N G van Kampen 1981 Amsterdam: North-Holland xiv + 419 pp price Dfl 180 This is a book which, at a lower price, could be expected to become an essential part of the library of every physical scientist concerned with problems involving fluctuations and stochastic processes, as well as those who just enjoy a beautifully written book. It provides an extensive graduate-level introduction which is clear, cautious, interesting and readable.

https://iopscience.iop.org/article/10.1088/0031-9112/34/4/040/pdf

Stochastic Processes in Physics and Chemistry

Stochastic thermodynamics as reviewed here systematically provides a framework for extending the notions of classical thermodynamics such as work, heat and entropy production to the level of individual trajectories of well-defined non-equilibrium ensembles. It applies whenever a non-equilibrium process is still coupled to one (or several) heat bath(s) of constant temperature. Paradigmatic systems are single colloidal particles in time-dependent laser traps, polymers in external flow, enzymes and molecular motors in single molecule assays, small biochemical networks and thermoelectric devices involving single electron transport. For such systems, a first-law like energy balance can be identified along fluctuating trajectories. For a basic Markovian dynamics implemented either on the continuum level with Langevin equations or on a discrete set of states as a master equation, thermodynamic consistency imposes a local-detailed balance constraint on noise and rates, respectively. Various integral and detailed fluctuation theorems, which are derived here in a unifying approach from one master theorem, constrain the probability distributions for work, heat and entropy production depending on the nature of the system and the choice of non-equilibrium conditions. For non-equilibrium steady states, particularly strong results hold like a generalized fluctuation–dissipation theorem involving entropy production. Ramifications and applications of these concepts include optimal driving between specified states in finite time, the role of measurement-based feedback processes and the relation between dissipation and irreversibility. Efficiency and, in particular, efficiency at maximum power can be discussed systematically beyond the linear response regime for two classes of molecular machines, isothermal ones such as molecular motors, and heat engines such as thermoelectric devices, using a common framework based on a cycle decomposition of entropy production. (Some figures may appear in colour only in the online journal) This article was invited by Erwin Frey.

/pdf/stochastic-thermodynamics-fluctuation-theorems-and-molecular-2epwyvmpf0.pdf

Stochastic thermodynamics, fluctuation theorems and molecular machines

Mathematical modelling of the movement of animals, micro-organisms and cells is of great relevance in the fields of biology, ecology and medicine. Movement models can take many different forms, but the most widely used are based on the extensions of simple random walk processes. In this review paper, our aim is twofold: to introduce the mathematics behind random walks in a straightforward manner and to explain how such models can be used to aid our understanding of biological processes. We introduce the mathematical theory behind the simple random walk and explain how this relates to Brownian motion and diffusive processes in general. We demonstrate how these simple models can be extended to include drift and waiting times or be used to calculate first passage times. We discuss biased random walks and show how hyperbolic models can be used to generate correlated random walks. We cover two main applications of the random walk model. Firstly, we review models and results relating to the movement, dispersal and population redistribution of animals and micro-organisms. This includes direct calculation of mean squared displacement, mean dispersal distance, tortuosity measures, as well as possible limitations of these model approaches. Secondly, oriented movement and chemotaxis models are reviewed. General hyperbolic models based on the linear transport equation are introduced and we show how a reinforced random walk can be used to model movement where the individual changes its environment. We discuss the applications of these models in the context of cell migration leading to blood vessel growth (angiogenesis). Finally, we discuss how the various random walk models and approaches are related and the connections that underpin many of the key processes involved.

https://royalsocietypublishing.org/doi/pdf/10.1098/rsif.2008.0014

Random walk models in biology.

The study of animal foraging behaviour is of practical ecological importance1, and exemplifies the wider scientific problem of optimizing search strategies2. Levy flights are random walks, the step lengths of which come from probability distributions with heavy power-law tails3, 4, such that clusters of short steps are connected by rare long steps. Levy flights display fractal properties, have no typical scale, and occur in physical3, 4, 5 and chemical6 systems. An attempt to demonstrate their existence in a natural biological system presented evidence that wandering albatrosses perform Levy flights when searching for prey on the ocean surface7. This well known finding2, 4, 8, 9 was followed by similar inferences about the search strategies of deer10 and bumblebees10. These pioneering studies have triggered much theoretical work in physics (for example, refs 11, 12), as well as empirical ecological analyses regarding reindeer13, microzooplankton14, grey seals15, spider monkeys16 and fishing boats17. Here we analyse a new, high-resolution data set of wandering albatross flights, and find no evidence for Levy flight behaviour. Instead we find that flight times are gamma distributed, with an exponential decay for the longest flights. We re-analyse the original albatross data7 using additional information, and conclude that the extremely long flights, essential for demonstrating Levy flight behaviour, were spurious. Furthermore, we propose a widely applicable method to test for power-law distributions using likelihood18 and Akaike weights19, 20. We apply this to the four original deer and bumblebee data sets10, finding that none exhibits evidence of Levy flights, and that the original graphical approach10 is insufficient. Such a graphical approach has been adopted to conclude Levy flight movement for other organisms13, 14, 15, 16, 17, and to propose Levy flight analysis as a potential real-time ecosystem monitoring tool17. Our results question the strength of the empirical evidence for biological Levy flights.

/pdf/revisiting-levy-flight-search-patterns-of-wandering-12hbphehb3.pdf

Revisiting Lévy flight search patterns of wandering albatrosses, bumblebees and deer.

https://scholarlypublications.universiteitleiden.nl/access/item%3A2733176/view

Front propagation into unstable states

Kinetics of target site localization of a protein on DNA: a stochastic approach.

The cell cytoskeleton is a striking example of an ‘active’ medium driven out-of-equilibrium by ATP hydrolysis1. Such activity has been shown to have a spectacular impact on the mechanical and rheological properties of the cellular medium2,3,4,5,6,7,8,9,10, as well as on its transport properties11,12,13,14: a generic tracer particle freely diffuses as in a standard equilibrium medium, but also intermittently binds with random interaction times to motor proteins, which perform active ballistic excursions along cytoskeletal filaments. Here, we propose an analytical model of transport-limited reactions in active media, and show quantitatively how active transport can enhance reactivity for large enough tracers such as vesicles. We derive analytically the average interaction time with motor proteins that optimizes the reaction rate, and reveal remarkable universal features of the optimal configuration. We discuss why active transport may be beneficial in various biological examples: cell cytoskeleton, membranes and lamellipodia, and tubular structures such as axons1.

/pdf/enhanced-reaction-kinetics-in-biological-cells-26y4w6le3f.pdf

Enhanced reaction kinetics in biological cells

L\'evy flights are known to be optimal search strategies in the particular case of revisitable targets. In the relevant situation of nonrevisitable targets, we propose an alternative model of two-dimensional (2D) search processes, which explicitly relies on the widely observed intermittent behavior of foraging animals. We show analytically that intermittent strategies can minimize the search time, and therefore do constitute real optimal strategies. We study two representative modes of target detection and determine which features of the search time are robust and do not depend on the specific characteristics of detection mechanisms. In particular, both modes lead to a global minimum of the search time as a function of the typical times spent in each state, for the same optimal duration of the ballistic phase. This last quantity could be a universal feature of 2D intermittent search strategies.

Two-dimensional intermittent search processes: An alternative to Lévy flight strategies.

We consider a random walk in confined geometry, starting from a site and eventually reaching a target site. We calculate analytically the distribution of the occupation time on a third site, before reaching the target site. The obtained distribution is exact and completely explicit in the case or parallelepipedic confining domains. We discuss implications of these results in two different fields: The mean first passage time for the random trap model is computed in dimensions greater than 1 and is shown to display a nontrivial dependence with the source and target positions. The probability of reaction with a given imperfect center before being trapped by another one is also explicitly calculated, revealing a complex dependence both in geometrical and chemical parameters.

Random walks and Brownian motion: a method of computation for first-passage times and related quantities in confined geometries.

In this paper we study the kinetics of diffusion-limited, pseudo-first-order $A+\stackrel{\ensuremath{\rightarrow}}{B}B$ reactions in situations in which the particles' intrinsic reactivities are not constant but vary randomly in time. That is, we suppose that the particles are bearing ``gates'' which fluctuate in time, randomly and independently of each other, between two states---an active state, when the reaction may take place between A and B particles appearing in close contact; and a blocked state, when the reaction is completely inhibited. We focus here on two customary limiting cases of pseudo-first-order reactions---the so-called target annihilation and the Rosenstock trapping model---and consider four different particular models, such that the A particle can be either mobile or immobile or gated or ungated, and ungated or gated B particles can be fixed at random positions or move randomly. All models are formulated on a d-dimensional regular lattice, and we suppose that the mobile species perform independent, homogeneous, discrete-time lattice random walks. The model involving a single, immobile, ungated target A and a concentration of mobile, gated B particles is solved exactly. For the remaining three models we determine exactly, in the form of rigorous lower and upper bounds showing the same N dependence, the large-$N$ asymptotical behavior of the probability that the A particle survives until the $N\mathrm{th}$ step. We also realize that for all four models studied here the A particle survival probability can be interpreted as the moment generating function of some functionals of random walk trajectories, such as, e.g., the number of self-intersections, the number of sites visited exactly a given number of times, the ``residence time'' on a random array of lattice sites, etc. Our results thus apply to the asymptotic behavior of corresponding generating functions which are not known as yet.

Michel Moreau

Papers

Kinetics of target site localization of a protein on DNA: a stochastic approach.

Enhanced reaction kinetics in biological cells

Two-dimensional intermittent search processes: An alternative to Lévy flight strategies.

Random walks and Brownian motion: a method of computation for first-passage times and related quantities in confined geometries.

Kinetics of stochastically gated diffusion-limited reactions and geometry of random walk trajectories