Diffusion Processes with Hidden States from ... - FU Berlin, FB MI

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3 Theoryall the dynamical states through which, as a result of the molecular motion, the systemis rapidly passing”, as pointed out by Enrico Fermi, [20, p. 3].In our case the dynamical states are given by the single trajectories while the ensemble state(thermodynamical state) can be represented by a probability distribution function, leading to thequestion how this distribution will evolve over time.The consequence of this notion by Fermi is that the single trajectories alone on the one hand, andthe ensemble state alone on the other hand will not suffice for a proper understanding of the system.By this fact, we will need both pictures. Thus in the next section a general introduction toDiffusion will be given. Afterwards as a physical motivation for the Stochastic Differential Equations(subsection 3.9.3), representing the dynamical-state approach (model 1), we introduce theLangevin Equation (subsection 3.9.1). This is denoting the dynamics of a single trajectory, drivenby noise. The ensemble-state approach (model 2) will be given by means of the Fokker-PlanckEquation (subsection 3.9.4), denoting the evolution of a probability density function, representingan ensemble of trajectories.3.9 DiffusionAccording to [4, p. 5] one can say that ”diffusion is the random migration of molecules or smallparticles arising from motion due to thermal energy”.Formally one can define Diffusion by differential equations of the following type:Definition 3.3 (Diffusion).∂ f (x,t)∂t= D ∂ 2 f (x,t)∂x 2 , (3.33)with:• f (x,t) denoting a function in space x and time t for an ensemble of several particles of thesame nature and• D denoting a constant, the so called diffusion constant, which is depending on the nature ofthe particle in consideration and the environment in which it is suspended.According to this, diffusion is a collective phenomenon. That is the motion of an ensemble,consisting of equal particles, while the single particles are moving completely independent fromeach other in a random manner due to thermal energy. We will see later in the text that this randommotion of the single particles is the so called Brownian motion.Furthermore diffusion is a deterministic phenomenon, deterministic with regard to the fact thatdiffusional motion of an ensemble always leads to equilibrium, provided that the system is isolatedfrom the surrounding environment, and will not be disturbed externally.28
3.9 DiffusionConsider a basin filled with water, and in the middle of this basin, within the fluid, let there bedistributed particles of the same kind in a comparatively small region (see Figure 3.6 (a)). After afinite time interval by monitoring our basin again, we observe that the area in which the particlesare distributed, is augmented. Some times later it is even much more enhanced and so on, until theparticles are distributed equally over the whole basin.Mathematically expressed: When the initial distribution is delta-shaped f (x,t 0 ) = δ(x − x 0 ), thenafter a time t 1 > 0, we have a Gaussian distribution function which is broadening over time (seeFigure 3.6 (b))[ ]f (x) = √ 1 (x − µ)2exp − 2πσ 2σ 2 ,with mean µ and variance σ.Basin with watert 0 =0 t 1 0 t 2 ≫t 1Area, in which the particles are distributedf x(a)x(b)Figure 3.6: Schematic representation of the diffusion phenomenon. (a) A basin filled with water,and a particle concentration in a small area of the basin which due to diffusion propagatesin time, until it is distributed equally over the whole basin. (b) Mathematicallythis phenomenon is represented by a Gaussian distribution function that is broadeningover time.29
Page 1: Diploma ThesisDepartment for Theore
Page 5 and 6: AbstractAnomalous diffusion is a ub
Page 7 and 8: Zusammenfassung 1Anomale Diffusion
Page 9 and 10: Contents1 Motivation 12 Visual Tran
Page 11 and 12: List of Figures2.1 Anatomy of the e
Page 13 and 14: 1 Motivation„NEC FASCES, NEC OPES
Page 15 and 16: Theoretically we profit from enormo
Page 17 and 18: 2 Visual Transduction”The eye owe
Page 19 and 20: (a)(b)(c)Figure 2.3: (a) The anatom
Page 21 and 22: effect of generating the photorecep
Page 23 and 24: 3 Theory”The theory is the net, t
Page 25 and 26: 3.2 Markov Chains, Markov Processes
Page 31 and 32: 3.3 Hidden Markov ModelsP[X(t + τ)
Page 33 and 34: 3.4 Maximum Likelihood Principlewit
Page 35 and 36: 3.5 Optimization3.5 OptimizationAcc
Page 37 and 38: 3.7 Baum-Welch-AlgorithmAlgorithm 3
Page 39: 3.8 Two Approaches on Stochastic Sy
Page 43 and 44: 3.9 Diffusion169].Rearranging (3.34
Page 45 and 46: 3.9 Diffusionwith τ = t −t ′ .
Page 47 and 48: 3.9 DiffusionD = 2k2 B T 2σ . (3.4
Page 49 and 50: 3.10 Hidden Markov Models with Stoc
Page 55 and 56: 3.11 Hidden Markov Model - Vector A
Page 57 and 58: 3.11 Hidden Markov Model - Vector A
Page 59 and 60: 3.12 Artificial Test Examples for H
Page 69 and 70: 3.13 Global Optimization Methods3.1
Page 71 and 72: 3.13 Global Optimization MethodsEve
Page 73 and 74: 4 Fluorescence Tracking Experiments
Page 75 and 76: 4.2 Fluorescence SpectroscopyAfter
Page 77 and 78: 4.3 Single Molecule Tracking via Wi
Page 79 and 80: 4.4 Total Internal Reflection Fluor
Page 81 and 82: 4.4 Total Internal Reflection Fluor
Page 83 and 84: 4.6 The expected range for the Tran
Page 85 and 86: 5 Modeling of the Experiment”The
Page 87 and 88: 5.1 Experimental Data(a)(b)(c)(d)Fi
Page 89 and 90: 5.2 Model Ansatz and Estimation of
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5.2 Model Ansatz and Estimation of
Page 93 and 94:
5.3 Testing the Modelalgorithm was
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5.5 Estimation of the Noise Intensi
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5.6 Estimation on the Basis of Diff
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6 Conclusion and OutlookThe main as
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7 Bibliography[1] R. C. Aster, B. B
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[36] C. U. M. Smith: Elements of Mo
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Index11-cis retinal isomer, 87TM se
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A AppendixA.1 From Copernicus to Ne
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A.2 Important Papers on the Rhodops
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A.3 Probability TheoryDefinition A.
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A.4 Definitions for OptimizationDef
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A.4 Definitions for OptimizationDef
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A.7 The Fluctuation-Dissipation-The
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A.7 The Fluctuation-Dissipation-The
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A.8 Kramers-Moyal Forward Expansion
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A.9 Deriving the Fokker-Planck Equa
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A.9 Deriving the Fokker-Planck Equa
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DanksagungenIch möchte folgenden M
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AffirmationHereby I, Arash Azhand,
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