Tech. Note: Survey of Simple, Continuous, Univariate Probability Distributions

Unimodal distributions

Version: 0.5 BETA

In a desperate attempt to preserve my own sanity, a survey of probability distributions used to describe a single, continuous, unimodal, univariate random variable.

Whats New: Added uniform product, half generalized Pearson VII, half exponential power distributions, GUD and q-Type distributions; Moved Pearson IV to own section; Fixed errors in Inverse Gaussian; Added random variate generation appendix. Fixed typos. Thanks to David Sivak, Dieter Grientschnig, Srividya Iyer-Biswas and Shervin Fatehi.

[ Full Text ]

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Article: Using nonequilibrium fluctuation theorems to understand and correct errors in equilibrium and nonequilibrium discrete Langevin dynamics

David A. Sivak, John D. Chodera, Gavin E. Crooks, Phys. Rev. X 3, 011007 (2013)

[ Full text | Journal]

Abstract: Common algorithms for computationally simulating Langevin dynamics must discretize the stochastic differential equations of motion. These resulting finite-time-step integrators necessarily have several practical issues in common: Microscopic reversibility is violated, the sampled stationary distribution differs from the desired equilibrium distribution, and the work accumulated in nonequilibrium simulations is not directly usable in estimators based on nonequilibrium work theorems. Here, we show that, even with a time-independent Hamiltonian, finite-time-step Langevin integrators can be thought of as a driven, nonequilibrium physical process. Once an appropriate worklike quantity is defined – here called the shadow work – Continue reading

Preprint: Time step rescaling recovers continuous-time dynamical properties for discrete-time Langevin integration of nonequilibrium systems

David A. Sivak, John D. Chodera, Gavin E. Crooks, arXiv:1301.3800

[ Full text | ArXiv ]

Abstract: While the numerical integration of deterministic equations of motion for molecular systems now has a well-developed set of algorithms with commonly agreed-upon desirable properties, the simulation of stochastic equations of motion lacks algorithms with a similar degree of universal acceptance. Part of the difficulty is in determining which of many properties should be satisfied by such a discrete time integration scheme, with additional difficulties in satisfying many properties simultaneously with a single scheme. The desire to use these integration schemes for nonequilibrium simulations and in conjunction with recent nonequilibrium fluctuation theorems adds additional complications. Here, we compare a number of discrete time integration schemes for Langevin dynamics, Continue reading

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In science, as in life, it is extremely dangerous to fall in love with beautiful models.

Via Vijay Pande

Article: The geometry of thermodynamic control

Abstract: A deeper understanding of nonequilibrium phenomena is needed to reveal the principles governing natural and synthetic molecular machines. Recent work has shown that when a thermodynamic system is driven from equilibrium then, in the linear response regime, the space of controllable parameters has a Riemannian geometry induced by a generalized friction tensor. Continue reading

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♬ Hey I just met you, and this is crazy, but here’s my paper – so cite me maybe? ♬

Article: Thermodynamics of prediction

Susanne Still, David A. Sivak, Anthony J. Bell and Gavin E. Crooks, Phys. Rev. Lett., 109, 120604 (2012)

[ Full Text | Journal | Press | arXiv ]

Hat Trick! And Editors’ Suggestion. Also a Nature News item by Philip Ball.

This paper is a melding of ideas about machine learning from Susanne Still and Tony Bell, with ideas from David and I about nonequilibrium thermodynamics. For a molecular scale machine with information processing capabilities, there’s a tradeoff between thermodynamic efficiency, memory and prediction. A prodigious memory allows more accurate prediction of the future, which can be exploited to reduce dissipation. But the persistence of memory is a liability, since information erasure leads to increased dissipation. A thermodynamically optimal machine must balance memory versus prediction by minimizing its nostalgia, the useless information about the past.

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Memory has great survival utility … but why would forgetting be adaptive. Forgetting well is almost as important as remembering well. Forgetting is about editing, about taking the flood– the ocean– of sense information coming at you and forgetting everything except what’s important. Life is not only about acquiring memories. Memory can cripple us too.
Michael Pollen, The Botany of Desire

WebLogo 3.3 Released

WebLogo 3.3 web-sever

WebLogo 3.3 code

3.3 (2012-07-02)
[Gavin Crooks, David Sivak]
* Improved the algorithm that guesses the sequence type (DNA, RNA or protein) (Kudos: Bug report, Roland Pache )
* Fixed an issue with reading transfac matrices with alternative alphabets(Kudos: Bug report, Nima Emami )
* Fixed Motif.reindex()
* Implemented Motif.reverse() and Motif.complement() (Can now reverse complement transfac matrix input on the command
line.) (Kudos: Feature request, Seth P Boudreaux )
* Command line interface now automagically recognizes transfac files.
* Add command line option “–reverse-stacks NO” which inverts the logo stacks so that the most conserved monomers are at the bottom of the stack, rather than the top. This ordering is consistent with the standard ordering for histograms, and is arguably a better representation. (Kudos: Luke Hutchison )
* Fixed an issue so that the correct color scheme is chosen for the specified sequence type.
* Miscellaneous minor bug fixes and refactoring.

Article: Thermodynamic metrics and optimal paths

David A. Sivak, Gavin E. Crooks, Phys. Rev. Lett., 108, 190602 (2012)

[ Full Text | Journal | arXiv ]

Excess Power
This is mostly David’s creation, from inception to conclusion, although I like to think I’ve contributed some elegant prose. Essentially, he shows that at the intersection of linear response theory and thermodynamic length analysis, there is a really nice framework for understanding optimal protocols for driving thermodynamic systems. Continue reading

Article: Near-equilibrium measurements of nonequilibrium free energy

David A. Sivak, Gavin E. Crooks. Phys. Rev. Lett. 108, 150601 (2012)

[ Full Text | Journal ]

Abstract: A central endeavor of thermodynamics is the measurement of free energy changes. Regrettably, although we can measure the free energy of a system in thermodynamic equilibrium, typically all we can say about the free energy of a non-equilibrium ensemble is that it is larger than that of the same system at equilibrium. Herein, we derive a formally exact expression for the probability distribution of a driven system, which involves path ensemble averages of the work over trajectories of the time-reversed system. From this we find a simple near-equilibrium approximation for the free energy in terms of an excess mean time-reversed work, which can be experimentally measured on real systems. With analysis and computer simulation, we demonstrate the accuracy of our approximations for several simple models.