Erwin Frey

Hauke Hinsch

Louis Reese

R. Jiang

English

Hinsch, Hauke

Reese, Louis

Frey, Erwin

Universität München: Elektronischen Publikationen

Hinsch, Reese, and Frey Reply: We have carefully re-
viewed our simulation algorithm and found that the origi-
nal implementation featured a bias on the passive part that
favored moves to the right by a small fraction of orderN1.
Repeating the simulations with the corrected algorithm, we
have now obtained density profiles in good agreement with
the simulations by Jiang et al. [1]. In particular, our revised
simulations also indicate that the mean field phase dia-
gram, calculated by our analytical approach [2], seems to
be exact.
Neither our corrected algorithm nor the simulations of
Jiang et al. [1] were designed to simulate the actual dy-
namics of the system, but are sequential updating algo-
rithms. Therefore, we decided to scrutinize these results
upon comparing them with simulations based on the
Gillespie algorithm, which gives an exact solution for
the stochastic dynamics; see, e.g., Ref. [3]. As shown in
Fig. 1(a), the obtained density profiles are in good quali-
tative agreement, but show slight quantitative deviations.
We have conducted simulations with different estab-
lished random number generators and have not found any
dependence of our results on the used sequence of random
numbers. We would like to point out, however, that devia-
tions between the results obtained from a Gillespie and an
updating algorithm for large systems are most likely due to
insufficient sampling in the updating algorithms. In order
to obtain valid averages, the time window for sampling
configurations has to be much larger than the typical
correlation time T. The latter is estimated as the mean
passage time of a particle through the active part, T  N.
As is well known, the expected error in the simulations
should then scale with the inverse square root of the
number of statistically independent configurations. In an
updating algorithm this implies that the number of moves
required for sampling should scale as N3 [see Fig. 1(b)].
This scaling is a peculiarity of the considered lattice gas, as
the diffusion rate scales with system size by construction.
This leads to the supplemental power of N in addition to
the usual N2 scaling for a diffusion process. We think that
in update algorithms this large number of relaxation steps
is important to note in the particular case of large system
sizes.
In conclusion, we attribute the deviation frommean field
theory claimed in the original Letter [2] to a faulty simu-
lation algorithm.
We thank R. Jiang and collaborators for their attentive
and careful revision of our original work.
Hauke Hinsch, Louis Reese, and Erwin Frey
Arnold Sommerfeld Center for Theoretical Physics (ASC)
and Center for NanoScience (CeNS)
Department of Physics
Ludwig-Maximilians-Universität München
Theresienstrasse 37, D-80333 München, Germany
Received 11 November 2010; published 18 February 2011
DOI: 10.1103/PhysRevLett.106.079602
PACS numbers: 64.60.Ht, 05.40.a, 05.60.k, 05.70.Ln
[1] R. Jiang et al., previous Comment, Phys. Rev. Lett. 106,
079601 (2011).
[2] H. Hinsch and E. Frey, Phys. Rev. Lett. 97, 095701 (2006).
[3] D. T. Gillespie, J. Comput. Phys. 22, 403 (1976); J. Phys.
Chem. 81, 2340 (1977).
FIG. 1 (color online). (a) Density profiles on the active part of
the lattice with N ¼ 500 and np ¼ 0:38 for various diffusion
strengths d. Data were acquired using our revised updating
algorithm (thick lines) and a Gillespie algorithm (thin lines),
of which we ran 100 simulations, each recording 100 indepen-
dent configurations of the particle density on the lattice. To
assure independence of consecutive configurations, we chose a
sampling time T ¼ N and equilibrated the system for a time
2N. (b) Number of updates as a function of system size in the
Gillespie algorithm. After an equilibration of the system (np ¼
0:38 and d ¼ 0:65) for a time 5N, the number of updates
performed in the Gillespie algorithm were counted over a time
interval 15N, and from that the average number of updates in the
sampling time T ¼ N were determined. This shows that in an
update algorithm the number of updates has to grow like N3 for
the observation time to be larger than the correlation time.
PRL 106, 079602 (2011) P HY S I CA L R EV I EW LE T T E R S
week ending
18 FEBRUARY 2011
0031-9007=11=106(7)=079602(1) 079602-1  2011 American Physical Society


Comment on "Bulk-Driven Nonequilibrium Phase Transitions in a Mesoscopic Ring" Reply

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Hinsch, Reese, and Frey Reply:

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Hinsch, Reese, and Frey Reply: We have carefully re-viewed our simulation algorithm and found that the origi-nal implementation featured a bias on the passive part thatfavored moves to the right by a small fraction of orderN1.Repeating the simulations with the corrected algorithm, wehave now obtained density profiles in good agreement withthe simulations by Jiang et al. [1]. In particular, our revisedsimulations also indicate that the mean field phase dia-gram, calculated by our analytical approach [2], seems tobe exact.Neither our corrected algorithm nor the simulations ofJiang et al. [1] were designed to simulate the actual dy-namics of the system, but are sequential updating algo-rithms. Therefore, we decided to scrutinize these resultsupon comparing them with simulations based on theGillespie algorithm, which gives an exact solution forthe stochastic dynamics; see, e.g., Ref. [3]. As shown inFig. 1(a), the obtained density profiles are in good quali-tative agreement, but show slight quantitative deviations.We have conducted simulations with different estab-lished random number generators and have not found anydependence of our results on the used sequence of randomnumbers. We would like to point out, however, that devia-tions between the results obtained from a Gillespie and anupdating algorithm for large systems are most likely due toinsufficient sampling in the updating algorithms. In orderto obtain valid averages, the time window for samplingconfigurations has to be much larger than the typicalcorrelation time T. The latter is estimated as the meanpassage time of a particle through the active part, T  N.As is well known, the expected error in the simulationsshould then scale with the inverse square root of thenumber of statistically independent configurations. In anupdating algorithm this implies that the number of movesrequired for sampling should scale as N3 [see Fig. 1(b)].This scaling is a peculiarity of the considered lattice gas, asthe diffusion rate scales with system size by construction.This leads to the supplemental power of N in addition tothe usual N2 scaling for a diffusion process. We think thatin update algorithms this large number of relaxation stepsis important to note in the particular case of large systemsizes.In conclusion, we attribute the deviation frommean fieldtheory claimed in the original Letter [2] to a faulty simu-lation algorithm.We thank R. Jiang and collaborators for their attentiveand careful revision of our original work.Hauke Hinsch, Louis Reese, and Erwin FreyArnold Sommerfeld Center for Theoretical Physics (ASC)and Center for NanoScience (CeNS)Department of PhysicsLudwig-Maximilians-Universita¨t Mu¨nchenTheresienstrasse 37, D-80333 Mu¨nchen, GermanyReceived 11 November 2010; published 18 February 2011DOI: 10.1103/PhysRevLett.106.079602PACS numbers: 64.60.Ht, 05.40.a, 05.60.k, 05.70.Ln[1] R. Jiang et al., previous Comment, Phys. Rev. Lett. 106,079601 (2011).[2] H. Hinsch and E. Frey, Phys. Rev. Lett. 97, 095701 (2006).[3] D. T. Gillespie, J. Comput. Phys. 22, 403 (1976); J. Phys.Chem. 81, 2340 (1977).FIG. 1 (color online). (a) Density profiles on the active part ofthe lattice with N ¼ 500 and np ¼ 0:38 for various diffusionstrengths d. Data were acquired using our revised updatingalgorithm (thick lines) and a Gillespie algorithm (thin lines),of which we ran 100 simulations, each recording 100 indepen-dent configurations of the particle density on the lattice. Toassure independence of consecutive configurations, we chose asampling time T ¼ N and equilibrated the system for a time2N. (b) Number of updates as a function of system size in theGillespie algorithm. After an equilibration of the system (np ¼0:38 and d ¼ 0:65) for a time 5N, the number of updatesperformed in the Gillespie algorithm were counted over a timeinterval 15N, and from that the average number of updates in thesampling time T ¼ N were determined. This shows that in anupdate algorithm the number of updates has to grow like N3 forthe observation time to be larger than the correlation time.PRL 106, 079602 (2011) P HY S I CA L R EV I EW LE T T E R Sweek ending18 FEBRUARY 20110031-9007=11=106(7)=079602(1) 079602-1  2011 American Physical Society

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