Exploration and Simulation of Entropic Forces

Wieger Steggerda 1

Supervised by Erik Verlinde 2

Insitute for Theoretical Physics

University of Amsterdam

1 wieger.steggerda@gmail.com2 e.p.verlinde@uva.nl

Bachelor thesis Physics, extent 12 EC, carried out between May 1 and August 31 2010.

Revised in July 2011.

1

Abstract

January this year (2010), Erik Verlinde publicated his paper, Onthe Origin of Gravity and the Laws of Newton[1]. In this paper theDutch professor from the Institute for Theoretical Physics Amsterdam(ITFA), and also supervisor of this thesis, suggests that gravity isan entropic force. Until this day there is no other theory which hasa reasonable explenation on why gravity excists, only the eects itcauses. In this project multiple entropic forces will be looked into,from which one specic. Elasticity will be discribed with probabilitytheory and simulated. Concluding that computers are very suit forcalculating and visualising entropic forces. Furthermore entropic forcesare able to give a very smooth and consistent force which makes thema possible candidate for the origin of fundamental forces.

Samenvatting (Dutch abstract)

Begin dit jaar (2010) publiceerde Erik Verlinde het artikel, On theOrigin of Gravity and the Laws of Newton[1]. In dit artikel suggereertde professor van het Instituut voor Theoretische Fysica Amsterdam(ITFA), tevens begeleider van dit project, dat zwaartekracht een en-tropische kracht is. Tot op heden is er nog geen andere goede theo-rie die de oorsprong van zwaartekracht bloot legt, slechts beschrijvin-gen van het eect. In deze scriptie worden verschillende entropischekrachten bekeken waarvan een in speciek. Met behulp van stochastieken computersimmulaties zal elasticiteit worden beschreven. Met alsconclusie dat de computer erg geschikt is voor het berekenen en visu-aliseren van entropische krachten. Verder kunnen entropische krachteneen erg gladde en consistente kracht op leveren wat ze een mogelijkekandidaat maakt voor de oorsprong van fundamentele krachten.

2

Contents

1 Introduction Into Entropic Forces 4

1.1 Examples of Entropic Forces . . . . . . . . . . . . . . . . . . . 51.1.1 Pressure . . . . . . . . . . . . . . . . . . . . . . . . . . 51.1.2 Osmosis . . . . . . . . . . . . . . . . . . . . . . . . . . 51.1.3 Hydrophobic Eect . . . . . . . . . . . . . . . . . . . . 6

2 Ideal Chain 7

2.1 Elasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.2 Ideal Chain Model . . . . . . . . . . . . . . . . . . . . . . . . 9

2.2.1 Probability Density Function of a Random Vector . . . 92.2.2 Expectation Value of a Random Vector . . . . . . . . . 102.2.3 Variance of a Random Vector . . . . . . . . . . . . . . 102.2.4 Central Limit Theorem in Random Vectors . . . . . . . 102.2.5 The Probability Density Function of the End to End

Vector . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.2.6 The Expectation Value of the End to End Vector Length 12

2.3 Elasticity Explained With the Ideal Chain Model . . . . . . . 13

3 Simulating the Ideal Chain 14

3.1 Input and Output of the Simulation . . . . . . . . . . . . . . . 153.2 The results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

4 Conclusion and Discussion 17

A Mathematical Probabilities 22

A.1 Probability Density Function . . . . . . . . . . . . . . . . . . . 22A.2 Expectation Value . . . . . . . . . . . . . . . . . . . . . . . . 22A.3 Variance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22A.4 Normal Distribution . . . . . . . . . . . . . . . . . . . . . . . 23

A.4.1 The derivative of the normal distribution . . . . . . . . 24A.4.2 Integrating the normal distribution . . . . . . . . . . . 24A.4.3 The expectation value of the normal distribution . . . 25A.4.4 The variance of the normal distribution . . . . . . . . . 25

A.5 Central Limit Theorem . . . . . . . . . . . . . . . . . . . . . . 26

B Pseudo Code For Simulation of the Ideal Chain 28

B.1 Creating a Random Vector . . . . . . . . . . . . . . . . . . . . 28B.2 The Basic Algorithm . . . . . . . . . . . . . . . . . . . . . . . 29B.3 Speeding Up the Basic Algorithm . . . . . . . . . . . . . . . . 30

3

1 Introduction Into Entropic Forces

Back in 1915, Einstein revolutionized physics with the general theory of rel-ativity. Because the gravitational force on an object is proportional to themass m and F = ma, the gravitational acceleration doesn't depend on m.Einstein therefor stated that gravity is a geometric phenomenon; that grav-ity is a mere ctitious force. Nearly a century after (2010), Erik Verlindepublished On the Origin of Gravity and the laws of Newton[1]. In this paper,the professor states that gravity is not a fundamental but an entropic force.This new insight not only describes gravity, it also explains why it exists.

In physics, especially theoretical physics, there isn't a lot of researchinto entropic forces. Entropic forces aren't fundamental and therefor notvery interesting to a purist. However, the new perspective on gravity makesentropic forces a lot more interesting.

Entropic forces are caused by the second law of thermal dynamics. Thelaw that states that entropy always increases. The force is present in systemswere some macro states are more likely to occur, while there is no preferredmicro state, or conguration. When, for example, N coins are ipped, andfor every heads a line of 1 cm is drawn, the line's size will be somewhereabout N/2 cm long, if N is large enough. There is, of course, no time andforce in this system, but it shows that the length of the line, the macro state,is likely to be somewhere around N/2 cm, whilst the results of the coins, themicro state, is completely random.

Let's take a look at the thermodynamic identity,

dE = TdS − PdV + µdN

where E is the energy, T the temperature, P the pressure, V the volume, µthe chemical potential and N the number of particles. The last two termscan often be neglected,

dE = TdS

If the entropy depends on some spatial variable x, the derivative to thatvariable can be made so that the left side of the equation will result in aforce,

F (x) =dE

dx= T

dS

dx(1)

This formula is the essence of an entropic force, notice the proportionalitywith the temperature.

4

Figure 1: Schematic representation of osmosis. The water ow to the highersolution causes the water to rise, which causes pressure on the membrane.

1.1 Examples of Entropic Forces

1.1.1 Pressure

Pressure is probably the most easy entropic force one can think about. In agas it is very unlikely that all molecules are close to each other, even if themolecules wouldn't interact with each other at all. The consequence is thatgasses expand and thus produce pressure. The ideal gas law shows that theforce is proportional to the temperature, as predicted in the introduction,

P =NkT

V

1.1.2 Osmosis

Osmosis is an eect that, like pressure, is caused by the small probability tohave high and low concentrations close to each other. Osmosis is the diusionof molecules from a place of high concentration to a place of low concentrationuntil the concentration on both sides is equal. When a partially-permeablemembrane has a solution of dierent concentrations on both sides, water willow trough the membrane, from the low to the high concentration, trying tomake the concentrations equal. In Figure 1 the water level rises and thereforcreates a pressure on the membrane.

5

Figure 2: The hydrophobic eect on grass. Photo taken by Michael Apel.

1.1.3 Hydrophobic Eect

The hydrophobic eect occurs when water comes into contact with a non-polar molecules, such as oil. It causes water to separate from oil. Someplants, especially the lotus ower, have hydrophobic leaves. Water cannotattach to the leaves and falls of the leave, making it hard for fungi to growon the leaves. A H2O molecule can accept and donate two bonds with otherwater molecules, making a network of bonds. Near a hydrophobic substance,the water bonds are broken, creating less congurations for the network toappear in. Therefor the multiplicity is the highest if there is the least amountof contact between the water and the hydrophobic substance.

6

2 Ideal Chain

A polymer, see Figure 3, is a molecule that consists of a chain of similarnodes called monomers. There are natural polymers like starch, DNA andrubber from the rubber tree; and there are synthetic polymers like plastic,nylon and synthetic rubber. A common feature to polymers is elasticity.Since dierent polymers consist of dierent monomers, it is likely that elastic-ity isn't emerged by the building materials but rather by the common shapeof the polymers. The structure can be captured in vectors.

Let

N + 1 be the number of monomers;

ri with i ∈ 1, . . . , N be the vector between monomers i and i+ 1;

l be the distance between monomers. So |ri| = l for all i;

R =∑

i∈1,. . . ,N ri be the end to end vector. It is the vector fromthe rst to the last monomer.

In this chapter it is essential to have a good understanding of probabilitytheory. In Appendix A is a quick overview of probability theory used in thisthesis.

2.1 Elasticity

In the previous section the end to end vector is dened as,

R =∑

i∈1,..,N

ri

Imagine the polymer to be stretched completely so that R = Nl, and then isput into a heath bath. Because of the interactions with the heath bath, theconguration (conformation in biology and chemistry) of the polymer will bemodied a bit. If a completely unfolded polymer is modied a bit, it will endup a bit shorter. In other words, there is an entropic force that contracts thepolymer if its completely stretched. Also if R = 0, there will be a force thatenlarges the molecule. Assuming this force, F (R), is continues, there mustbe a point where there is no force at all.

7

Figure 3: Representations of a polymer. Created with a self written computerprogram described in Section 3.

8

2.2 Ideal Chain Model

The ideal chain model describes a polymer in which there is only a forcebetween the adjacent monomers. Therefor, in a polymer in thermal equi-librium, the vectors ri are independent for all i ∈ 1, .., N and have nopreference for certain angles and thus there is no preferred micro state. Let'slook at the components of the end to end vector R.

Rk =N∑i=1

rki , k ∈ x, y, z

These components are the sum of independent random variables. So if themean, 〈rki〉, and variance, σ2

ki, are nite for all i, the Central Limit Theorem

A.5 can be applied. This means the x, y, z components of R will be nor-mally distributed if N is large enough. The next subsections are devoted tocalculate the mean and variance of random vectors and to calculate a proba-bility density function for R. As stated earlier a random vector doesn't havea preference for certain angles. In other words, this problem has a spheri-cal symmetry. Because r has a xed length, it is useful to transition intospherical coordinates.

rx = l sin θ cosφry = l sin θ sinφrz = l cos θ

With φ ∈ [0, 2π) and θ ∈ [0, π]. With surface element: dA = l2 sin θdθdφ.

2.2.1 Probability Density Function of a Random Vector

To calculate the probability density function fr of a random vector r it isimportant to know the domain of this function. Since the vector has xedlength l the domain of fr is the sphere with radius l, lS2. If two surfaces arethe same size the chance that they are hit by a random vector is the same.For this reason the surface element dA = l2 sin θ · dθdφ is used in fr,

Pr(A) =

ˆA

l2 sin θ

4πl2dθdφ ⇒ Pr(lS

2) = 1

Where Pr(A) is the chance a random vector hits the surface A.This makes the probability density function,

fr(r) =sin θ

4π

9

2.2.2 Expectation Value of a Random Vector

Due to the spherical symmetry 〈rx〉 = 〈ry〉 = 〈rz〉. Because rz = l cos θdoesn't depend on φ, 〈rz〉 is the least dicult to work out. Calculating themean, Eq. (6), is easy now,

〈rz〉 =

ˆlS2

rz · f(x) · dθdφ

=

ˆlS2

l cos θ · sin θ

4π· dθdφ

=l

8π

ˆ 2π

φ=0

ˆ π

θ=0

sin 2θ · dθdφ

=l

4

ˆ π

0

sin 2θ · dθ = 0

Not very surprising for a spherical symmetric problem.

2.2.3 Variance of a Random Vector

Now for a more interesting value, the variance, Eq. (8),

σ2z =

⟨r2z

⟩− 〈rz〉2 =

⟨r2z

⟩=

=

ˆlS2

l2 cos2 θ · sin θ

4π· dθdφ

=l2

2

ˆ π

0

dθ · 1

4(sin 3θ + sin θ)

=1

3l2

This result is obtained by writing out the trigonometric functions into expo-nents.

2.2.4 Central Limit Theorem in Random Vectors

Because the spherical symmetry,

〈rx〉 = 〈ry〉 = 〈rz〉 = 0

σ2x = σ2

y = σ2z =

1

3l2

10

It turns out that both the expectation value and variance are indeed nite,so we can apply the Central Limit Theorem on,

Rk =N∑i=1

rk with k ∈ x, y, z

The theorem states that the probability density function fRk of Rk, is nor-

mally distributed with mean µk = N · 〈rk〉 = 0 and variance σ2k = Nl2

3.

So,

fRk =

1√2πNl

2

3

e−3R2k

2Nl2

2.2.5 The Probability Density Function of the End to End Vector

Now all parts are ready to be assembled into the probability density functionof R. The domain of fR(R) is of course R3. Let´s dene a volume V =∆X × ∆Y × ∆Z where ∆X,∆Y and ∆Z are intervals. The probability ofnding R in V is equal to the probability of nding Rx in ∆X and Ry in∆Y and Rz in ∆Z,

PR(V ) = PRx(∆X) · PRy(∆Y ) · PRz(∆Z) ⇒

ˆV

fR(R)dv =

ˆ∆X

fRx(x)dx ·ˆ

∆Y

fRy(y)dy ·ˆ

∆Z

fRz(z)dz

=

ˆ∆X

ˆ∆Y

ˆ∆Z

fRx(x) · fRy(y) · fRz(z) · dxdydz

=

ˆV

fRx(x) · fRy(y) · fRz(z) · dv ⇒

fR(R) = fRx(Rx) · fRy(Ry) · fRz(Rz)

fR(R) =

(3

2πNl2

) 32

e−3(R2

x+R2y+R

2z)

2Nl2

Notice that fR only depends on the magnitude R of R,

fR(R) =

(3

2πNl2

) 32

e−3R2

2Nl2 =

(1

2πσ2

) 32

e−R2

2σ2 (2)

11

2.2.6 The Expectation Value of the End to End Vector Length

Finally it is possible to calculate the expectation value of |R| by using σ2 =13Nl2 in fR(R),

〈|R|〉 = 〈R〉 =

ˆR3

r · fR(r) · dv

=

ˆ 2π

0

ˆ π

θ

ˆ ∞0

rfR(r) · r2 sin θdθdφdr

Using the derivative rfR(r) = −σ2f ′R(r), Eq. (10),

〈R〉 = −4πσ2

ˆ ∞0

r2f ′R(r)dr

Integration by parts,

〈R〉 = −4πσ2r2fR(r)∣∣∞0

+ 4πσ2

ˆ ∞0

2rfR(r)dr

= 0 + 4πσ2

ˆ ∞0

2rfR(r)dr

And again using the derivative f ′R(r),

〈R〉 = −8πσ4

ˆ ∞0

f ′R(r)dr

= −8πσ4fR(r)∣∣∞0

= 2πσ4

(3

2πNl2

) 32

= 2

√2

πσ

=

(2

√2

3π

)√Nl ≈ 0.92

√Nl

The result predicts that a polymer in a heath bath will shrink to(

2√

23π

)√Nl

for large N , which is only a fraction of its stretched size Nl.Some sites on the Internet[2] and other information sources use the root

mean square,√〈R2〉, to calculate the expectation value 〈R〉. The root mean

square is however an approximation and doesn't give the exact answer.√〈R2〉 =

√〈R2

x〉+⟨R2y

⟩+ 〈R2

z〉 =√Nl

12

2.3 Elasticity Explained With the Ideal Chain Model

With the probability density function of the end to end vector in our grasp,we are but one step away to determine the dynamics of the ideal chain. Let'stake a look at the formula for entropic forces, Eq. 1,

Fr(r) =dE(r)

dr= T

dS(r)

dr

Using the denition for entropy S = kblogΩ, where kb is the Boltzmannconstant and Ω the number of micro states,

dS(r) = kb log Ω(r + dr)− kb log Ω(r) = kb logΩ(r + dr)

Ω(r)

Since this isn't a discrete model the multiplicity, Ω(x), is innite for almostall of the cases. There is a way to deal with this, the probability densityfunction is proportional to the multiplicity. So we may conclude,

dS(r) = kb logΩ(r + dr)

Ω(r)= kb log

fR(r + dr)

fR(r)

Now by substituting fR(x),

dS(r) = kb log(e

−1

2σ2(r2+2rdr+dr−r2)

)= kb log exp

−2rdr

2σ2= − kb

σ2rdr

Finally the entropic force can be calculated,

Fr(r) = TdS

dr(r) = −kbT

σ2r = −3kbT

Nl2r (3)

And because of the spherical symmetry there are no angular terms,

F (r) = −3kbT

Nl2r

This is of course Hooke's law, the formula for a harmonic oscillator where3kbTNl2

is the force constant. This means that when a mass M is put at the endof a polymer, and the rst monomer is in a xed position, the polymer willgo into a spring like motion with a frequency of,

ω =

√3kbT

Nl2M

Also note that the force is indeed proportional to the temperature, as seenin Eq. 1 on page 4.

13

3 Simulating the Ideal Chain

It would be great to see this force in action to see how the polymer reactsto a heath bath and how this leads to an oscillation, Eq. (3). Luckily this ispossible with computer simulations. In this section a computational methodis explained which is used to simulate an ideal chain. The simulation programis written in c++. OpenGL is the graphics library used to render the graphicsof the ideal chain. The graphics don't add anything to the measurement, butgives insight into the dynamics of the ideal chain, and makes it possible tocheck if the program runs naturally, so that there are no unrealistic eventsoccurring in the chain. The rigid rods of xed length between the monomersare replaced by springs with rest length l = 1, for it would be very dicultto program and computational intensive to use rigid rods.

To simulate the ideal chain, it has to be translated to a physical repre-sentation. Let

ρi,vi,ai respectively be the absolute position, the velocity and theacceleration of the i'th monomer with i ∈ 0, .., N ;

m be the mass of the monomers;

M be the mass of the last monomer;

k be the spring constant between the monomers.

Remember that N + 1 is the number of monomers. That's why the rangeof i goes from 0 to N . To calculate ρi, the velocity needs to be calculate;to calculate vi the acceleration needs to be calculated. The rst monomeris stationary, and the last one has mass M . For the others the accelerationonly depends on the two springs that connect the monomer. Remember thatthe springs have stationary for length l = 1.

a0 = 0

aN = − k

M(rN−1 − rN−1)

ai = − km

(ri−1 − ri−1 + ri − ri)

Where ri = ρi+1 − ρi. And v = v|v| for every vector.

The basic algorithm for the simulation is to make small time steps, framesof length dt, and update the positions of the monomers,

ri(t+ dt) = ri(t) + vi(t)dt

14

The velocities are also updated every frame by the acceleration. There isalso a term added to simulate a heath bath. In a heath bath, energy istransported by colliding molecules. To simulate this, the velocities of themonomers are changed a bit per time step by a impulse dp.

vi(t+ dt) = vi(t) + ai(t)dt+dp

m

Where dp is a random vector of length dp.Take a look at Appendix A for a more detailed discussion about this

algorithm.

3.1 Input and Output of the Simulation

With the use of OpenGL it is possible to render the Ideal Chain every frame,and thus create a short video of the simulation. A video is put up on YouTubeat: http://www.youtube.com/user/WiegerSteggerda.

The output of the program is a data plot, with the length of the endto end vector set out against time. If indeed the simulated ideal chain actsas predicted in Section 3, it will result in an sinus, but length can only bepositive so it will be an absolute sinus.

The inputs parameters for the simulation are:

N , the number of monomers;

m, the mass of one monomer;

M , the mass of the last monomer;

dt, the duration of one time step;

k, the spring constant between the monomers;

dp, the impulses of the colliding atoms in the heath bath;

Q, the ratio of impulses dp per time step per monomer.

To get the most accurate results a large N , k; and small dp, dt are required.The only problem is that these requirements add signicantly to the compu-tational time, so a good balance needs to be found.

15

3.2 The results

Let's try to link the theory to the newly dened parameters for the simulation,so that a prediction can be made on how the ideal chain behaves underdierent variables/inputs. The theory states that the angular frequency is,

ω =

√3kbT

Nl2M

This means that when the temperature T is quadrupled, the frequency isdoubled. When the mass M or the number of monomers N is quadrupled,the frequency is halved. The temperature isn't really directly visible in thesimulation, but there are variables that represents the heath bath, namely dp

and Q. Both kbT and (dp)2

2mhave the dimension of energy, so they are probably

proportional. This means that when dp is doubled, T is quadrupled, and thusthe frequency also doubles.

The next three graphs have the number of frames passed, or time, on thex-axis, and the end to end distance ratio |R|

Nlon the y-axis. The rst graph,

see Figure 4, tests the dependency of dp. For the three plots dp1 = 0.001,dp2 = 0.002 and dp3 = 0.004. The expected result was that the frequencydoubled if dp doubled. Because two periods of the dp2 plot t into one periodof dp1; and two periods of the dp3 plot t into one period of dp2, the frequencyis indeed doubled.

The second graph, Figure 5, tests the eect of quadrupling M . M1 =2500, M2 = 10000 and M3 = 40000, clearly the period of M1 ts twice in theperiod of the M2 case, and the period of M2 ts twice in the period of M3.So again the theory and the simulations agree.

In the third and last graph, Figure 6, the eect of quadrupling N is tested.N1 = 75, N2 = 300 and N3 = 1200. And again the predictions hold. Thefrequency is doubled, if N is multiplied by four.

So in all of the cases, the simulation veries the theory with a goodprecision.

16

4 Conclusion and Discussion

There is of course a very big dierence between the simulated model and thetheoretical model. The rigid rods of the theoretical model are replaced bysprings. One might argue that it is not very surprising to get an oscillationfrom a lot of connected springs, and that we must not speak of an entropicforce, because the force is put into the model directly. If this were thecase, there would be no dependence on the temperature (dp) which there is.There are also measurements made to verify that the spring constant k, ofthe springs between the monomers, has no inuence of the results. There issome dierence because the springs store energy from the heath bath, andthis energy is dierent for dierent spring constants. If k is a hundred timessmaller, the frequency gets only about 3.4 times bigger. So the eect of kis very small compared to the other results. For even smaller k the idealchain becomes very elastic and unrealistic. For bigger k the program bugsout, because the forces on the monomers get too big. Combined with adiscontinuity in time, this will result in enormous, unrealistic translations.Therefor k is chosen to be as high as possible for all the measurements, forthe most realistic ideal chain simulation.

There are still a few interesting things that could be researched but weresimply not possible in the short time span of a bachelor thesis. For example:When will the oscillation die out, and how long does it take? And when is Ntoo small to give accurate results? It would also be nice to retrieve a formulafor the frequency in which all the input variables of the program are used.Also it is not very accurate to measure the frequencies by hand and makesit impossible to get a good margin of error. Though it looks like the resultare very accurate.

It comes as a surprise that the plots are so incredibly smooth. One mightforget that there are actually N monomers busy with pushing and pulling thechain at every frame. And this is the point this project has been all about.In nature, the fundamental forces are seen as indivisible, because they are soaccurate and consistent. This project showed that from entropic principles,very smooth forces can be created, especially for large N . This might be aclue that there is more under the engine cowl of nature.

17

0

0.02

0.04

0.06

0.080.1

0.12

050

0000

1000

000

1500

000

2000

000

2500

000

3000

000

3500

000

4000

000

dp =

0.0

01dp

= 0

.002

dp =

0.0

04

Figure

4:Threeplots

of|R|

Nlas

functionof

timefordierentvalues

ofdp.

(N=

1000,M

=10

000,Q

=0.

1,k

=0.

01,dp

=0.

001)

18

0

0.02

0.04

0.06

0.080.1

0.12

050

0000

1000

000

1500

000

2000

000

2500

000

3000

000

3500

000

4000

000

M =

250

0M

= 1

0000

M =

400

00

Figure

5:Threeplots

of|R|

Nlas

functionof

timefordierentvalues

ofM.

(N=

1000,Q

=0.

1,k

=0.

01,dp

=0.

001)

19

0

0.02

0.04

0.06

0.080.1

0.12

0.14

0.16

050

0000

1000

000

1500

000

2000

000

2500

000

3000

000

3500

000

4000

000

N =

120

0N

= 3

00N

= 7

5

Figure

6:Threeplots

of|R|

Nlas

functionof

timefordierentvalues

ofN.

(M=

1000

0,Q

=0.

1,k

=0.

01,dp

=0.

001)

20

Acknowledgements

A huge thanks to Erik Verlinde for his insight and for welcomingthe idea to use a computer in this thesis. I have enjoyed this projecta lot. And because the subject wasn't pinned down from the start, itgave me the freedom to research a lot of dierent aspects of entropicforces and also throwing my own ideas in once in a while. And despitebeing a busy man, Erik always could sit down to take the time andhelp me with my progress. I wish him bests of luck in the furtherdevelopment of his theory.

References

[1] http://arxiv.org/pdf/1001.0785

[2] http://en.wikipedia.org/wiki/Ideal_chain

[3] http://en.wikipedia.org/wiki/Central_limit_theorem

[4] B. van Es, C. Klaassen and M. Nuyens, Syllabus Stochastiek(2007-2008)

[5] http://www.youtube.com/WiegerSteggerda

21

A Mathematical Probabilities

A.1 Probability Density Function

The Probability Density Function f(x) is useful when dealing with con-tinues random variables. It is used to calculate the probability to measurethe random variable in a certain interval, [a, b].

P ([a, b]) =

ˆ b

a

f(x)dx (4)

where we integrate over the possible results of a random experiment andP ([a, b]) denotes the probability this experiment results in a value betweena and b.

This function doesn't has to be bound by only one variable. For exam-ple, measuring the position of a particle in three dimensions results in threenumbers. The next function gives the probability of measuring a randomvector in the volume V ,

P (V ) =

ˆV

f(r)dv (5)

The most used probability density function in physics is probably the squaredabsolute wave function|Ψ|2 in quantum mechanics.

Integrating over the whole space results, of course, in a probability of 1,

P (]−∞,∞[) = 1

P (All space) = 1

A.2 Expectation Value

In a probability density function the expectation value or mean 〈g(x)〉 ofg(x) is,

〈g(x)〉 :=

ˆ ∞−∞

g(x) · f(x)dx (6)

where f(x) is a probability density function.

A.3 Variance

The variance σ2x is dened as,

σ2x =

⟨x− 〈x〉2

⟩(7)

22

It depends on the spread of a probability density function around the expec-tation value. This can be derived further,

σ2x =

⟨(x− 〈x〉)2

⟩=⟨x2 + 〈x〉2 − 2x 〈x〉

⟩=⟨x2⟩

+ 〈x〉2 +−〈x〉 〈x〉=⟨x2⟩− 〈x〉2 (8)

This formula is a very useful tool to calculate the variance of probabilitydensity functions.

A.4 Normal Distribution

A Normal Distribution is a special probability density function.

f(x) =

(1√

2πσ2

)e−

x2

2σ2 (9)

-Σ1 -Σ2 Σ2 Σ1

Figure 7: Normal distribution with σ1 = 1.0 and σ2 = 0.5

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A.4.1 The derivative of the normal distribution

It's not very dicult to determine the derivative of this function,

f ′(x) = −(

1

σ3√

2π

)· xe−

x2

2σ2 (10)

= − x

σ2· f(x)

A.4.2 Integrating the normal distribution

Determining the primitive of the normal distribution is not possible. How-ever, it is possible to evaluate the surface under the function with the use ofdouble integrals,

(ˆ ∞−∞

e−x2

2σ2 dx

)2

=

(ˆ ∞−∞

e−x2

2σ2 dx

)·(ˆ ∞−∞

e−y2

2σ2 dy

)=

ˆ ∞−∞

ˆ ∞−∞

e−x2+y2

2σ2 dxdy

=

ˆR2e−

x2+y2

2σ2 dA

Making the transition to polar coordinates,

ˆR2

e−x2+y2

2σ2 dA =

ˆ ∞0

ˆ 2π

0

e−r2

2σ2 · rdrdθ

Substituting s = r2

2σ2 ⇔ rdr = σ2ds results in,ˆ ∞0

ˆ 2π

0

e−r2

2σ2 r · drdθ = 2π

ˆ ∞0

σ2e−sds

= 2πσ2[−e−s

]∞0

= 2πσ2

And thus, ˆ ∞−∞

e−x2

2σ2 dx =√

2πσ2 (11)

This means that the surface under the normal distribution is:ˆ ∞−∞

f(x) =

ˆ ∞−∞

(1√

2πσ2

)· e−

x2

2σ2 =

(1√

2πσ2

)·√

2πσ2 = 1 (12)

The area under f(x) equals 1, so the normal distribution is a well denedprobability density function.

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A.4.3 The expectation value of the normal distribution

Applying the denition of the expectation value gives:

〈x〉 :=

ˆ ∞−∞

x · f(x)dx = 0

Because x · f(x) is anti-symmetric around zero and nite everywhere, inte-grating from −∞ to ∞ gives 0.

A.4.4 The variance of the normal distribution

Since 〈x〉 = 0 the variance is equal to,

σ2x =

⟨x2⟩− 〈x〉2 =

⟨x2⟩

Using the denition of the expectation value,⟨x2⟩

=

ˆ ∞−∞

x2f(x)dx

This can be rewritten in a form containing the derivative of f(x), Eq. (5),⟨x2⟩

= −σ2

ˆ ∞−∞

xf ′(x)dx

Integration by parts gives,⟨x2⟩

= −σ2xf(x)∣∣∣∞−∞

+ σ2

ˆ ∞−∞

f(x)dx

Integrating over a probability density function results in an area of 1, Eq.(7), ⟨

x2⟩

= −σ2xf(x)∣∣∣∞−∞

+ σ2

And nally, to calculate the rst term,

−σ2xf(x)∣∣∣∞−∞

=

(−σ2

√2πσ2

)· xe−

x2

2σ2

∣∣∣∞−∞

= 0

This concludes the calculation,

σ2x =

⟨x2⟩

= σ2

Obviously, the letterσ in the normal distribution was neatly chosen from thebeginning to mach the variance σ2

x.

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A.5 Central Limit Theorem

The Central Limit Theorem is a very important theorem in statistics.If X1, X2, . . . , Xn are independent random distributed variables with nite

mean µi and nite variance σ2i and large enough n, the sum Sn =

n∑i=1

Xi is

normally distributed with mean µ =n∑i=1

µi and variance σ2 =n∑i=1

σ2i .

Proofs of this theorem can be found on the web, in statistics books andsyllabi[3, 4].

Lets put this theory to the test with the use of Mathematica. Four datasets of size 100000 are created with n1 = 1, n2 = 2, n3 = 8 and n4 = 16.Each element in these sets is the sum of N random by a quadratic distributionfunction f(x). Where

f(x) =

32x2 if x ∈ [−1, 1]

0 otherwise

To create a distribution in Mathematica a random number R ∈ [0, 1] ispicked. R represents the following probability,

R = P ([−1, x]) =

ˆ x

−1

f(y)dy =1

2x3 +

1

2

x = 3√

2R− 1

The random measurement 3√

2R− 1 is now distributed by f(x).

The mathematical code for this measurement:

Z[] := 2*Random[] - 1X[] := Sign[Z[]] * (Abs[Z[]]^(1/3))(* X[] is actually just the real part of (Z[])^(1/3) *)

(* To sum n measurments *)S[n_] := Sum[X[], i, n]

The mean of these random numbers is µi = 〈x〉 = 0, and the variance is

σ2i = 〈x2〉−〈x〉2 = 〈x2〉 =

´ 1

−132x4dx = 3

10+ 3

10= 3

5. The central limit theorem

states that these sums are normally distributed with a mean µ =n∑i=1

µi = 0

and a variance σ2 =n∑i=1

σ2i = σ2

i · n = 3n5.

26

Let's nd out whether the central limit theorem holds for this example. Math-

ematica is used to create data sets of S[n] for dierent values of n. Afterthat, the variance is calculated, and histograms are made from the data sets.

data1 = Table[S[1], 100000];data2 = Table[S[2], 100000];data3 = Table[S[8], 100000];data4 = Table[S[16],100000];

Variance[data1] / 1Variance[data2] / 2Variance[data3] / 8Variance[data4] /16

Histogram[data1]Histogram[data2]Histogram[data3]Histogram[data4]

The results for the variance are,

σ2data1/1 = 0.600233

σ2data2/2 = 0.599631

σ2data3/8 = 0.600507

σ2data4/16 = 0.600865

So the variance stays close to the expected value from the central limit the-orem: σ2 = 0.6n.

The Histograms are listed in Figure 8. Indeed the higher n the more itresembles a normal distribution.

So even if the probability density function is completely dierent from anormal distribution, the sum of only a few measurements becomes normallydistributed.

27

-0.5 0.0 0.5 1.0

2000

4000

6000

8000

-1 0 1 2

2000

4000

6000

8000

-6 -4 -2 0 2 4 6

1000

2000

3000

4000

-10 -5 0 5 10

500

1000

1500

2000

2500

Figure 8: The histograms of data1, data2, data3 and data4 convergeto a normally distributed function

B Pseudo Code For Simulation of the Ideal Chain

Pseudo code is not code for a program to read, it is code so every humanwith a basic understanding of programming can understand it. It is there-for not in a specic language, but it should be familiar if the reader hasprogrammed in an (imperative) programming language before. If you havenever programmed before you could understand it as well. It is importantto know that the equals sign, "=", has to be read as becomes. For example,the expression a = a + 1 is a common thing to state in a programminglanguage, and has to be read as, "a becomes a plus one", so that the newvalue for a, is one higher than before.

B.1 Creating a Random Vector

Creating a random unit vector (spherical symmetrical), is less straight for-ward as it might seem. Simply creating two random angles θ and φ is notsymmetric because the probability density function will be denser at thepoles. The following method is simplistic, but it works:

DO-- create random vector in [-1,1]^3

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v.x = (rand() * 2.0) - 1.0v.y = (rand() * 2.0) - 1.0v.z = (rand() * 2.0) - 1.0

WHILE |v| > 1.0 -- accept vector if it is-- inside the unit sphere

-- normalize the vectorv = v / |v|

Where rand() returns a random value between zero and one and v is avector.

B.2 The Basic Algorithm

In Section 3, the basic algorithm is already mentioned. Let

a[i], v[i] and rho[i] be the acceleration, velocity and the posi-tion of the i'th monomer respectively;

N be the number of monomers;

k be the spring constant between the monomers;

m be the mass of each monomer;

M be the mass of the last monomer;

dp() return a random vector of length dp;

r(i) := rho[i+1] - rho[i].

Here it is in pseudo code for the simulation,

FOR EACH frame do-- calculate acceleration loopa[0] = 0FOR i = 1 TO N-1

a[i] = -k/m * (r(i-1) - norm(r(i-1))) +k/m * (r(i) - norm(r(i)))

END FORa[N] = -k/M * (r(N-1) - norm(r(N-1)))

-- calculate velocity loopFOR i = 0 TO N

29

v[i] = v[i] + a[i] * dt

-- collide with an atom every once in a whileIF rand() < Qv[i] = v[i] + dp()/m

END FOR

-- calculate position loopFOR i = 0 TO Nrho[i] = rho[i] + v[i] * dt

END FOREND FOR EACH

B.3 Speeding Up the Basic Algorithm

The algorithm in the previous section is short and understandable, which isgood. It preforms badly on a computer however, because there are for loopsover a, v and r. It turns out that the acceleration doesn't really need tobe remembered for it is changed completely for every frame. Therefor it ispossible to calculate the acceleration in the velocity loop. This takes care ofone of the loops.

FOR EACH frame do-- calculate velocity loopv[0] = 0FOR i = 1 TO N-1a = -k/m * (r(i-1) - norm(r(i-1))) +

k/m * (r(i) - norm(r(i)))v[i] = v[i] + a * dt

-- collide with an atom every once in a whileIF rand() < Qv[i] = v[i] + dp()/m

END FORa = -k/M * (r(N-1) - norm(r(N-1)))v[N] = v[i] + a * dt

-- calculate position loopFOR i = 0 TO N

30

rho[i] = rho[i] + v[i] * dtEND FOR

END FOR EACH

Combining the velocity loop with the position loop is a little bit harder. Itis not possible to just put them together in one loop, like this:

FOR EACH frame dov[0] = 0FOR i = 1 TO N-1a = -k/m * (r(i-1) - norm(r(i-1))) +

k/m * (r(i) - norm(r(i)))v[i] = v[i] + a * dt

-- collide with an atom every once in a whileIF rand() < Qv[i] = v[i] + dp()/m

rho[i] = rho[i] + v[i] * dtEND FOR

a = -k/M * (r(N-1) - norm(r(N-1)))v[N] = v[N] + a * dtrho[N] = rho[N] + v[N] * dt

END FOR EACH

When calculating the acceleration, the information r(i) is used. This mem-ory is still waiting to be updated in the last line of the code. This might notseem too bad, it is however, a very serious problem. First r(1) will becalculated incorrectly, this will have an added eect on r(2), which will addextra error to r(3). So it creates a chain reaction of small errors, and thechain will end up in limbo.

The solution to this problem is to calculate the rst value of the velocity,and after that start to calculate the positions r(i) simultaneously withv[i+1]. In this case the velocities will be updated with the old positions,so that the new positions can be calculated from the just acquired velocities:

FOR EACH frame dov[0] = 0a = -k/m * (r(0) - norm(r(0))) +

k/m * (r(1) - norm(r(1)))v[1] = v[1] + a * dt

31

FOR i = 2 TO N-1a = -k/m * (r(i-1) - norm(r(i-1))) +

k/m * (r(i) - norm(r(i)))v[i] = v[i] + a * dt

-- collide with an atom every once in a whileIF rand() < Qv[i] = v[i] + dp()/m

rho[i-1] = rho[i-1] + v[i-1] * dtEND FOR

a = -k/M * (r(N-1) - norm(r(N-1)))v[N] = v[N] + a*dt

rho[N-1] = rho[N-1] + v[N-1] * dtrho[N] = rho[N] + v[N] * dt

END FOR EACH

It is hideous, hard to read, code, but it runs like a beast. Three timesfaster then the rst naive algorithm. The program that was used to calculatethe results was running for 45 minutes just to retrieve one plot. If it wasn'tfor the improved algorithm this would be 2 hours and 15 minutes. Thatwould be unbearable because measurements don't always go right the rsttime.

32