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High Efficiency Configuration Space Sampling  probing the distribution of available states
by Paweł T. Jochym, Jan Łażewski
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Submission summary
As Contributors:  Paweł Jochym · Jan Łażewski 
Preprint link:  scipost_202101_00011v2 
Date submitted:  20210426 20:27 
Submitted by:  Jochym, Paweł 
Submitted to:  SciPost Physics 
Academic field:  Physics 
Specialties: 

Approaches:  Theoretical, Computational 
Abstract
Substantial acceleration of research and more efficient utilization of resources can be achieved in modelling investigated phenomena by identifying the limits of system's accessible states instead of tracing the trajectory of its evolution. The proposed strategy uses the MetropolisHastings MonteCarlo sampling of the configuration space probability distribution coupled with physicallymotivated prior probability distribution. We demonstrate this general idea by presenting a high performance method of generating configurations for lattice dynamics and other computational solid state physics calculations corresponding to nonzero temperatures. In contrast to the methods based on molecular dynamics, where only a small fraction of obtained data is used, the proposed scheme is distinguished by a considerably higher, reaching even 80%, acceptance ratio and much lower amount of computation required to obtain adequate sampling of the system in thermal equilibrium at nonzero temperature.
Current status:
Author comments upon resubmission
Your reference: scipost_202101_00011v1
Corresponding author: Paweł T. Jochym Address: Institute of Nuclear Physics, Radzikowskiego 152, 31342 Cracow, Poland email: pawel.jochym@ifj.edu.pl
Title: High Efficiency Configuration Space Sampling  probing the distribution of available states
Authors: Paweł T. Jochym and Jan Łażewski Type: regular article
Dear Dr Attaccalite,
Thank you for arranging the review of our paper. We are glad that criticism of both referees contributed to the improvement of quality of our work. In fact, both referees noticed strong aspects of our idea and highlighted weak points of the presentation causing possible confusion of the reader. None of the key assumptions of the method have been questioned. The second referee went even further and in his summary rating recognized our approach as of the "high validity and significance" and "top originality".
We thank both referees for careful reading of our text and their valuable remarks. We regret a few mistakes and some deficiencies in presentation pointed by the referees. Following their advice we have made substantial revision of the text and figures correcting all mistakes and omissions as well as extending the explanations to make the presentation clearer.
Please find included a detailed response to both referees. We have already submitted an early response to the first review. This initial response is still valid, but we have reformulated it to closely reflect changes we have made in the text.
We have addressed all points raised by the referees, applied all their suggestions and answered all questions. We believe that after these corrections the manuscript is ready for publication in SciPost Physics without delay.
This resubmission includes:  the revised text,  the detailed reply to both referees (submitted as reply on the submission page),  list of changes
Sincerely Yours, Paweł T. Jochym Jan Łażewski
List of changes
Summary of changes in scipost_202101_00011v1/Jochym&Lazewski:

In response to the referee reports we made the following changes
in the manuscript:
* The Eq. 5 has been corrected.
* The definition of the method was supplemented and the scope of its applicability was more precisely specified.
* Calculations are extended up to 2000 K.
* Figures 3, 4, 5 and 6 were split to several panels to separate data for different temperatures and increase readability of the contents.
* Imprecise statements in our presentation (like "too wild", "very quick", "hardly visible" etc.) were quantified.
* We have modified the text as recommended by the referees  these changes are described in the replies.
* Minor modifications of the text were made in a few other places in order to improve the reading and to remove some typographical errors.
Submission & Refereeing History
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Reports on this Submission
Anonymous Report 2 on 2021519 (Invited Report)
 Cite as: Anonymous, Report on arXiv:scipost_202101_00011v2, delivered 20210519, doi: 10.21468/SciPost.Report.2938
Report
The authors of the manuscript "High Efficiency Configuration Space Sampling – probing the distribution of available states" have addressed all the points suggested in my initial report.
The scope and current limitations of the HECSS scheme are now clearly presented. All the previously missing details required for reproducibility are now reported in the manuscript.
I appreciate that Fig. 5 and 6 show the increasing loss of accuracy of the current version of the HECSS as the temperature is increased.
I agree with the authors that hightemperature properties are challenging also using existing MD schemes.
Even if the CLT holds, i.e., even if the variance is finite, the convergence to the expected normal distribution can get rather slow.
Perhaps this can be better understood in terms of the tails of the potential energy distribution  not clearly visible in Fig. 2  which get "heavier" at hightemperature because of the anharmonicity.
Investigating the role of these tails, e.g., perturbatively, may reveal a strategy to improve the agreement between the HECSS and MD without resorting to a somehow arbitrary "temperature calibration".
Report 1 by Bjorn Wehinger on 2021512 (Invited Report)
 Cite as: Bjorn Wehinger, Report on arXiv:scipost_202101_00011v2, delivered 20210512, doi: 10.21468/SciPost.Report.2910
Report
Referee report for the revised manuscript entitled "High Efficiency Configuration Space Sampling – probing the distribution of available states".
The authors have carefully addressed all my comments in the revised version of the manuscript. The new version of the manuscript properly describes the new method and nicely illustrates its application and limitations to lattice dynamics calculations at finite temperatures.
As I have mentioned in my first report, the idea is highly original and clearly opens a new pathway in the study of thermal properties at finite temperatures applicable to large systems that are difficult to address otherwise.
The revised version of the manuscript fulfills all general acceptance criteria. I believe that it is of great interest to a broad audience and thus recommend publication in SciPost Physics.
I only have a few minor comments:
1. How does the applied temperature adjustment of 90K mentioned for the calculation of phonon life times at a temperature of 600K compare to an estimated correction obtained by "deriving the $C_n$ coefficients from the force constants matrices" as explained in Section 2?
2. Labels in Fig. 5 disagree with description in the figure caption.
Author: Paweł Jochym on 20210515 [id 1426]
(in reply to Report 1 by Bjorn Wehinger on 20210512)
We would like to thank Dr Wehinger for a quick response and his favourable outlook on our revised manuscript. We are glad that Dr Wehinger found our reply and corrections satisfactory and convincing and would like to immediately address his remaining comments:
1) How does the applied temperature adjustment of 90K mentioned for the calculation of phonon life times at a temperature of 600K compare to an estimated correction obtained by "deriving the $C_n$ coefficients from the force constants matrices" as explained in Section 2?
The temperature correction shown in Fig. 6 is intended only as an illustration that our hypothesis from section 2, that the leading correction to the proposed scheme is a temperature shift, is indeed plausible. The derivation of the correction term from the forceconstant matrix requires developing of a selfconsistent correction procedure based on the higher order (above quadratic) force constants matrices. We agree that this is a very interesting direction of the future research, but it is beyond the scope of the current paper which is intended to establish a baseline of the core HECSS method to be extended in the future.
2) Labels in Fig. 5 disagree with description in the figure caption.
This unfortunate mistake will be corrected on resubmission.
Paweł T. Jochym Jan Łażewski
Paweł Jochym on 20210503 [id 1405]
The updated reply to the report of the first referee is included as a PDF attachment.
Attachment:
reply_1.pdf
Paweł Jochym on 20210503 [id 1404]
The reply to the report of Dr Wehinger is included as a PDF attachment.
Attachment:
reply_2.pdf
Paweł Jochym on 20210503 [id 1403]
Updated reply to the report of the First Referee
We thank the referee for reading our paper and spotting omissions and mistakes in our text. We hope that the following clarifications and corrections will enable full understanding of our approach. We address the comments paragraph by paragraph quoting the referee before our response.
The text was intended as a demonstration of the proposed method not as a reproduction or prediction of a particular experimental result. This may perhaps explain unfortunate omission of sampled temperature in the text. While the temperature can in fact be extracted from the average energy in Fig. 3 and 4, it should also be stated in the text explicitly. The sampling temperature for all data in first version of the text is 300 K, chosen as standard ambient temperature. In the resubmitted version of the text we have corrected this omission. Furthermore, following suggestions of both referees, we have extended the presented data to higher temperatures  up to 2000K. The temperatures are indicated in the text.
The shape of the energy distribution is not determined by harmonicity of the potentials but by the Central Limit Theorem and the size of the system. The difference between prior distribution (which comes from our approximation of the displacement distribution) and the target distribution does not stem from the anharmonicity of the potential but mainly from the fact that the displacements of the atoms in the crystal are not independent (as stated in our text). Please note that we have no direct access to the potential energy of the system  we can only specify the geometry and calculate the resulting energy instead of directly generating the target energy distribution from Fig. 3. The high acceptance ratio comes from our selection of the displacement distribution and tuning algorithm described in the paper. The MetropolisHastings (MH) algorithm can generate a target distribution from any prior which is nonzero over the domain. However, the acceptance ratio may be very low if you fail to use the prior which is a good approximation of target distribution. It is a wellknown fact in the numerical statistics community that the good selection of the prior distribution is the key to an effective use of probability distribution sampling algorithms. To further demonstrate the effectiveness of the algorithm for wide range of temperatures we have included results up to T=2000K. We have also extended explanation of the procedure used to generate Fig. 3 (which clearly has acceptance ratio below 80% claimed in the text for a typical values of delta). This figure was generated with artificially large value of delta (0.1 instead of 0.0050.02) to make difference between prior and posterior distributions easily visible.
The issue of sample correlation would be indeed important if we used a 'random walk'type algorithm for the prior generation (which is a popular variant of the MH algorithm). Instead, we use independent samples and the only possible correlation between them arises from a very small change (less than 2%) in the variance of the position distribution. We discuss this issue in the paragraph starting in line 192 (212 in revised text). On the other hand, the MD derived data has obvious autocorrelations  note that we can derive phonon frequencies from the Fourier transform of the velocity autocorrelation function along the MD trajectory. Thus, it is necessary to separate sampling points on the trajectory by substantial intervals allowing for these correlations to die out. Nevertheless, all time steps between the sampling points still need to be calculated, which leads to large inefficiency of the MD as a configuration generator. To further clarify the issue we have expanded the explanation in the text in paragraph starting at line 212 (revised text).
Indeed, for the purely harmonic system the phonon frequency test is not useful since phonon frequencies are independent from the displacement size. However, if the system considered in the paper were close to harmonic, we would expect to obtain very long phonon lifetimes (since they are infinite in the harmonic system). The data in Fig. 6 demonstrates that many of the phonon modes exhibit lifetimes below 10ps  showing nonnegligible anharmonicity in the model. This fact provides justification for the validity of the phonon frequency test. Furthermore, expanded temperature data of new Fig. 5 (up to 2000K), demonstrates that anharmonicity induced by high temperatures has some small influence on the convergence of phonon data but not on the converged results (lower row of Fig. 5) which shows good agreement between MD and HECSS data in full range of temperatures. Additionally, we have included in the text RMS errors for frequencies obtained with both methods. The phonon lifetimes are very sensitive to the accuracy of the model. This is especially true in case of large values which indicate small deviations from harmonicity and usually carry large error bars. Unfortunately, the large range (close to four orders of magnitude) of the values of lifetimes makes the simple RMS measure of differences very misleading  since the differences at high end of the range will dominate the sum. However, we agree that the previous Figure 6 was indeed not very clear. Thus, we have replaced it with the separate plot for three temperatures (T=100, 300, and 600K) splitting the smallsample data set to a separate row. We think that the new Fig.6 clearly demonstrates good agreement between data obtained with MD and HECSS procedure over 4 orders of magnitude in phonon lifetime.
The number of steps (30 000) used in the introduction was a typical relaxation time of a longrun MD suggested by the often used "rule of thumb" in MD calculations (50 times period of typical vibrations in the system). For 3CSiC: $f\approx 10$THz = $10^{13}$Hz $ \Rightarrow t=10^{13}$s = 100fs; 50 * 100fs = 5ps. With 1fs time step that equals 5000 steps minimum run where we can use at most half of it for actual data (you need to provide time to obtain thermal equilibrium). If we need approx. 30 data points (as required by anharmonic calculations, see Fig. 6) and they should be separated by at least 1ps interval (at least 10 typical vibrations) we get approximately 30 000 steps. The cited number itself has no 'magical' value and results from the setup of the calculations presented in the paper. To avoid impression that the number 30 000 has any special meaning, we have replaced the number by the phrase: "thousands of MD steps".
Eq. (1): we have added to the revised text a sentence introducing the missing symbols: $x_n$  generalized coordinate or momentum, $H$  Hamiltonian, $T$  temperature, $k_B$  Boltzmann constant, $\delta_{mn}$  Kronecker delta.
Eq. (2): We make no twobody assumption, neither implied nor explicit. The formulation of equipartition theorem, Eq. (1), explicitly concerns single coordinates (the only nonzero term due to the Kronecker delta) and makes no assumption on the form of the Hamiltonian $H$. The Taylor expansion, Eq. (2), is not a complete expansion in all coordinates $q$ (note the scalar $q$ symbol). It is the Taylor expansion in a single coordinate with coefficients ($C_n$  proportional to partial derivatives of energy with respect to this coordinate) which are functions of all the other coordinates in the system. What is more, the calculations presented in the paper use the mentioned Tersoff potential developed in refs 17, 18. We understand that due to the formulation of the surrounding text this may not be entirely clear and may confuse the reader. To avoid this, we have added a clarifying sentence below Eq. (2).
The unfortunate sentence 8081 brings nothing of importance to the text. Thus we have removed it in the resubmitted version and reformulated the surrounding paragraph.
Variance of several quantities is indeed divergent in some phase transitions. In cases where the transition involves divergent heat capacity this includes energy variance. Thus, our phrase :"...all physically interesting cases..." was indeed wrong. The sentence has been corrected and we clearly state that in cases where energy variance diverges, the procedure cannot be used. We thank the referee for spotting this important fact.
The Eq. (5) was an attempt to formally write asymptotic relation of the Central Limit Theorem described in the paragraph 99102. The CLT is indeed not a limit relation but asymptotic distribution convergence relation and the Eq. (5) should use appropriate notation for such relations as convergence in distribution:
$$ \sqrt{3N}\left(\frac{1}{N} \sum_i E_i \langle E\rangle \right) \xrightarrow{d}\mathcal{N}(0, \sigma). $$The mistake in notation has no consequences for the arguments and conclusions presented in the text. The Eq. (5) has been corrected in the revised text.
"Single server" mentioned in line 125 was used as a rough indication of the computational effort involved in the described task. It is nothing out of ordinary: 2x4 cores CPU and 32GB RAM. This is actually a fairly underpowered and old machine, less powerful than some of newer generation laptops. The information has been added to the sentence in revised text.
In some cases the random number generation may be indeed fairly expensive but in the case of typical systems of tens of atoms, the energy and forces evaluation is much more timeconsuming. For instance, the random number generator we have used (from SciPy.stats library) takes 180$\mu$s to generate 3000 random numbers required to create one sample for the 5x5x5 supercell of 3CSiC. The single evaluation of energy for the same cell (1000 atoms) takes 4ms (20 times longer) using ASAP3 with OpenKIM model from our calculations. A more sophisticated interaction model is bound to be even more timeconsuming. Furthermore, molecular dynamics requires calculating multiple time steps per every generated sample. Considering this facts, we maintain that the proposed HECSS approach offers substantial advantage over MD as a source of configuration data.
Paweł Jochym on 20210430 [id 1400]
Reply to the report of Dr Wehinger
We would like to thank Dr Wehinger for careful reading of the manuscript and his positive opinion on our work.
We would like to point out that the presented approach does not depend on strict harmonicity of the system. The Eq. (4) and its description (l. 7887) explicitly point to the impact of the anharmonicity on the formulas used in the proposed method. In particular, the normality of the distribution is not impacted  since it originates from the Central Limit Theorem (CLT, Eq. 5). What may be influenced is the value of the mean and the variance of the distribution  which will skew the temperature scale and possibly diminish the fidelity of our approximation of the thermal equilibrium state. Since both referees missed this point we have expanded our explanation of this issue to make it more clear to the reader.
Additionally, while we use lattice dynamics as an example in the text, the potential applicability of the proposed method is broader  it may be useful in other places where we need to reproduce the configuration of the system of atoms in thermal equilibrium in nonzero temperature. This fact is mentioned in the abstract but we will expand the conclusions by mentioning it there as well.
Indeed, the chosen system is not strongly anharmonic at T=300K. But still there is enough anharmonicity in the model to produce 5ps phonon lifetimes plotted in Fig. 6. Also, the Tersoff potential selected for the study is not a simplistic, harmonic, twobody potential. It is a published, effective model of interactions in the SiC compounds.
We would like to stress that the closeness of the prior distribution to the target (Figs 3 and 4) originates from the size of the system and careful selection of the prior generating algorithm (Eq. 6 and description in l. 172183). As we noted in the reply to the first referee the extreme cases of anharmonicity dominating the right hand side of the Eq. 4 for all, or most coordinates may be beyond the direct applicability of the proposed method. To illustrate the point, we have added to Figures 3, 4, 5, and 6 the calculations performed for higher temperatures (up to T=2000K) closer to the melting point of 3CSiC demonstrating effectiveness of the proposed approach even in high temperatures.
The computational cost is essentially proportional to the number of requested configurations plus necessary burnin samples (110, can be limited to 12 with careful selection of initial displacement variation). Due to the details of the MetropolisHastings algorithm this cost is independent of the acceptance ratio. Low acceptance ratio leads to low quality of the generated distribution, not a direct increase in computational cost. This increase stems from the fact that with low acceptance more samples are required for the reasonable fidelity of the produced distribution. In comparison with MD calculations, each generated configuration is equivalent to one time step in trajectory. However, in case of the DFTbased calculations, the MD procedure can be optimized by starting each step from the charge density/wave functions converged in the previous step. Due to the fact that samples generated by HECSS are independent, this optimization is not easily available in DFTbased calculations. This amounts to approximately twice as many electronic SCF steps per evaluated configuration. Thus, nconfigurations HECSS run is equivalent to approximately 2*(n+10) time steps of the MD calculation. In our experience this is not enough to provide even single, wellthermalized sample for n<500.
Regarding the parallel computation: In current implementation each configuration evaluation may be run on multiple cores but the sample generation is strictly serial. The nearindependence of generated samples provides opportunity for future splitting of the computation to multiple processes. Naturally, each temperature scan may be run as a separate process with full linear scaling.
We will add analysis of the computational cost of the HECSS approach to the final paragraph of the text.
We will expand the abstract to better reflect our focus  which is indeed, at this moment, on lattice dynamics applications. The other applications mentioned in the abstract are our suggestions of other fields where this type of procedure may be beneficial.
Following the comment of the first referee we have expanded the description of probable limitations of the proposed method (phase transitions, highly anharmonic systems).
We have replaced these imprecise phrases with quantitative description showing the speed of convergence and cited the appropriate literature.
The correlation between variances of the kinetic and potential distribution comes directly from energy conservation and statistical mechanics. Both energies are part of the Hamiltonian and sum up to the total energy. Thus, due to the energy conservation their variances should match. We have added the appropriate sentence to the discussion at the end of section 2.
Appropriate citation has been added to the list of references.
The independence from the system (supercell) size stems from the connection with the displacement distribution  it is our conclusion drawn from the experience gained during the development of the HECSS scheme. If the interactions are reproduced reasonably well in the small supercell (e.g. single crystallographic unit cell) the average size of thermal displacement is expected to be the same as in larger supercell due to the same energy per degree of freedom (i.e. temperature) and very similar shape of the potential. The independence and the practical ranges of the parameters cited in the text are derived from the multiple tests run during the development of the HECSS code. We have added a sentence explaining this property and rephrased the surrounding text to make this issue more clear to the reader.
The phonon frequencies presented in Fig. 5 are derived by fitting of a third order anharmonic model to both datasets and the frequencies are derived from this model. The lifetimes from the Fig. 6 are obtained from the same model using ALAMODE to compute anharmonic selfenergy and phonon lifetimes from the third order coefficients in the fit (using relaxation time approximation). We have corrected a misleading description of Figs 5 and 6 and expanded the description to make the point clear. We have also included the RMS differences between phonon frequencies derived by both methods.
The access to the vicinity of the zonecenter is limited by the supercell size used in the calculation. The closest point provided by the supercell used in the paper (5x5x5, 1000 atoms) and reciprocal space sampling grid (20x20x20) is located at 1/10 of the zone size from the center. All data between this point and the zone center are interpolated from the fitted force constant matrices in real space. We will add information about the reciprocal space sampling to the text. Additionally we have expanded the presented data to include more temperatures: 100K 600K and 2000K.
Figure 5 is intended to show small difference between frequencies computed from both data sets. We agree that the presentation in both Fig. 5 and 6 will benefit from such split and we have replaced both figures by separate panels containing data sets of the same size. We have also added additional temperatures  as mentioned above. The description has been modified appropriately.
We hope that the above explanations and corrections to the text make our paper convincing and clarify all the issues raised by Dr Wehinger.
Paweł T. Jochym, Jan Łażewski