Bachelor Thesisat

Faculty of SciencesDepartment of Physics

Equilibrium statistics of a slave estimator inLangevin processes

by

Robert Clemens Löffler

research atBiophysics Lab,

Department of Physics,Simon Fraser University, Burnaby, Canada

supervised by Prof. Dr. John Bechhoefer

referees:

Prof. Dr. Georg Maretand

Prof. Dr. John Bechhoefer

Konstanz, 2016

Konstanzer Online-Publikations-System (KOPS) URL: http://nbn-resolving.de/urn:nbn:de:bsz:352-2--1a7j9xn2llv3v5

3

Abstract

In statistical mechanics, the equilibrium susceptibility of a one-dimensional Langevin process 𝑥𝑡is linked to the system’s variance by the fluctuation-dissipation theorem. Another approach tocalculate it uses a slave estimator 𝜉𝑡, which was predicted to become power-law distributed and,as a result, to break down in potentials with concave regions [1, 2].

In this work, we have shown this breakdown experimentally by trapping charged 1.49 μm silicabeads, suspended in water, with a camera-driven feedback trap. By applying one-dimensionalelectric double-well potentials of the form 𝜙(𝑥) = −√𝐸𝐵𝑥2 + 1

4 𝑥4, we were able to smoothlychange the barrier height and thereby the properties of the concave region. Our results showa growing power-law tail in the slave estimator’s distribution, making it less precise and finallybreaking down for power-law exponents 𝛼∗ < 2, confirming the predictions.

We also investigated the power-law tail of the slave’s distribution with methods from extreme valuestatistics. For correlated, finite data series – the ones found in physical experiments – analysismethods are not well developed. Here, using slave-estimator data, we test some of the methodsproposed to handle such data [3–5].

German abstract / Deutsche Zusammenfassung

In der statistischen Mechanik ist die Suszeptibilität eines eindimensionalen Langevin-Prozesses 𝑥𝑡durch das Fluktuations-Dissipations-Theorem mit seiner Varianz verknüpft. Eine weitere Möglich-keit sie zu berechnen, bietet ein sogenannter Slave-Estimator 𝜉𝑡. In Potentialen mit konkavenBereichen wird dessen Verteilung allerdings hyperbolisch abfallend und führt zum Versagen dieserMethode [1, 2].

In der vorliegenden Arbeit wurde dieses Versagen experimentell untersucht. Dazu wurden ge-ladene, 1.49 μm große und in Wasser suspendierte Glasperlen in einer kameragesteuerten Feed-backfalle gehalten. Durch anlegen eines elektrischen Doppelmuldenpotentials der Form 𝜙(𝑥) =−√𝐸𝐵𝑥2+ 1

4 𝑥4, konnten wir dabei die Barrierenhöhe und dadurch die Eigenschaften des konkavenBereichs fliesend ändern. Unsere Ergebnisse zeigen einen wachsenden hyperbolisch abfallendenRandwahrscheinlichkeit der Slave-Estimator-Verteilung, der die Methode unpräziser macht undschließlich, bei einem Exponent 𝛼∗ < 2, zum Versagen führt.

Weiterhin haben wir die Slave-Estimator-Verteilung mit Mitteln der Extremwertstatistik ausgew-ertet. Für korrelierte endliche Datenreihen – wie sie in physikalische Experimenten zu findensind – sind entsprechende analytische Methoden noch nicht vollständig entwickelt. Mit unserenSlave-Estimator Daten, haben wir manche der Vorgeschlagenen Methoden [3–5] untersucht.

4

Acknowledgments

This work would not have been possible without the support of various people.

First of all I have to thank my supervisor Professor John Bechhoefer, who gave me lots of adviceand helped me all along the way. It was a great pleasure to be a member of his group and I amlooking back at a time that was full of instructive experiences.

Another great help in daily work was Momčilo Gavrilov, Phd student in the group, who introducedme to the lab and helped me to build the new setup. Many things would have taken me longerwithout his physical as well as technical advice.

Special thanks go to Professor Joel Stavans from the Weizmann Institute in Israel, who suggestedthese experiments and who visited us for a week.

Lastly I also have to thank Professor Georg Maret, for making the contact between John and me,which made this work possible for me.

5

Contents

1 Introduction and motivation 6

2 Theoretical background 72.1 Overdamped Brownian motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.2 Susceptibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.3 Breakdown of the slave-estimator method . . . . . . . . . . . . . . . . . . . . . . . 82.4 Extreme value theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

3 Experiment 113.1 Overall setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113.2 Design of the feedback trap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123.3 Position estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153.4 Analysis methods – statistics of correlated data sets . . . . . . . . . . . . . . . . . 16

4 Results 174.1 Slave-estimator breakdown . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174.2 Extreme value statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

5 Conclusion 22

6 Future work 23

List of Figures 24

List of Tables 25

Bibliography 26

6

1 Introduction and motivationThe susceptibility, the response of a physicalproperty to an external field, is a very basicphysical property and widely used. While itmostly relates to the electric or magnetic sus-ceptibility in classical mechanics, it is usedmore generally in statistical mechanics. If defin-ed as the response of a property ⟨𝑥⟩, it is alsorelated to that property’s variance by the fluc-tuation-dissipation theorem [6, 7].

The importance of the susceptibility has mo-tivated people to investigate into more waysto determine a process’s susceptibility, as dir-ect measurements of the variance can be proneto numerical errors. One variant, used mainlyin numerical simulations, is a slave estimator𝜉, which estimates the equilibrium susceptibil-ity of a Langevin process 𝑥. It is derived anddriven by the process, but has no influence onit. Dean et al. [1] as well as Drummond etal. in a following paper [2], have argued thatthis slave method works well for arbitrary con-vex potentials but becomes unstable in poten-tials with concave regions. These regions causelarge excursions in the slave estimator, thatare distributed with a power-law tail. If thattail becomes too heavy, the method breaksdown completely.

One such potential is a simple double-well po-tential. Motivated by the above-mentionedwork, Ben-Zion [8] performed a first set of ex-

periments, using a silica bead trapped by twooptical tweezers that created a double-well po-tential. These experiments showed a clear bre-akdown of the slave method for the observedpotential. However, in the approach taken inRef. [8], only a fixed potential could be ex-plored. In the work below, we present a dif-ferent experimental approach that gives con-siderable flexibility in choosing the form andthen varying smoothly the parameters describ-ing the potential.

Our approach is based on a feedback trap, de-scribed below that allows us to apply arbit-rary potentials to a trapped silica bead and toobserve not only the slave behavior for differ-ent barrier heights, but also for small externalfields, measuring the susceptibility for compar-ison in a very basic way.

The long times series of “instantaneous sus-ceptibilities” that we measure turn out to bepower-law distributed, and we take advantageof that fact to explore extreme-value statist-ics. Extreme value theory has been mainly inthe interest of insurance companies and otherfinancial applications in the past, but more re-cently it also came into focus of various physicsapplications [3–5]. In particular, the case ofcorrelated data and the finite amount of meas-urements are still an open field to be investig-ated further.

7

2 Theoretical background

2.1 Overdamped Brownianmotion

A colloidal particle suspended in a fluid is sub-ject to several physical forces: a Newtonianforce, due to the potential 𝜙(𝑥), a friction force𝐹𝑓 due to the damping by the surroundingfluid, and a random force 𝛿𝐹 due to the thermalkicks given to the colloid by the surroundingfluid molecules.

For one dimension, the balance of these forcesresults in the Langevin equation

𝑚 𝑥 = −𝜙(𝑥, 𝑡) + 𝐹𝑓( 𝑥) + 𝛿𝐹(𝑡) , (2.1)

which is an inhomogeneous stochastic differen-tial equation [6, 9].

The friction term

𝐹𝑓 = −𝛾 𝑥 = −6𝜋𝜂𝑟 𝑥 (2.2)

represents the Stokes resistance due to fluidflow around the sphere and depends on the vis-cosity 𝜂 of the fluid, the radius 𝑟 of the colloidand its speed 𝑥. For spheres close to a surface,the hydrodynamic forces are larger, increasingthe numerical factor 6𝜋 [10].

The random force 𝛿𝐹 is normally modelledas Gaussian white noise and should have zeromean and be uncorrelated at different times.Thus,

⟨𝛿𝐹(𝑡)⟩ = 0 (2.3a)

⟨𝛿𝐹(𝑡)𝛿𝐹(𝑡′)⟩ = 2𝛾𝑘𝐵𝑇 𝛿(𝑡 − 𝑡′) . (2.3b)

The noise variance 2𝛾𝑘𝐵𝑇 is set by the fluctu-ation-dissipation theorem [6, 9].

For an overdamped particle, which is the caseconsidered in this thesis, the effects of inertiamay be neglected and 𝑚 𝑥 set to zero. Fur-thermore, we denote the inverse temperature

𝛽 = 1/𝑘𝐵𝑇 and use the Einstein-Smoluchow-ski relation [11],

𝐷 = 𝑘𝐵𝑇𝛾 . (2.4)

We can then simplify Eq. (2.1) to

𝑥𝑡 = −𝛽𝐷𝜙′(𝑥𝑡) + 𝛽𝐷ℎ + 𝜂𝑡 , (2.5)

where we have included an external field ℎ.The fluctuating thermal term 𝜂𝑡 is modeledas white noise, with

⟨𝜂𝑡𝜂𝑡′⟩ = 2𝐷𝛿(𝑡 − 𝑡′) . (2.6)

The subscript 𝑡 emphasises that 𝑥𝑡 is a time-dependent process.

2.2 Susceptibility

In general, a susceptibility is the linear responsefunction of a physical property to an externalfield [6, 7]. Given Eq. (2.5), a static suscept-ibility ⟨𝜉⟩ for the average position ⟨𝑥⟩ in equi-librium in an external field ℎ can be definedas

⟨𝜉⟩ ≡ 𝜕𝜕ℎ∣

ℎ=0⟨𝑥⟩ . (2.7)

In thermal equilibrium, the distribution of 𝑥is governed by the Boltzmann distribution

𝑝(𝑥) = exp(−𝛽𝜙(𝑥) + 𝛽ℎ𝑥)∫d𝑥 exp(−𝛽𝜙(𝑥) + 𝛽ℎ𝑥) . (2.8)

h = 0.2h = 0 h = 0.1

Figure 2.1: As an external field “leans”against the potential, it gets tilted, shiftingthe mean position of a trapped particle.

2 Theoretical background 8

Calculating the average in Eq. (2.7) explicitly,we can derive the fluctuation-dissipation rela-tion

⟨𝜉⟩ = 𝛽 (⟨𝑥2⟩ − ⟨𝑥⟩2) , (2.9)

which connects the particle’s response to anexternal field to its fluctuations in equilibrium,here, the variance of its position [6, 7].

Another method to calculate the susceptibil-ity, often used in numerical simulations, is theslave estimator. Considering Eq. (2.7), one candefine a slave variable

𝜉𝑡 ≡ 𝜕𝜕ℎ∣

ℎ=0𝑥𝑡 . (2.10)

whose average should be again the susceptibil-ity [1].

Differentiating Eq. (2.5) by ℎ and using thedefinition in Eq. (2.10), leads to the slave pro-cess

𝜉𝑡 = −𝛽𝐷𝜙″(𝑥𝑡)𝜉𝑡 + 𝛽𝐷 , (2.11)

which is driven by the process 𝑥𝑡 but has noinfluence on it. The main advantage of thisapproach is its potential for greater accuracy.It avoids the subtraction of two possibly veryhigh numbers that can occour during the cal-culation of the variance. Also the uncertaintyof ⟨𝜉⟩, denoting the variance of the main pro-cess 𝑥𝑡, is given by the variance of 𝜉𝑡, whichis easier to calculate than the statistical uncer-tainties of ⟨𝑥2⟩ and ⟨𝑥⟩2 [1].

2.3 Breakdown of theslave-estimator method

The slave estimator works well for processes inarbitrary potentials, as long as they have a con-vex shape. This is true for many applications,and these benefit from the advanced charac-teristics of the slave estimator. However, asproposed by Dean et al. [1], the method canlead to problems when used for processes inpotentials with a concave region, for examplea double-well potential. The reasoning is that,in this case, there is a negative curvature 𝜙″(𝑥)

≈ −𝑔 implying that the slave estimator inEq. (2.11) grows exponentially

𝜉 ≈ 𝜉0 exp(𝛽𝐷𝑔𝜏) , (2.12)

while the particle is in the concave region. Giv-en that the process is expected to be memory-less, the time intervals 𝜏 spent in the concaveregion should follow an exponential distribu-tion

𝑝(𝜏) ≈ 𝜇 exp(−𝜇𝜏). (2.13)

Combining Eqs. (2.12) and (2.13) results in adistribution for 𝜉, with power-law tails

𝑝(𝜉) ≈ 𝛼∗

𝜉0( 𝜉

𝜉0)

−1−𝛼∗

(2.14)

for large 𝜉, with

𝛼∗ = 𝜇𝛽𝐷𝑔 . (2.15)

From a physical point of view, Eq. (2.14) im-plies that the slave estimator stays mostly be-low the average but has spikes that rise highabove the average. The spikes arise when thedriving process 𝑥𝑡 is within the concave re-gions of the potential 𝜙(𝑥). As discussed in[1] and [2], it is quite difficult in most cases,if not impossible, to analytically calculate theexact exponent 𝛼, as the time 𝜏 spent in theconcave region depends on the overall shape ofthe potential. The only physical constraint isthat 𝛼 > 1 with 𝛼 → 1 for 𝑇 → 0.

As discussed by Newman in [12], first two mo-ments of a power-law-tailed distribution aregiven by

⟨𝜉⟩ ≈ 𝛼∗

𝛼∗ − 1 𝜉0 , (2.16a)

⟨𝜉2⟩ ≈ 𝛼∗

𝛼∗ − 2 𝜉20 . (2.16b)

Notice that the second moment and hence thevariance of the susceptibility estimator divergeand become undefined for 𝛼∗ ≤ 2. Thus, theslave estimator method breaks down for thesecases. Several examples of this breakdown arediscussed analytically and numerically in [1]and [2].

9 2.4 Extreme value theory

Positions over time Slave estimator over timePotential and histogram

(a)

(b)

2/3 kT

Figure 2.2: Slave estimator in an (a) convex potential, fluctuating around a mean value, and ina (b) double-well potential, where the concave central region produces large spikes each time theparticle crosses the barrier.

2.4 Extreme value theory

In order to understand the behavior of theselarge spikes, one approach is to examine theirpeak values. Most scientists are familiar withthe central limit theorem, where the values ofindependent identically distributed (iid) vari-ables form (in most cases) a normal distribu-tion centered at the average value. In extremevalue theory, a similar procedure is used toexamine the extremes of iid variables. Histor-ically, interest in extreme value statistics hasbeen driven by its applications to the insur-ance industry and to the understanding of fin-ancial phenomena such as stock-market crashes.

The starting point of extreme value statisticsis to note that the overall shape of the value’sparent distribution is less important when itcomes to extreme excursions. Indeed, whatmatters is the decay law of its tail. Dependingon that tail, extreme value distributions canbe classified into three families:

CDF(𝑥) = exp(− exp(−𝑥)) (2.17a)

CDF(𝑥) = 0 (𝑥 < 0)exp(−𝑥−𝛼) (𝑥 ≥ 0) (2.17b)

CDF(𝑥) = exp(−(−𝑥)𝛼) (𝑥 < 0)1 (𝑥 ≥ 0) (2.17c)

for 𝛼 > 0. The distributions in Eqs. 2.17 arereferred to as the Gumbel, the Fréchet and theWeibull distributions, respectively. The Gum-bel distribution represents the extreme valuedistribution for a parent distribution whosetail decays faster than exponential, for examplea Gaussian. The Fréchet distribution, on theother hand, represents the case where the tailof the parent distribution decays as a powerlaw. An example of the latter is the Cauchydistribution [13].

These cases are of special interest for our ap-plication, as potentials with no concave regionlead to a susceptibility time series 𝜉𝑡 whosedistribution 𝑝(𝜉) decays more quickly than anexponential whereas potentials with concaveregions should lead to 𝑝(𝜉) that decay with apower-law tail [1, 2]. Thus, we expect that theextreme-value statistics should follow a Gum-bel distribution in the first case but a Fréchetdistribution in the second. Moreover, since wecan smoothly deform our potential from onecase to the other by reducing the barrier height𝐸𝐵, we can investigate the crossover betweenthese two universality classes of extreme valuedistributions.

Consider 𝑛 iid variables 𝑋1, ..., 𝑋𝑛. We candefine the probability of their maximum, with

𝐹 𝑛(𝑥) = Pr(𝑥 ≥ max𝑋1, ..., 𝑋𝑛) (2.18)

2 Theoretical background 10

2

1

0

prop

abili

ty p

(x)

1.51.00.50.0rescaled extreme values x / <x>

α = 1.5 α = 2.5

Figure 2.3: Mean scaled Fréchet distribu-tions, 𝑠 = 1/Γ(1 − 1

𝛼).

being the cumulative probability distributionfunction. If, for proper scaling factors 𝑎𝑛, 𝑏𝑛,this maximum value CDF converges under thelaw of large numbers, following

𝐹 𝑛(𝑎𝑛𝑥 + 𝑏𝑛) → 𝐺(𝑥) (𝑛 → ∞) , (2.19)

then 𝐺(𝑥) is an extreme value distribution ofone of the mentioned families [13, 14].

For a Fréchet distribution, these scaling factorsare

𝑎𝑛 = 𝑛1/𝛼, (2.20a)

𝑏𝑛 = 0 . (2.20b)

In the physical systems we study, the positions𝑥𝑡 are correlated and hence not iid. Althoughthere are no general results for the extremevalue distribution of correlated random vari-ables [15, 16], systems with weak correlationsare better understood. For example, if theautocorrelation of a series of length 𝑛 decaysexponentially with a correlation range 𝑛𝐶/2in both directions, then it behaves similarlyto the uncorrelated series obeying = 𝑛/𝑛𝐶points.

In a real experiment, the data are always fi-

nite. Efforts to better understand normaliza-tion and scaling of such finite size time serieswere made recently [3–5]. One approach is torescale by the distribution’s mean, which con-verges to the theory extreme value distributioneven when the block size is small. Another ad-vantage of this approach is that it is independ-ent of the correlation of the system, as is notneeded for the rescaling. This is useful for usbecause the relation = 𝑛/𝑛𝐶 is only approx-imate.

Given a parent power-law distribution of theform

PDF(𝑥) = 𝛼𝑥min

( 𝑥𝑥min

)−1−𝛼

, (2.21a)

CDF(𝑥) = ( 𝑥𝑥min

)−𝛼

, (2.21b)

with 𝑥 ≥ 𝑥𝑚𝑖𝑛 > 0 and 𝛼 > 0. The associatedFréchet distribution is

PDF(𝑥) = 𝛼𝑠 (𝑥

𝑠 )−1−𝛼

𝑒(−(𝑥/𝑠)−𝛼), (2.22a)

CDF(𝑥) = 𝑒(−(𝑥/𝑠)−𝛼), (2.22b)

with a scaling factor 𝑠. The mean is given by

⟨𝑥⟩ = 𝑠 Γ(1 − 1𝛼) . (2.23)

We can now compare the Fréchet distribution,scaled by its mean and using an estimated al-pha, to histograms of experimental data, scaledwith their means as well. We therefore we areable to avoid the dependency of for correl-ated data we study, as the mean for differentblock sizes includes the scaling 𝑎𝑛 [3, 4]. 1

1Another approach given in [3] is to rescale by thestandard deviation. However, as the variance of thedistributions is undefined for 𝛼 < 2, this approach

is not suitable in our case.

11

3 Experiment

3.1 Overall setup

In the experiment, I observe the one-dimensio-nal position of a particle, subject to Browni-an motion and a double-well potential, usinga feedback trap. The trap, discussed in moredetail in the next section, enables us to approx-imate various potentials in two dimensions, aslong as the first derivative is continuous andthe feedback gain is low. This is a significantimprovement over previous experiments on thesubject, where the double well potential wasfixed, given by the potentials of two opticaltweezers [8].

As the temperature is approximately constantwhile the experiment is running and as thesoftware loop uses units based on the camerapixels (px), it is worthwhile to rescale to di-mensionless parameters:

𝑥 = 𝑥𝑥0

, (3.1)

𝜙( 𝑥) = 𝜙(𝑥)𝑘𝐵𝑇 , (3.2)

where, usually, 𝑥0 = 5 px = 0.557(3) μm. Thesame is done for other quantities such as thebarrier height 𝐸𝐵 of the double-well in unitsof 𝑘𝐵𝑇 and the strength ℎ of an external fieldin units of 𝑘𝐵𝑇 /𝑥0. In these units, 𝛽 = 1.

This rescaling allows us to define the applieddouble-well potential

𝜙( 𝑥) = −√ 𝐸𝐵 𝑥2 + 14 𝑥4 (3.3)

in a dimensionless, temperature-independentform. Adding the potential ℎ 𝑥 of an externalfield, the resulting feedback force is

𝐹 ( 𝑥) = 2√𝐸𝐵 𝑥 − 𝑥3 + ℎ . (3.4)

To confine the particle in the transverse direc-tion, a simple harmonic potential is applied in

1.0

0.5

0.0 sla

ve ξ

t [µ

m²/

kT]

600550500time t [s]

<ξ>

-0.5

0.0

0.5pr

oces

s x t

[µm

]

600550500time t [s]

<x>

1.0

0.5

0.0

kT

-0.5 0.0 0.5position x [µm]

(a)

(b)

(c)

Figure 3.1: A potential (a) is applied to thetrap, giving position measurements (b) fromwhich the slave estimator (c) is generated.

the 𝑦-direction of the trap.

To measure the susceptibility of a trapped par-ticle, a weak external field ℎ is applied at thebeginning of each experiment and then reducedin discrete steps. At each step, we measurethe position average. This gives an estimatefor ⟨𝜉⟩ by directly fitting to ⟨𝑥⟩/ℎ for smallvalues of ℎ. Afterwards, the particle stays in

3 Experiment 12

a)

b)

x

ξ

x

ξ

t t+1 t+2

Figure 3.2: Illustration of (a) a normal Eulermethod and (b) the Euler-like method I used.

a static double-well potential with no externalfield, to get good estimates for the positionand its variance, along with enough statisticsfor the slave estimator. The latter is generatedfrom the position measurements by numericalintegration, using an Euler-like method basedon a time step, which is half of the time stepof the position measurements (Fig. 3.2)1,

𝜉𝑡+ 12

= 𝜉𝑡 + Δ𝑡2 𝐷𝜙″(𝑥𝑡) 𝜉𝑡 + Δ𝑡

2 𝐷 .

𝜉𝑡+1 = 𝜉𝑡+ 12

+ Δ𝑡2 𝐷𝜙″(𝑥𝑡+1) 𝜉𝑡+ 1

2+ Δ𝑡

2 𝐷 ,

The initial value of 𝜉𝑡 can be simply set tozero, as the estimate of its average convergesrapidly, when the slave method is valid.

We can then analyze the distribution 𝑝(𝜉) toinvestigate both the slope of its tail and itsextreme values for different barrier heights.

3.2 Design of the feedbacktrap

The feedback trap we used to observe a particlein a given potential can be viewed as a mi-croscope with attached electrodes. Figure 3.3shows the conceptual setup, consisting of illu-mination, the sample box, and a microscopewith camera read out. Within the sample box,silica beads settle near the bottom surface, be-cause of gravitational force, and can be moved

1In retrospect, it would have been better to useHeun’s method [17], as it has a better, second-orderprecision but still works without intermediate stepsof 𝑥, which are not provided by the measurements.

LED

CAMERA

V(F)

Cali-bration

COMPUTER

Vx

Figure 3.3: Conceptual experiment setup.

locally by two pairs of electrodes in 𝑥 and 𝑦direction.

Similar feedback traps have already been builtearlier by members of Professor Bechhoefer’sgroup [18, 19]; however, in order to increasethe feedback loop rate, we decided to build anew version.

Microscopy The observation of the bead isdone by darkfield microscopy: A powerful LEDlight source2, focused on the sample box, isused to illuminate the particle. Behind theobjective3, the main beam is blocked, and thescattered light from the particle is focused ontothe camera through the objective and a tubelens.

The sample box itself is a square box, contain-ing a solution of beads in distilled, deionizedwater4. Bottom to top, it consists of an alu-

2Thorlabs M660L4: Deep Red (660 nm) MountedLED, 1200 mA, 940 mW.

340x magnification air objective, 0.65 numerical aper-ture.

4𝑅 ≈ 18 MΩ cm−1, 0.22 µm particle filtered.

13 3.2 Design of the feedback trap

Figure 3.4: 3d-printed sample box with con-nected electrodes, cover and illumination.

minum plate with a hole for the objective, a mi-croscope coverglass, four platinum electrodes(assembled as two pairs on each of the oppos-ite sites of the square), and a 3d-printed waterreservoir5. All elements are separated by rub-ber layers and sealed by screwing the reservoiragainst the bottom plate. To prevent evap-oration, we clamp a microscope slide to thetop of the water reservoir box (Fig. 3.4). Wehave mounted the base plate on a piezoelec-tric stage, capable of moving in all directions6,to be able to find a particle in the horizontalplane and to bring it into focus.

The particles we observe are 1.49 μm siliconbeads, which are dense enough to settle nearthe cell’s bottom surface and perform two-dim-ensional Brownian movement there. They havea small surface charge, which prevents the beadsfrom sticking to the similarly charged substrate.We can also apply horizontal forces to the beads,using the electric field created by the electrodepairs. To have electrode voltages that are highenough, the analogue voltage output from thedata acquisition board7 connected to the com-puter needs to be amplified by an externalamplifier up to ±120 V.

To reduce stray light, the entire experiment iscovered by an opaque box.

5Lultzbot Taz4, using ABS filament.6Melles Griot NanoMax-TS and associated Melles

Groit controller.7National Instruments PCIe-6353, X Series DAQ

ExposureImage transfer

0 ms 2 ms

Calculation

Figure 3.5: Time diagram of the feedbackloop, with 300 μs of exposure, ca. 300 μs totransfer the images area of interest, ca. 500 μsfor calculation and a comparable big buffer forunexpected process interruptions by Windows.

Feedback loop The microscope image is cap-tured by a USB3-driven CMOS camera8 andtransferred to a nearby computer, where it isprocessed by a LabVIEW program that imple-ments a feedback loop.

In each iteration of the feedback loop, the part-icle’s position in the image is estimated, usinga mass centroid-algorithm based on the intens-ity of the pixels. Next, the forces in the 𝑥 and 𝑦directions are calculated, according to the po-sition of the bead in the applied potential andthe recent state of the measurement protocol.Using calibrations, the force is then convertedto 𝑥 and 𝑦 voltages and sent to the data ac-quisition board.

The data acquisition board also generates twotrigger signals: One is used to trigger the cam-era exposure, the other to trigger the softwareloop and the voltage update in the middle ofthe next exposure (Fig. 3.5). For this purpose,it is shifted by half the exposure time, relativeto the camera trigger.

The position, the iteration counter, and vari-ous other measured quantities and calibrationvariables of each iteration are saved to disk atthe same time for later evaluation.

In the new trap, I carried out an extensive coderefactoring to speed up sub components. Also,with the implementation of the hardware-timedsoftware loop, as mentioned earlier, we wereable to check if the correct timing was achievedspecifically for each iteration. Thus, we couldgather exact statistics of when and how long

8Basler acA800-510um, ace Series

3 Experiment 14

V t

Δ xt

V 0

⏟∼D

∼µ−1

D

Figure 3.6: Simplified model of the trap cal-ibration, assuming perfect measurement andsame weighting of previous time steps. Thefull algorithm can be found in [18].

delays occurred in the system. These improve-ments enabled us to run the new setup at 500 Hzinstead of 200 Hz, which was the maximum re-liable speed of the previous trap.

Higher frequencies, up to 1 kHz and beyond,are possible, but the loop timing is unstableon the Windows 10 machine in use. We triedto run the program on a Linux system with akernel patched to support hard real-time pro-cess scheduling, but unfortunately the only of-ficially distributed standalone version of Lab-VIEW for Linux lacks some of the main fea-tures we need9.

Running calibration To calculate the out-put voltage, we need to calibrate between elec-trode voltage and applied force on the bead.We expect a relation of the form

𝐕 = 𝝁−1𝐅 + 𝐕𝟎, (3.5)

where 𝝁 is the 2×2 mobility matrix, which de-pends on the surface charge of the silica beadand the electric field in the cell. The voltage𝐕𝟎, is due to offsets in the high-voltage ampli-fier and chemical reactions at the electrodes.

To calibrate these values, we compare the ob-served displacement of the particle (Δ𝑥, Δ𝑦)𝑇

to the expected displacement. Based on run-ning averages, the inverse mobility and the off-set voltage can be calculated for each time step.

9There is a version of LabVIEW for Linux with allfeatures, we need, but it is only distributed prein-stalled on NI real-time hardware.

The fluctuations observed around the expectedposition give an estimator for the diffusion ofthe bead. The calibration is done at each timestep, while the experiment is running, to reactto slow shifts in these values, caused by chem-ical reactions and thermal drifts. The overallprocess is described in detail by Gavrilov et al.[18], taking also the update delay and meas-urement noise into account.

If the observed displacement is much higherthan expected, a second particle may have ent-ered the area of interest. Likewise, if the ob-served intensity drops to zero, the observedparticle is presumed lost. In both cases, theiteration counter is reset to zero, the measure-ment protocol restarted, and the calibrationmechanism returned to predefined values.

Timescales Depending on the bead and dirtconcentration around the selected area of in-terest, these observation restarts can happenwithin minutes. However, with good samplepreparation, runs with observation lengths ofmultiple hours and even beyond one day with-out interruption are possible. In practice, thedata sets we used had 10 million or more points,corresponding to five or more hours’ worth ofdata.

Another important consideration is the wellseparation, as the process 𝑥𝑡 of the particleis weakly correlated. The correlation time 𝜏𝐶of the system can be interpreted as the timethe particle needs to explore the whole poten-tial. It grows quadratically with the well sep-aration, according to the bead’s mean-squareddisplacement, as well as with a rising barrierhigh. Typical correlation times varied from0.1–10 s depending on the parameters.

Autofocus loop Another software loop I im-plemented used the gradient of observed in-tensity to move the 𝑧 axis of the piezo stage.The goal is to maximize the intensity depend-ing on 𝑧 position to keep the particle in fo-cus over long time scales. In fact, we foundthat the mechanical drifts due to, for example,overnight temperature changes in the lab, were

15 3.3 Position estimation

small enough that we did not use this functionfor the data reported here.

3.3 Position estimation

In order to trap the particle in a given po-tential, its position must be estimated fromthe recorded camera image. The main issue inthis process was that we tried to minimize ourwell distance, to shorten the position’s correl-ation time. However, we discovered that ouralgorithm for position estimation has subtlebiases at the single-pixel scale.

The algorithm we mainly used is the mass-centroid algorithm, which calculates the centerof intensity for both lateral positions (𝑥,𝑦). Toget the algorithm to work properly, we neededto subtract a threshold from the image matrixto remove the background noise, which wasabout 20% in our case. The mass-centroid al-gorithm is known to be biased towards themiddle of the image if the background levelis non-zero. In addition, there is a pixel-scalebias [20]. Although these biases were not aproblem in earlier experiments, they were sig-nificant when we used well spacings on the or-der of a few pixels, as shown in Fig. 3.8.

As an alternative algorithm we tried to useLabVIEW’s built in circular-edge detection10.The results were even worse. Other alternat-ives with better accuracy and still acceptablespeed, are discussed in literature, as for ex-ample in [21], but unfortunately I did not haveenough time to try them.

All measurements series discussed in this thesisare therefore measured with the mass-centroidalgorithm. I chose a relatively large lengthscale of 𝑥0 = 5 px and a large enough areaof interest, to be in a regime where pixel bi-ases can be ignored. For future work, this isan important issue to address.

10IMAQ Find Circular Edge 3.vi

50403020100

px

Figure 3.7: Snapshot and cross section of theregion of interest, which is used to calculatethe mass centroid of intensity, after subtract-ing a threshold.

0.10

0.05

0.00

His

togr

am

-6 -3 0 3 6px

1.0

0.5

0.0

kT

Figure 3.8: Pixel bias in 𝑥 positions (bin size= 10−1); in the measurement shown, the re-gion of interest was too small and cut off non-zero pixels, increasing the bias seen in the his-togram.

3 Experiment 16

0.1

1

302520151050s

intra-well correlation

inter-well correlation

Figure 3.9: Autocorrelation function for𝐸𝐵 = 0.656(18) 𝑘𝑇 , showing intra-well andinter-well correlations.

3.4 Analysis methods –statistics of correlateddata sets

The positions we measured had mainly twocorrelation time scales, the “intra-well” cor-relation time of the particle within one welland the “inter-well” correlation time of theparticle passing back and forth between thetwo wells. For high barriers, we can identifythese two time scales in the autocorrelationfunction (Fig. 3.9). However, as the barrierdecreases to zero, these two time scales merge.A good way to define the correlation time forsuch systems is the integral of the autocorrela-tion function. For a perfect exponential decay,the integral equals the the inverse exponential

exponent, but it also is the proper way to es-timate the correlation time when the autocor-relation function has a more complicated timedependence [22].

Knowing the correlation time of the system isimportant, as the standard deviation for thecalculated moments depends on the numberof independent measurements = 𝑛/𝑛𝐶 [22],where the number of correlated measurementsresult from the correlation time and the usedtime step 𝑛𝐶 = 2𝜏𝐶/Δ𝑡. The factor of two canbe understood intuitively as reflecting that apoint in a time series is correlated both withits future and with its past. See Ref. [22] fora derivation.

The standard deviation of correlated data isdiscussed in the literature; an analytic methodis, for example, given in [23]. But for higher-order central moments, things become quitecomplicated. Another method that is used instatistics to estimate uncertainties is re-samp-ling, where data sets of the same length aregenerated by randomly drawing from the ori-ginal data set. Afterwards the re-sampled setsfor the same quantities are compared to eachother [24]. Using this approach for weakly cor-related data, we just have to re-sample with = 𝑛/𝑛𝐶 values, as the re-sampled data setswill be uncorrelated. This gives a comparat-ively easy method to estimate the uncertain-ties of different quantities, mean and variancein our case.

17

4 Results

4.1 Slave-estimator breakdown

As discussed in the theoretical background,the slave estimator works well as long as itis applied to a concave potential. Figure 4.1shows such a case for our potential, with 𝐸𝐵 =0. Although the parent distribution 𝑝(𝜉) is notGaussian, its tail still decays faster than expo-nential, allowing no big excursions from theexpected value.

However, when we add a barrier to our poten-tial, the particle can get into concave regionsof the potential. The time spent there is nearlyexponentially distributed, as predicted by [1]and clearly visible in Fig. 4.2. This leads to thepredicted high excursions in the slave estim-ator, as also shown in Fig. 4.2. This highly in-creases the uncertainty of the slave estimator,and, if the excursions become to high, causesa complete breakdown.

0.6

0.4

0.2

0.0

slav

e ξ

[µm

²/kT

]

200150100500time t [s]

(a) 𝜉𝑡 time series and mean

10-5

10-4

10-3

10-2

p(ξ)

0.80.60.40.20.0slave ξ [µm²/kT]

(b) 𝜉𝑡 histogram and mean

Figure 4.1: Slave estimator in a convex potential (𝐸𝐵 = 0, 𝑥0 = 5.50 μm). Notice that thehistogram tail decays faster than exponential.

10-6

10-4

10-2

p(τ)

10005000residence time τ [s]

µ = 0.00799(12) 1/s

(a) 𝜏 histogram

20

10

0

slav

e ξ

[µm

²/kT

]

3600350034003300time t [s]

(b) 𝜉𝑡 sequence

Figure 4.2: Slave estimator in a potential with concave region (𝐸𝐵 = 0.365 𝑘𝑇 , 𝑥0 = 5.85 μm);the distribution of time intervals 𝜏 spend in the concave region has an exponential tail, whichcauses the large spikes in the slave estimator.

4 Results 18

50

0

p(x)

[x1

0-3 ]

-1 0 1position x [µm]

EB = 0

10-8

10

-6

10

-4

10

-2

p(ξ)

0.01 0.1 1 10 100

slave ξ [µm²/kT]

α → ∞

50

0

p(x)

[x1

0-3 ]

-1 0 1position x [µm]

EB = 0.177(16) kT

10-8

10

-6

10

-4

10

-2

p(ξ)

0.01 0.1 1 10 100

slave ξ [µm²/kT]

α ≈ 6.5

50

0

p(x)

[x1

0-3 ]

-1 0 1position x [µm]

EB = 0.364(21) kT

10-8

10

-6

10

-4

10

-2

p(ξ)

0.01 0.1 1 10 100slave ξ [µm²/kT]

α ≈ 2.4

50

0

p(x)

[x1

0-3 ]

-1 0 1position x [µm]

EB = 0.656(18) kT

10-8

10

-6

10

-4

10

-2

p(ξ)

0.01 0.1 1 10 100

slave ξ [µm²/kT]

α ≈ 1.55

Figure 4.3: Comparison of different potentials and the power-law tails of the resulting slave-estimator distributions; notice that the histograms are plotted on a log-log scale.

19 4.1 Slave-estimator breakdown

0.6

0.4

0.2

0.0

Sus

cept

ibili

ty [µ

m²/

kT]

0.60.40.20.0

EB [kT]

β <x²> <ξ> <x> / h

α = 6.5

α = 2.4

α = 1.5

Figure 4.4: Behavior of different methods to estimate susceptibility.

𝐸𝐵 𝑥0 𝐷 𝜏𝐶 ⟨𝑥⟩/ℎ 𝛽⟨𝑥2⟩ ⟨𝜉⟩ 𝛼∗

[kT] [µm] [µm²/s] [s] [µm²/kT] [µm²/kT ] [µm²/kT]

0 0.5492(37) 0.2063(41) 1.66(5) 0.22(7) 0.2038(57) 0.205(3) —0.177(16) 0.5848(45) 0.2088(55) 2.40(5) 0.41(11) 0.3318(48) 0.335(5) 6.5(1.0)0.364(21) 0.5852(40) 0.2258(44) 2.53(5) 0.42(12) 0.3932(57) 0.398(10) 2.39(10)0.656(18) 0.5479(25) 0.2171(50) 4.80(5) — 0.4181(35) 0.533(44) 1.55(5)

Table 4.1: Results of the different measurement approaches for different barrier heights

4 Results 20

4

2

0

-2

<x>

0.080.060.040.020.00

h

Figure 4.5: Large uncertainties for averagepositions in external fields (here for 𝐸𝐵 ≈0.36𝑘𝑇 ). Measurement time = 60 s / datapoint.

Table 4.1 and Fig. 4.3 summarize the datafor the different potentials we measured. Weclearly see that the increasingly heavy tail ofthe slave-estimator distribution degrades thesusceptibility estimates as 𝐸𝐵 increases.

When we compare the different estimators forthe susceptibility (Tab. 4.1 and Fig. 4.4), wecan see that for a potential that is everywhereconvex, the uncertainty of the slave estimatoris lower than that of the variance, even thoughthe mean of the raw data is zero here (whichimproves the variance estimate). But whilethe uncertainty of the variance depends onlyon the effective number of measurements (𝜎2 ∼1/

√), the uncertainty of the slave estimator

increases as 𝑝(𝜉) acquires a power-law tail ofever decreasing exponent alpha.

Notice, also, that not only does the uncertaintyof the slave estimator grow, but we can seethat the slave estimator agrees with the sus-ceptibility as calculated from the variance with-in error bars for 𝛼 > 2 and disagrees for smal-ler 𝛼, showing the predicted breakdown

Another point is the relatively large error barsfor the ⟨𝑥⟩/ℎ measurements. The size of theerror bars is not due to an unreliable method,but due to the lack of statistics for non-zero ex-ternal fields in my experiment. Since each non-zero field required its own special data set, weused shorter time series (minutes, not hours)for the measurements shown in Fig. 4.5. Tak-ing into account the differences in time-series

𝛼∗ 𝑔 𝜙″(0)[kT/µm²] [kT/µm²]

6.5(1.0) 0.0076(15) 0.615(37)2.39(10) 0.0154(12) 0.881(38)1.55(5) 0.0216(15) 1.349(31)

Table 4.2: Comparison of 𝜙″(0) = √𝐸𝐵/2𝑥20

and 𝑔 = 𝜇/𝛼∗𝛽𝐷 for different barrier heights

length, we estimate that the direct methodfor measuring the susceptibility by imposinga field would lead to uncertainties comparableto those observed for the slave-estimator andvariance methods. But lack of time preventedus from confirming this point directly.

In Tab. 4.2, we can compare the curvature𝜙″(0) at the peak of the barrier with the “ob-served” curvature calculated from the time ser-ies. To evaluate the times 𝜇 within the con-cave region, we use the analytic form of theimposed potential, recording the entrance andexit times from the concave region. The es-timate of the exponent alpha is from the tailof p(xi). Intuitively one would assume, thatwhile passing the barrier, the particle spendsmost of its time at the top, where it experi-ences the weakest forces, which would implythat 𝑔 ≈ 𝜙″(0). In fact, 𝑔 and 𝜙″(0) differ bytwo orders of magnitudes. One explanation forthis discrepancy is that the curvature equals𝜙″(0) only at the top of the barrier. Perhaps,most of the time, the particle is near the edgesof the convex region, where the curvature goesto zero. Another noticeable property is the av-erage time of these extreme spikes ⟨𝜏ext⟩ =1/𝜇, which is around 100 s in our experimentand thus longer than the correlation time. Thisdemonstrates that these events should not beseen as smooth crossings of the barrier butas random stays that are distributed over thewhole barrier.

4.2 Extreme value statistics

Figure 4.6 shows the extreme value distribu-tions for two different barrier heights. The

21 4.2 Extreme value statistics

2

1

0

prob

abili

ty p

(max

(ξ))

1.51.00.50.0normalized maxima max(ξ)

Theory 5s blocks 10s blocks 50s blocks

(a) 𝛼=2.39(10) , 𝜏𝐶 =2.53(5) s , 𝑡𝑡𝑜𝑡 ≈5.5 h

2

1

0

prob

abili

ty p

(max

(ξ))

1.51.00.50.0normalized maxima max(ξ)

Theory 10s blocks 20s blocks 100s blocks

(b) 𝛼=1.55(5) , 𝜏𝐶 =4.80(5) s , 𝑡𝑡𝑜𝑡 ≈26 h

Figure 4.6: Extreme value distributions for two different barrier heights, plotted for block sizesblock of approximately 1, 2 and 5. We can see how the curves are converging towards the predictedFréchet distribution with increasing block size, but also are getting more noisy, due to the lack ofenough statistics.

histograms are generated by splitting up theoriginal time series into equal blocks of a cer-tain size and then extracting the maximumvalue of each block. The series of these max-ima are then divided by there mean value, tonormalize them to ⟨max(𝜉)⟩ = 1. Next, his-tograms, normalized in 𝑝(max(𝜉)), are gener-ated to compare the data to the theory distri-bution, which is also rescaled by the mean, asmentioned earlier.

Doing so, we need not account for the correla-tion of our data. Nevertheless, we can give aqualitative discussion on how fast this method

is converging, as we have an estimate for ourcorrelation time. By using the correlation time𝜏𝐶, we can define the number of independentmeasurements per block as

block = block ≈ 2 𝜏𝐶

𝑡block. (4.1)

Compared to the very small block sizes thatwe have used in Fig. 4.6, the histograms areconverging remarkably fast; however, we alsosee that they are becoming quite noisy.

22

5 Conclusion

We have shown experimentally the breakdownof the slave-estimator method in potentialswith concave regions, as predicted in [1, 2].In contrast to the preceding experiment [8],we were able to observe a series of differentbarrier heights for predefined double-well po-tentials and were also able to include smallexternal fields into the potential. The resultsshow a clear transition from the case of a fullyconvex potential, where the slave method isworking well, to the case of a small concaveregion, for which the slave estimator is becom-ing unstable, and finally to the case of a barrierhigh enough to cause the complete breakdownof the slave-estimator method.

One remarkable observation is, that even inour zero-mean potential, the slave estimatorgives a slightly better value for the suscept-ibility, than measuring the variance directly.A qualitative evaluation of the performance ofmeasuring the susceptibility directly by apply-ing small external fields cannot be given basedon this work, as the timescales used for this ap-proach were much shorter than the ones usedto measure the variance and the slave estim-ator.

We also showed that the slave estimator is

a good source of experimental data having apower-law-tail distribution. By changing theshape of the underlying potential, the power-law exponent 𝛼∗ can easily be influenced, al-lowing the investigation of extreme value stat-istics for different cases.

At the same time, it shows the difficulties of ex-treme value statistics, for correlated data andfinite sizes. One issue is that even when theform of correlations is well known, it is notclear how to analyze the approach to the Gum-bel or Fréchet distributions. For iid data, theanalysis is relatively straightforward. For cor-related data – even weakly correlated data –the required analysis is less clear. Simply di-viding into blocks gives only a qualitative ana-lysis. The methods developed in Refs. [3–5]are a potential way forward. On the otherhand, such methods should converge to thetheoretical extreme value distribution as fastas possible, to still provide good estimates inthe absence of enough statistics.

For the Fréchet distribution, scaling data bythe mean proved to be a good approach for par-ent distributions with a power-law exponentdown to 𝛼∗ ≥ 1 [3]. However, the field is stillopen for further investigation.

23

6 Future work

There are several things to be addressed in fu-ture work.

First of all, there are still many issues to beaddressed regarding the experimental setup.One important step is to implement a betteralgorithm for the position estimation. Also,the optics of the microscope could be improvedto use a higher magnification, allowing smal-ler potentials with shorter correlation times,so that the image of a particle is spread overmore pixels.

Another idea is to add an optical line trapwith optical tweezers, which would trap theparticle in the 𝑦 direction, allowing for a one-dimensional electronic feedback trap in 𝑥 dir-ection. The optical line trap would also trapthe particle at a certain height away from thebottom surface, which would solve issues withhaving too much dirt and other particles inthe cell, and the particle becoming eventually

stuck to the surface, while being trapped hori-zontally.

Changing to a software loop that is implemen-ted on a system that supports hard real-timescheduling could improve the loop rate further.Together with shorter correlation times, thiswould increase the statistics gathered in thesame time and thus improve the precision ofthe results. This would especially help withextreme value statistics, as a lot of data is re-quired to be in “universal” regimes.

On the other hand, there is of course also theinterest to investigate further into methods ofextreme value statistics, handling finite sizedata as well as possible. As already mentionedearlier, extreme value statistics on correlated,finite series of data is still a quite new field inphysics and needs to be covered more deeply,theoretically as well as experimentally.

24

List of Figures

2.1 Influence of an external field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.2 Slave estimator in a convex and double well potential . . . . . . . . . . . . . . . . . 92.3 Mean scaled Fréchet distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

3.1 Experiment example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113.2 Illustration of (a) a normal Euler method and (b) the Euler-like method I used. . . 123.3 Conceptual experiment setup. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123.4 3d-printed sample box with connected electrodes, cover and illumination. . . . . . 133.5 Time diagram of the feedback loop . . . . . . . . . . . . . . . . . . . . . . . . . . . 133.6 Simplified model of the trap calibration . . . . . . . . . . . . . . . . . . . . . . . . 143.7 Snapshot and cross section of the region of interest . . . . . . . . . . . . . . . . . . 153.8 Pixel bias in 𝑥 positions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153.9 Auto correlation function, showing intra-well and inter-well correlations . . . . . . 16

4.1 Slave estimator in a convex potential . . . . . . . . . . . . . . . . . . . . . . . . . . 174.2 Slave estimator in a concave region . . . . . . . . . . . . . . . . . . . . . . . . . . . 174.3 Comparison of different potentials and slave-estimator distributions . . . . . . . . . 184.4 Behavior of different methods to estimate susceptibility. . . . . . . . . . . . . . . . 194.5 Large uncertainties for average positions in external fields . . . . . . . . . . . . . . 204.6 Extreme value distributions for two different barrier heights . . . . . . . . . . . . . 21

25

List of Tables

4.1 Results of the different measurement approaches for different barrier heights . . . . 194.2 Comparison of 𝜙″(0) and 𝑔 for different barrier heights . . . . . . . . . . . . . . . . 20

26

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