Confinement-induced superdiffusion

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Table of contents

0 Main results 
0.1 Introduction
0.2 Biased tracer diffusion in a high-density lattice gas
0.2.1 General method
0.2.2 One-dimensional lattice
0.2.3 Confinement-induced superdiffusion
0.2.4 Velocity anomaly in quasi-one-dimensional geometries
0.2.5 Universal formula for the cumulants
0.2.6 Simplified description
0.3 Biased tracer diffusion in a hardcore lattice gas of arbitrary density
0.3.1 General formalism
0.3.2 One-dimensional situation
0.3.3 Higher-dimensional lattices
0.4 Conclusion
1 Introduction 
I Biased tracer diffusion in a high-density lattice gas
2 Motivation and general presentation 
2.1 Introduction
2.1.1 Statement of the problem
2.1.2 Experimental works
2.1.3 Theoretical descriptions
2.1.4 Objectives
2.2 Presentation of the model
2.2.1 Model
2.2.2 Evolution rules
2.3 Single-vacancy situation
2.3.1 Single-vacancy propagator
2.3.2 Calculation of the conditional first-passage time densities F
2.4 Finite low vacancy density
2.4.1 Average over the initial positions of the vacancies
2.4.2 Computation of the quantities F0
2.4.3 Thermodynamic limit and expression of the cumulants
2.5 Conclusion
3 One-dimensional geometry 
3.1 Introduction
3.1.1 Single-file diffusion
3.1.2 Case of a biased TP
3.1.3 Objectives of this chapter
3.2 Resolution
3.2.1 Model
3.2.2 Cumulant generating function
3.2.3 Calculation of b F1
3.2.4 Calculation of h()
3.2.5 Expression of the cumulants
3.3 Results
3.3.1 Cumulants in the long-time limit
3.3.2 Numerical simulations
3.3.3 Case of a symmetric TP
3.3.4 Full distribution
3.4 Conclusion
4 Confinement-induced superdiffusion 
4.1 Introduction
4.1.1 Context
4.1.2 Objectives of this Chapter
4.2 Main results of this Chapter
4.3 Stripe-like geometry
4.3.1 Expression of b (k1; )
4.3.2 Calculation of the conditional FPTD F
4.3.3 Calculation of the quantities F0
4.3.4 Expression of the second cumulant in the long-time limit
4.3.5 Comments
4.3.6 Numerical simulations
4.4 Capillary-like geometry
4.4.1 Introduction
4.4.2 Expression of b (k1; )
4.4.3 Conditional FPTD b F and sums F0
4.4.4 Expression of the second cumulant in the long time limit
4.4.5 Numerical simulations
4.5 Two-dimensional infinite lattice
4.5.1 Computation of the second cumulant
4.5.2 Long-time expansion
4.5.3 Subdominant term
4.5.4 Remarks and numerical simulations
4.6 Three-dimensional infinite lattice
4.7 Crossover to diffusion – Stripe-like geometry
4.7.1 Introduction
4.7.2 Determination of the conditional FPTD
4.7.3 Propagators
4.7.4 Ultimate expression of the second cumulant
4.7.5 Scaling function
4.8 Crossover to diffusion – Capillary-like geometry
4.8.1 Introduction
4.8.2 Ultimate expression of the second cumulant
4.8.3 Scaling function
4.9 Crossover to diffusion – Two-dimensional lattice
4.9.1 Ultimate expression of the second cumulant
4.9.2 Scaling function
4.10 Conclusion
5 Velocity anomaly in quasi-one-dimensional geometries 
5.1 Introduction
5.2 Quasi-one-dimensional geometries
5.2.1 Stripe-like geometry
5.2.2 Capillary-like geometry
5.3 Two-dimensional lattice
5.4 Conclusion
6 Universal formulae for the cumulants 
6.1 Introduction
6.2 First cumulants of the TP position
6.2.1 Mean position of the TP
6.2.2 Fluctuations of the position of the TP
6.3 Higher-order cumulants in the longitudinal direction
6.3.1 Method
6.3.2 Recurrent lattices
6.3.3 Transient lattices
6.4 Cumulants in the transverse direction
6.4.1 Method
6.4.2 Recurrent lattices
6.4.3 Transient lattices
6.5 Extension to fractal lattices
6.6 Conclusion
7 Simplified continuous description 
7.1 Introduction
7.2 General formalism
7.3 Two-dimensional lattice
7.4 Stripe-like lattice
7.5 One-dimensional lattice
7.6 Conclusion
II Biased tracer diffusion in a lattice gas of arbitrary density
8 General formalism and decoupling approximation 
8.1 Introduction
8.2 Model and master equation
8.2.1 Model
8.2.2 Master equation
8.3 Equations verified by the first cumulants
8.3.1 Mean position
8.3.2 Fluctuations of the TP position
8.3.3 Stationary values
8.4 Cumulant generating function
8.4.1 Governing equations
8.4.2 Evolution equations of the quantities ewr
8.4.3 Application: Third-order cumulant
8.5 Conclusion
9 One-dimensional lattice in contact with a reservoir 
9.1 Introduction
9.2 First cumulants of the TP position in one dimension
9.2.1 Solution of the equation on kr in one dimension
9.2.2 Solution of the equation on egr in one dimension
9.2.3 Solution of the equation on emr in one dimension
9.3 Results and discussion
9.3.1 Algorithm and numerical methods
9.3.2 Velocity
9.3.3 Diffusion coefficient
9.3.4 Third cumulant
9.4 Cumulant generating function and propagator
9.4.1 Calculation
9.4.2 Numerical simulations
9.5 Conclusion
10 Resolution on higher-dimensional lattices 
10.1 Introduction
10.2 Mean position of the TP
10.2.1 Basic equations
10.2.2 Infinite lattices
10.2.3 Generalized capillaries
10.2.4 General solution
10.2.5 High-density limit
10.2.6 Low-density limit and fixed obstacles
10.3 Negative differential mobility
10.3.1 Introduction
10.3.2 Simple physical mechanism
10.3.3 Method and results
10.3.4 Summary
10.4 Fluctuations of the TP
10.4.1 General equations
10.4.2 High-density limit
10.5 Conclusion
11 Conclusion 
12 Publications