Note: Cover -- Contents -- 1 The Fickian theory of diffusion -- 1.1 Fick's Law and the diffusion equation -- 1.2 Some one-dimensional examples -- 1.3 The road ahead -- Note to Chapter 1 -- 2 A review of random variable theory -- 2.1 Probability -- 2.2 Definition of a random variable -- 2.3 Some commonly encountered random variables -- 2.4 Multivariate random variables -- 2.5 Functional transformations of random variables: the RVT theorem -- 2.6 Some useful consequences of the RVT theorem -- 2.7 The bivariate normal random variable -- 2.8 Generating numerical samples of random variables -- 2.9 Integer-valued random variables -- Notes to Chapter 2 -- 3 Einstein's theory of diffusion -- 3.1 Einstein's derivation of the diffusion equation -- 3.2 A critique of Einstein's derivation -- 3.3 Einstein's new perspective -- 3.4 The covariance and correlation -- 3.5 The relative diffusion coefficient -- 3.6 The probability flux: boundary conditions -- 3.7 The stochastic bimolecular chemical reaction rate: Part I -- Notes to Chapter 3 -- 4 Implications and limitations of the Einstein theory of diffusion -- 4.1 Numerical simulation strategies -- 4.2 A serious problem -- 4.3 Proof of Eqs (4.12) and (4.13) in two dimensions -- 4.4 Implications of Eqs (4.12) and (4.13) -- 4.5 A hint of a quantitative lower bound on Δt in Eqs -- 4.6 The small-scale motion of a solute molecule -- 4.7 Collision probability of a solute molecule with a surface -- 4.8 The stochastic bimolecular chemical reaction rate: Part II -- Notes to Chapter 4 -- Appendix 4A: Proof of the reflecting boundary point simulation procedure -- Appendix 4B: Proof of the absorbing boundary point simulation procedure -- Appendix 4C: The Maxwell-Boltzmann distribution -- 5 The discrete-stochastic approach -- 5.1 Specification of the system -- 5.2 The key dynamical hypothesis. , 5.3 Connection to the classical Fickian model -- 5.4 Connection to the Einstein model -- 5.5 Constraints on l and δt -- 5.6 A more accurate formula for K[sub(l)] -- 5.7 The discrete-stochastic model's version of Fick's Law -- 5.8 Does the concentration gradient "cause" diffusion? -- 5.9 A microfluidics diffusion experiment -- Notes to Chapter 5 -- 6 Master equations and simulation algorithms for the discrete-stochastic approach -- 6.1 The single-molecule diffusion master equation -- 6.2 Relation to the Einstein model of diffusion -- 6.3 Solutions to the single-molecule master equation -- 6.4 Simulating the discrete-stochastic motion of a single solute molecule -- 6.5 Some examples of single-molecule simulations -- 6.6 The many-molecule diffusion master equation -- 6.7 The case M = 2: an exact solution of a different kind -- 6.8 The moments of the cell populations: recovering the diffusion equation -- 6.9 Simulating the discrete-stochastic motion of an ensemble of solute molecules -- 6.10 Some examples of many-molecule simulations -- 6.11 A simulation study of Fick's Law -- Appendix 6A: General solution to the single-molecule master equation -- Appendix 6B: Confidence intervals in Monte Carlo averaging -- Appendix 6C: Derivation of the first moment equation (6.31) -- 7 Continuous Markov process theory -- 7.1 The Chapman-Kolmogorov and Kramers-Moyal equations -- 7.2 The process increment and its PDF -- 7.3 The self-consistency requirement -- 7.4 Derivation of the Langevin equation -- 7.5 Implications of the Langevin equation -- 7.6 The forward Fokker-Planck equation -- 7.7 Multivariate continuous Markov processes -- 7.8 The driftless Wiener process -- 7.9 The Ornstein-Uhlenbeck process -- 7.10 The time-integral of the Ornstein-Uhlenbeck process -- 7.11 Numerically simulating the driftless Wiener process. , 7.12 Numerically simulating the Ornstein-Uhlenbeck process and its integral -- 7.13 The backward Fokker-Planck equation -- Notes to Chapter 7 -- 8 Langevin's theory of diffusion -- 8.1 Langevin's key assumption -- 8.2 A physical rationale for Langevin's assumption -- 8.3 Fixing the factor f: the fuctuation-dissipation theorem -- 8.4 The Langevin diffusion formulas -- 8.5 The correlation between position and velocity -- 8.6 Two-time auto-correlations -- Note to Chapter 8 -- 9 Implications of Langevin's theory -- 9.1 The Langevin mean-square displacement formulas -- 9.2 The coeffcient of diffusion: the connection to Einstein's theory -- 9.3 The relaxation time and the characteristic diffusion length -- 9.4 Implications for the discrete-stochastic model of diffusion -- 9.5 The Langevin picture of V[sub(x)](t) -- 9.6 The Langevin simulation formulas -- 9.7 Examples of trajectory simulations in the Langevin and Einstein theories -- 9.8 The relative motion of two solute molecules -- 9.9 The velocity auto-covariance formula for D -- 9.10 The energetics of diffusion -- 9.11 Are there "overdamped diffusing systems"? -- Notes to Chapter 9 -- 10 Diffusion in an external force field -- 10.1 The Smoluchowski equation-a Fickian derivation -- 10.2 An application: a rudimentary type of gradient-sensing chemotaxis -- 10.3 The Langevin equation for a solute molecule in an external force field -- 10.4 The Kramers equation -- 10.5 Energetics revisited -- 10.6 Some interesting aspects of the uninteresting limit & -- #947 -- & -- #8594 -- 0 -- 10.7 The large-& -- #947 -- limit: the Smoluchowski equation revisited -- 10.8 A constant external force field in the Langevin picture -- Notes to Chapter 10 -- 11 The first-passage time approach -- 11.1 The basic first-passage time problem -- 11.2 Limitations of the usual simulation approaches. , 11.3 A direct analytical approach -- 11.4 A little help from the backward Fokker-Planck equation -- 11.5 Formulas for the moments of the first-passage time -- 11.6 Explicit solutions for the mean and variance -- 11.7 Implications for the discrete-stochastic approach -- 11.8 An "averaged" first-passage time -- 11.9 Some conclusions -- Index -- A -- B -- C -- D -- E -- F -- G -- H -- I -- J -- K -- L -- M -- N -- O -- P -- R -- S -- T -- U -- V -- W.