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

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A Appendixregardless of any applied voltage. Here the FDT is given by the mean squared voltage 〈V 2 〉 inrelation to the resistance R:〈V 2 〉 = 4Rk B T ∆ν, (A.13)while ∆ν denote the bandwidth over which the voltage is measured.But a prove to general FDTs was initially delivered in 1951 by Herbert B. Callen and Theodore A.Welton [61].In the year 1957 the Japanese mathematical physicist Ryogo Kubo provided an ingenious piece ofwork [62] in which he presented a groundbreaking derivation of FDT in the framework of the socalled ”Linear Response Theory” (LRT), also developed by him. He is stating in the abstract:”A general type of fluctuation-dissipation-theorem is discussed to show that thephysical quantities such as complex susceptibility of magnetic or electric polarizationand complex conductivity for electric conduction are rigorously expressed in terms oftime-fluctuation of dynamical variables associated with such irreversible processes.This is a generalization of statistical mechanics which affords exact formulation asthe basis of calculation of such irreversible quantities from atomistic theory. Thegeneral formalism of the statistical-mechanical theory is examined in detail. The response,relaxation, and correlation functions are defined in quantum mechanical wayand their relations are investigated. The formalism is illustrated by simple examplesof magnetic and conduction problems. Certain sum rules are discussed for these examples.Finally it is pointed out that this theory may be looked as a generalization ofthe Einstein relation”.Kubo initially introduces the notion of the linear response function, before he derives the generalFDT from it.He applies a simple but elegant motivation of the liner response of a system due to a perturbationfor Hamiltonian systems. Kubo’s considerations are so ingenious that we should present themsimply here, just in the way he is introducing it in his seminal paper from 1957.Consider an isolated system, described by a Hamiltonian H 0 which is the energy of the system,externally unperturbed. Suppose furthermore an external force field f (t) that is switched on at adefinite time and disturbing the system. This external force field will lead to a new HamiltonianH = H 0 − x f (t).The new quantity x(t) that is introduced here, denote an arbitrary observable of the system that weconsider here.In the following, Kubo makes the assumption of weak perturbation that allows it to take intoaccount linear approximation of the whole case. The aim is to derive a relation for the responseof the system on this weak perturbation. One has to consider now an arbitrary physical quantity,denoted by y(t). Then we obtain the response of the system through the change of this quantity,given by ∆y.Kubo makes preliminary a treatment in the scope of classical mechanics by introducing a statisticalensemble with the associated initial distribution function ρ 0 (p k ,q k ) in the phase space of the110
A.7 The Fluctuation-Dissipation-Theoremrelated system. Then the dynamical motion of the unperturbed system is given by the Liouvilleequation:(∂ρ 0 ∂ρ0 ∂H 0=∂t−∑ − ∂ρ )0 ∂H 0= {H 0 ,ρ 0 }. (A.14)k∂q k ∂ p k ∂ p k ∂q kWe get this equation by building the total differential of the distribution function ρ 0 (p k ,q k ) andassuming this total differential to be zero, what is nothing else than the classical Liouville assumptionthat for Hamiltonian systems the phase space volume is conserved.Here the q k , p k are giving the set of the canonical momenta and coordinates, and {·,·} is denotingthe Poisson bracket, defined for two arbitrary phase space functions A(q k , p k ) and B(q k , p k ) in thefollowing manner:( ∂A ∂B{A,B} = ∑ − ∂A )∂B.k∂q k ∂ p k ∂ p k ∂q kWhen the perturbation H p = −x f (t) is inserted adiabatically at t = −∞, the perturbed distributionfunction obeys the following general time evolution equation:∂ρ p∂tThe assumption of linear approximation leads to= {H 0 ,ρ p } + {H p (t),ρ 0 }. (A.15)ρ p = ρ 0 + ∆ρ,and further with the assumption of equilibrium which is meaning∂ρ 0= {H 0 ,ρ 0 } = ! 0,∂tto∂ (∆ρ)= {H 0 ,∆ρ} − f (t){x,ρ 0 }. (A.16)∂tHere the time evolution of the perturbance ∆ρ is connected to unperturbed quantities like theHamiltonian H 0 of the unperturbed system and the unperturbed density function ρ 0 . The solutionof this evolution equation is easily to get and is given byˆt∆ρ = −−∞dt ′ e i(t−t′ )L {x,ρ 0 } f (t ′ ),with −∞ < t ′ < t and the Liouville operator L , defined throughiL • = {H,•}.Thus the change ∆y of the dynamical quantity y(t) is given byˆ∆y(t) = dΓ∆ρ y(p,q)ˆ= −ˆ= −ˆtdΓ−∞ˆtdΓ−∞[ ]dt ′ e i(t−t′ )L {x,ρ 0 } f (t ′ )y(p,q)dt ′ {x,ρ 0 } y(t −t ′ ) f (t ′ ),(A.17)111
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Diploma ThesisDepartment for Theore
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AbstractAnomalous diffusion is a ub
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Zusammenfassung 1Anomale Diffusion
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Contents1 Motivation 12 Visual Tran
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List of Figures2.1 Anatomy of the e
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1 Motivation„NEC FASCES, NEC OPES
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Theoretically we profit from enormo
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2 Visual Transduction”The eye owe
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(a)(b)(c)Figure 2.3: (a) The anatom
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effect of generating the photorecep
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3 Theory”The theory is the net, t
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3.2 Markov Chains, Markov Processes
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3.3 Hidden Markov ModelsP[X(t + τ)
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3.4 Maximum Likelihood Principlewit
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3.5 Optimization3.5 OptimizationAcc
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3.7 Baum-Welch-AlgorithmAlgorithm 3
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3.8 Two Approaches on Stochastic Sy
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3.9 DiffusionConsider a basin fille
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3.9 Diffusion169].Rearranging (3.34
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3.9 Diffusionwith τ = t −t ′ .
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3.9 DiffusionD = 2k2 B T 2σ . (3.4
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3.10 Hidden Markov Models with Stoc
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3.11 Hidden Markov Model - Vector A
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3.11 Hidden Markov Model - Vector A
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3.12 Artificial Test Examples for H
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3.13 Global Optimization Methods3.1
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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
Page 91 and 92: 5.2 Model Ansatz and Estimation of
Page 93 and 94: 5.3 Testing the Modelalgorithm was
Page 95 and 96: 5.5 Estimation of the Noise Intensi
Page 97 and 98: 5.6 Estimation on the Basis of Diff
Page 103 and 104: 6 Conclusion and OutlookThe main as
Page 105 and 106: 7 Bibliography[1] R. C. Aster, B. B
Page 107 and 108: [36] C. U. M. Smith: Elements of Mo
Page 109 and 110: Index11-cis retinal isomer, 87TM se
Page 111 and 112: A AppendixA.1 From Copernicus to Ne
Page 113 and 114: A.2 Important Papers on the Rhodops
Page 115 and 116: A.3 Probability TheoryDefinition A.
Page 117 and 118: A.4 Definitions for OptimizationDef
Page 119 and 120: A.4 Definitions for OptimizationDef
Page 121: A.7 The Fluctuation-Dissipation-The
Page 125 and 126: A.8 Kramers-Moyal Forward Expansion
Page 127 and 128: A.9 Deriving the Fokker-Planck Equa
Page 129 and 130: A.9 Deriving the Fokker-Planck Equa
Page 131 and 132: DanksagungenIch möchte folgenden M
Page 133: AffirmationHereby I, Arash Azhand,
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