The present thesis is thematically divided into two parts: the stochastic dynamics of particles and the stochastic energetics. Let us now address both of them, respectively.

1.1 Single-File Diffusion

In various situations stochastic motion of particles takes place in confined spaces. The confinement can be of a rather different nature, depending on system in question. Examples range from macroscopic systems to processes in nano-world, including traffic flow dynamics [1], customers waiting in tandem queues [2] (a well known situation when just after waiting to be served in one queue a person is immediately sent to another), movement of pedestrians in a pedestrian zone [3] or ants following trails [4] (the two very similar phenomena where confinement is not static since trails may evolve in time). On micro- and nano-meter scales, we encounter numerous systems which are of great interest in modern biophysics and chemistry like propagation of bacteria through confined spaces [5] and a broad spectrum of processes involved in intracellular transport [6, 7] (see below).

In the thesis we focus on Brownian motion taking place under, in a sense, the most extreme case of the external confinement. We assume that Brownian particles move in narrow channels, the channels being so narrow that their diameter is comparable with the diameter of Brownian particles. The second important ingredient of the model is the interparticle interaction. We consider only the hard-core interaction between the particles (also known as the excluded-volume or steric interaction), which means that the volume occupied by a single particles is inaccessible to other particles. As a consequence, the Brownian motion of particles will be restricted to a one-dimensional domain (infinite line, half-line, or finite interval) and, during the diffusion, the neighboring particles are not allowed to pass each other.

Diffusion in such conditions is known as the single-file diffusion (SFD). The concept of SFD has been originally introduced in 1955 in biophysics to explain anomalous properties of transport of ions through molecular-sized channels in membranes [8]. Since that time many systems has been discovered where SFD is the basic mechanism of mass transport. For example, the processes from cell biology like motion of proteins on double-stranded DNA [9, 10] and sliding of ribosomes along messenger RNA (transcription of genetic information) [6]. Further examples of SFD comprise one-dimensional conductors [11, 12], polymers translocating by reptation [13], diffusion in zeolites (important catalysts and molecular sieves) [14–18], and inside nanotubes [19]. Recently several artificial systems, where the motion of colloids is constrained to one dimension, has been realized experimentally in order to test the basic properties of molecules involved in SFD [20–26].

In mathematical literature SFD has been introduced in 1965 by Harris [27] who derived the basic law which nowadays is considered to be the hallmark of SFD. Harris has shown that the mean squared displacement of a given marked particle (a tagged particle or a tracer) grows with time as \(t^{1/2}\) in contrast to the linear time-dependence observed for a single noninteracting Brownian particle. The slowdown of the diffusion emerges from the hindering of the motion of a tagged particle caused by collisions with its nearest neighbors. From a general perspective, this result illustrates that in low-dimensional nonequilibrium systems even the simplest interactions (like the hard-core one) can lead to rich physical behavior [28–32]. Of course, this is in sharp contrast to what is known from the equilibrium statistical physics, where classical one-dimensional systems nowadays serve mainly as pedagogical tools.

In the thesis we address the motion of the tracer in the single-file system with absorbing boundaries. The emphasis is on an interplay between the hard-core interparticle interaction and the absorption process. Exact probability density functions (PDFs) for a position of the tracer diffusing under different conditions are derived. Starting from these exact PDFs, the dynamics and the first-passage properties of the tracer are discussed for different geometries and initial conditions.

1.2 Stochastic Energetics

Above, we have focused our attention on the stochastic dynamics of particles. The dynamics, however, represents only one part of the whole physical picture. An equally important part concerns with energy transformations in small nonequilibrium systems. A theoretical framework which has been designed to study energy flows in systems governed by stochastic evolution equations (in our case by the Langevin equation [33]) is known as the stochastic energetics [34] (or the stochastic thermodynamics [35], we will use the both terms interchangeably).

The Langevin equation for a Brownian particle immersed in a fluid is, in itself, consistent with well established laws of the classical thermodynamics. The equation contains the damping term (dissipation) and the noise term (fluctuations) which physically originate from the same source (interaction with the molecules of the surrounding liquid) and hence the two terms are not independent. They are connected by the Einstein’s (fluctuation-dissipation) relation for the diffusion coefficient. As a result, for any time-independent confining potential, the system described by the Langevin equation will eventually reach a Gibbsian canonical equilibrium state. Thus the consistency with the well established results of equilibrium statistical mechanics is achieved.

Stochastic energetics, introduced by Sekimoto [36, 37], goes far beyond above considerations. Its main goal is to provide a direct link from the stochastic dynamical equations to the thermodynamic description of the nonequilibrium process. Within the framework of the stochastic energetics, the quantities known from the classical thermodynamics, like work, heat and entropy, are identified along individual stochastic trajectories of the system. Thus defined (generalized) thermodynamic formalism holds for small systems, where fluctuations are inseparable from the dynamics, and for arbitrarily far-from-equilibrium processes. One of the advantages of the stochastic energetics (as compared e.g. to a more fundamental thermostated Hamiltonian dynamics) is that the analysis based on the Langevin equation (or on the Markovian master equation for discrete-state systems [35]) has proven to be particularly suitable for description of experiments on small systems (see Chap. 5 for details).

The paradigmatic system in the field of stochastic energetics is the Brownian particle diffusing in a confining external potential, which can be realized e.g. by the optical trap. Although the properties of the PDF for the position of the particle are relatively well understood [33], PDFs that characterize energetic quantities remain less explored. In the thesis we investigate a distribution of work performed on the Brownian particle diffusing in a time-dependent asymmetric potential well. The potential consists of a harmonic component with a time-dependent force constant and of a time-independent logarithmic barrier at the origin. The model is exactly solvable. The exact result for the characteristic function of the work allows us to extract essential properties of the work PDF, e.g., all its moments and the both tails. In particular, the results could be of interest for experimental determination of free energies using the Jarzynski equality (as discussed in detail in Chap. 5), where the tail of the work PDF for large negative values of work has two properties: (1) it corresponds to rare events which are almost never observed in experiments; (2) it significantly contributes to the value of the exponential average occurring in the Jarzynski equality (cf. Eq. (5.7)) and thus also to the value of the estimated free energy.

1.3 Thesis Organization

The first part of the thesis (Chaps. 2–4) is devoted to the single-file diffusion. Classical approaches and new directions in the theory of tracer dynamics are reviewed in Chap. 2. We would like to emphasize that the focus here is on the properties of a tagged particle. Which means that collective phenomena like nonequilibrium phase transitions and other intriguing topics are left without comment (we refer to Refs. [28–32] for more details).

In Chap. 3 we discuss the dynamics and the first-passage properties of the tracer in a semi-infinite system with a single absorbing boundary for two qualitatively different initial conditions. First, we consider the system with (initially) finite number of particles (Sect. 3.2), and, second, the system in the thermodynamic limit where the number of particle is infinite, but the initial mean density is constant (Sect. 3.3). In the both cases the first-passage properties (survival probabilities, PDFs for times of absorption) and the tracer dynamics (time-dependence of PDFs and their moments for both the unconditioned dynamics and the dynamics conditioned on nonabsorption) are deduced from the exact PDF of the tracer position. The latter is constructed using the mapping between the SFD system and the corresponding system of noninteracting particles (which is a direct generalization of ideas for a system without absorption as reviewed in Chap. 2).

Chapter 4 generalizes the analysis of Chap. 3 to the case of a finite interval with two types of boundary conditions: (i) both boundaries are absorbing (Sect. 4.2); (ii) one boundary is absorbing and the second boundary is reflecting (Sect. 4.3). The focus is on the first-passage properties and on their scaling behavior for large system size and for large initial number of particles. Sect. 4.2.3 accounts for possibility of random interval length.

The second part of the thesis (Chaps. 5 and 6) is devoted to the stochastic thermodynamics. In Chap. 5 we first define (stochastic) work and heat, and second, we review the two most widely known fluctuation theorems (the Crooks theorem and the Jarzynski equality) and their roles in determination of free-energy landscapes of macromolecules.

Chapter 6 addresses a nontrivial model for which the work characteristic function can be obtained exactly. Using the Lie-algebraic approach, the task to solve the Fokker-Planck equation for the joint PDF of work and position is reduced to the solution of a Riccati equation and to the evaluation of two quadratures (Sect. 6.2). PDF for particle position is derived in a closed form for any external driving (Sect. 6.3). On the other hand, it is only for a specific driving protocol that the Riccati equation is solved exactly in terms of elementary functions (Sect. 6.4) yielding desired information about work PDF including all its moments and the both its tails (Sects. 6.4, 6.5).

The thesis is concluded by a brief summarizing chapter (unnumbered). Notice that full-length concluding sections discussing main physical features of individual models are presented at the ends of Chaps. 3, 4, and 6, see Sects. 3.4, 4.4, and 6.5, respectively.