**From:** Jeff Comer (*jeffcomer_at_gmail.com*)

**Date:** Fri Sep 11 2015 - 08:12:15 CDT

**Next message:**Parag Khuraswar: "node file for NAMD"**Previous message:**Jeff Comer: "Re: Error in NAMD Controller Class Reference- Adaptive Tempering"**In reply to:**Jeff Comer: "Re: Error in NAMD Controller Class Reference- Adaptive Tempering"**Messages sorted by:**[ date ] [ thread ] [ subject ] [ author ] [ attachment ]

Oops! I forgot the radical in the normalization constant of the

probability density. It should be: 1/sqrt(4πΔt) exp(–x^2/(4Δt))

–––––––––––––––––––––––––––––––––––———————

Jeffrey Comer, PhD

Assistant Professor

Institute of Computational Comparative Medicine

Nanotechnology Innovation Center of Kansas State

Kansas State University

Office: P-213 Mosier Hall

Phone: 785-532-6311

On Fri, Sep 11, 2015 at 8:10 AM, Jeff Comer <jeffcomer_at_gmail.com> wrote:

*> Dear Sachin Natesh,
*

*>
*

*> I'm not that familiar with adaptive tempering, but I noticed that this
*

*> equation bears similarity with integrators for overdamped Brownian
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*> dynamics. What you have pointed out is not actually an error, but the
*

*> correct discretization of the Gaussian noise.
*

*>
*

*> First, a quick dimensional analysis. Notice that the Gaussian noise
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*> has the correlation function <ζ(t')ζ(t)> = δ(t-t'), which is stated in
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*> the paper you cite
*

*> (http://www.ncbi.nlm.nih.gov/pmc/articles/PMC2905458/). The delta
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*> function has units of the inverse of its argument since ∫ dt δ(t) = 1.
*

*> So the noise, ζ(t), has units of 1/sqrt(t). Thus, ζ(t) Δt must have
*

*> units of sqrt(t).
*

*>
*

*> You might want to do a search on the "Wiener process", which will give
*

*> you more insight about what is going on. Basically, we are integrating
*

*> the Gaussian noise ζ(t) over a finite time Δt. The integral of a
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*> zero-mean unit-variance Gaussian noise is the Wiener process, which is
*

*> basically Brownian dynamics with a diffusion constant=1 (which obeys
*

*> the diffusion equation with D=1 if you average over many instances).
*

*> Integrating ζ(t) from 0 to Δt gives a Gaussian probability density
*

*> p(x)=1/(4πΔt) exp(–x^2/(4Δt)). (See
*

*> https://en.wikipedia.org/wiki/Heat_kernel ) If you compare this to the
*

*> standard normal distribution, you see that the standard deviation is
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*> sqrt(2Δt). So to discretize Eq. 1 of the reference you cited, you need
*

*> a Gaussian random number with a standard deviation of sqrt(2Δt).
*

*>
*

*> random->gaussian()*sqrt(2.*adaptTempDt) is just that.
*

*>
*

*> Regards,
*

*> Jeff
*

*>
*

*> –––––––––––––––––––––––––––––––––––———————
*

*> Jeffrey Comer, PhD
*

*> Assistant Professor
*

*> Institute of Computational Comparative Medicine
*

*> Nanotechnology Innovation Center of Kansas State
*

*> Kansas State University
*

*> Office: P-213 Mosier Hall
*

*> Phone: 785-532-6311
*

*>
*

*>
*

*> On Thu, Sep 10, 2015 at 7:13 PM, Sachin Natesh <snatesh_at_uchicago.edu> wrote:
*

*>> Hello all,
*

*>>
*

*>> In the NAMD controller class reference, which can be found here, I notice a
*

*>> possible error in the computation of the adaptive tempering temperature
*

*>> update (lines 1875-1878; see below)
*

*>>
*

*>> 01875 //dT is new temperature
*

*>> 01876 BigReal dT =
*

*>> ((potentialEnergy-potEnergyAverage)/BOLTZMANN+adaptTempT)*adaptTempDt;
*

*>> 01877 dT += random->gaussian()*sqrt(2.*adaptTempDt)*adaptTempT;
*

*>> 01878 dT += adaptTempT;
*

*>>
*

*>>
*

*>> On line 1877, the stochastic portion of the temperature update is added to
*

*>> what has been computed for the new temperature thus far, though the
*

*>> integration time-step, adaptTempDt, is placed within the sqrt function. This
*

*>> seems particularly erroneous to me as equation 1 of the paper upon which the
*

*>> NAMD implementation is based (here) calls for the integration time-step to
*

*>> fall outside of the radical, which can be realized through simple cross
*

*>> multiplication of the dt differential.
*

*>>
*

*>> Does the NAMD implementation of the adaptive tempering algorithm actually
*

*>> contain this error, is it the online documentation that is flawed, or am I
*

*>> missing something?
*

*>>
*

*>> Any information in this regard would be much appreciated.
*

*>>
*

*>>
*

*>> Best,
*

*>>
*

*>> Sachin Natesh
*

*>>
*

*>>
*

*>>
*

*>>
*

*>>
*

*>>
*

*>>
*

*>>
*

*>>
*

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