n","protected":false},"author":6,"featured_media":330,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[22,11,19],"tags":[],"class_list":["post-446","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-exercises","category-general-biophysics","category-non-equilibrium-physics"],"_links":{"self":[{"href":"https:\/\/statisticalbiophysicsblog.org\/index.php?rest_route=\/wp\/v2\/posts\/446","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/statisticalbiophysicsblog.org\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/statisticalbiophysicsblog.org\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/statisticalbiophysicsblog.org\/index.php?rest_route=\/wp\/v2\/users\/6"}],"replies":[{"embeddable":true,"href":"https:\/\/statisticalbiophysicsblog.org\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=446"}],"version-history":[{"count":2,"href":"https:\/\/statisticalbiophysicsblog.org\/index.php?rest_route=\/wp\/v2\/posts\/446\/revisions"}],"predecessor-version":[{"id":448,"href":"https:\/\/statisticalbiophysicsblog.org\/index.php?rest_route=\/wp\/v2\/posts\/446\/revisions\/448"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/statisticalbiophysicsblog.org\/index.php?rest_route=\/wp\/v2\/media\/330"}],"wp:attachment":[{"href":"https:\/\/statisticalbiophysicsblog.org\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=446"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/statisticalbiophysicsblog.org\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=446"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/statisticalbiophysicsblog.org\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=446"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}