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We discuss the early investigations of Brownian motion as a stochastic process by surveying contributions by Fick and Rayleigh developed later in works of Einstein, Smoluchowski, Langevin, Fokker, Planck, Klein, Kramers and Pauli. In particular, we are interested in the influence of the theory of probability in the development of the kinetic theory and we briefly analyze the origin of fundamental equations used in the mathematical description of the stochastic processes.

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The phenomenon of giant suppression of activation, when two or more correlated noise signals act on the system, was found a few years ago in dynamical bistable or metastable systems. When the correlation between these noise signals is strong enough and the amplitudes of the noise are chosen correctly — the life time of the metastable state may be longer than in the case of the application of only a single noise even by many orders of magnitude. In this paper, we investigate similar phenomena in systems exhibiting several chaotic transitions: Pomeau–Manneville intermittency, boundary crisis and interior crisis induced intermittency. Our goal is to show that, in these systems the application of two noise components with the proper choice of the parameters in the case of intermittency can also lengthen the mean laminar phase length or, in the case of boundary crisis, lengthen the time the trajectory spends on the pre-crisis attractor. In systems with crisis induced intermittency, we introduce a new phenomenon we called quasi-deterministic giant suppression of activation in which the lengthening of the laminar phase lengths is caused not by the action of two correlated noise signals but by a single noise term which is correlated with one of the chaotic variables of the system.

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By means of the recently developed ergometer, Birkhoff’s theorem is made physically useful. In particular, the conditions of the theorem are given an interpretation through a many-body model which exhibits both ergodic and nonergodic behavior depending on the range of a certain parameter of the model. To our knowledge these illustrations are the first known examples of the use of Birkhoff’s theorem in many-body theory.

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Competition between subdiffusion and Lévy flights is conveniently described by the fractional Fokker–Planck equation with temporal and spatial fractional derivatives. The equivalent approach is based on the subordinated Langevin equation with stable noise. In this paper we examine the properties of such Langevin equation with the heavy-tailed noise belonging to the class of geometric stable distributions. In particular, we consider two physically relevant examples of geometric stable noises, namely Linnik and Mittag–Leffler. We describe in detail a numerical algorithm for visualization of subdiffusion coexisting with Lévy flights. Using Monte Carlo simulations we demonstrate the realizations as well as the probability density functions of the considered anomalous diffusion process.

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We propose a diffusion, continuous-time random walk (CTRW) scenario which is based on generalization of processes counting the number of jumps performed by a walker. We substitute the renewal counting process, used in the classical CTRW framework, by a compound counting process. The construction of such a compound process involves renormalized clustering of random number of walker’s spatio-temporal subsequent steps. The family of the renormalized steps defines a new class of coupled CTRWs. The diffusion front of the studied process exhibits properties which help us to enlarge the class of relaxation models discussed yet in the classical CTRW framework.

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A network of Greenberg–Hasting cellular automata with cyclic intrinsic dynamics \({F}\rightarrow {R} \rightarrow {A}\rightarrow {F}\rightarrow \dots \) is shown to be a reliable approximation to the cardiac pacemaker. The three possible cell’s states \(F\), \(R\), \(A\) are characterized by fixed timings \(\{n_{F}, n_{R}, n_{A}\}\) — time steps spent in each state. Dynamical properties of a simple line network are found to be critical with respect to the relation between \(n_{F}\) and \(n_{R}\). The properties of a network arisen from a square lattice where some edges are rewired (locally and with the preference to link to cells which are more connected to other cells) are also studied. The resulted system evolves rhythmically with the period determined by timings. The emergence of a small group of neighboring automata where the whole system activity initiates is observed. The dominant evolution is accompanied with other rhythms, characterized by longer periods.

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We give meaning to the first and second laws of thermodynamics in case of mesoscopic out-of-equilibrium systems which are driven by diffusion-type, specifically Smoluchowski, processes. The notion of entropy production is analyzed. The role of the Helmholtz extremum principle is contrasted to that of the more familiar entropy extremum principles.

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Stochastic resonance is investigated in the Ising model with ferromagnetic coupling on scale-free networks with various scaling exponents \(\gamma \gt 2\) of the degree distributions \(p(k)\propto k^{-\gamma }\), subjected to a weak oscillating magnetic field. In the case \(2 \lt \gamma \lt 3\) and for slow to moderate frequencies of the input signal the linear response theory and numerical simulations in the mean-field approximation predict the occurrence of stochastic multiresonance, with the spectral power amplification as a function of temperature exhibiting double maxima in the vicinity of and below the crossover temperature for the ferromagnetic transition. In the case \(\gamma \gt 3\) the spectral power amplification is expected to exhibit single maximum close to the critical temperature. These predictions are qualitatively confirmed by Monte Carlo simulations of the Ising model on scale-free networks obtained from a preferential attachment growing procedure.

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We investigate an optimal regime of negative-valued conductance, emerging in a resistively and capacitively shunted Josephson junction, which is driven simultaneously by both, a time-periodic (ac) and a constant (dc) current. We analyze the current-voltage characteristics in the regime of absolute negative conductance. We additionally explore the stability of the negative response with respect to the ac-current frequency.

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We consider the deterministic escape of a chain of harmonically coupled units from a metastable state over a cubic potential barrier. The underlying dynamics is conservative and noise-free. The supply of a sufficient total energy transforms the chain into the nonlinear regime from which an initially, nearly uniform lattice configuration becomes unstable, yielding a redistribution of energy. In an early stage of the dynamics, we estimate the degree of energy exchange enabling the coupled system to form strongly localized modes which eventually grow into a critical nucleus. Upon passing this transition state, the nonlinear chain performs a collective, deterministic escape. We analyze the associated nonlinear dynamics in phase space and relate the escape to diffusion in a separatrix layer.

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The reduced 1D Poisson–Nernst–Planck (PNP) model of artificial nanopores in the presence of a permanent charge on the channel wall is studied. More specifically, we consider the limit where the channel length exceed much the Debye screening length and channel’s charge is sufficiently small. Ion transport is described by the nonequillibrium steady-state solution of the PNP system within a singular perturbation treatment. The quantities, \(1\)/\(\lambda \) — the ratio of the Debye length to a characteristic length scale and \(\epsilon \) — the scaled intrinsic charge density, serve as the singular and the regular perturbation parameters, respectively. The role of the boundary conditions is discussed. A comparison between numerics and the analytical results of the singular perturbation theory is presented.

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The influence of concentrations of reagents on the rate of reaction: \(A+B\to C+B\) has been investigated by performing large scale computer simulations on systems of spherically symmetric molecules. The problem has been analyzed both for deterministic systems (gas, liquid) and for the stochastic ones where the particles are immersed in the Brownian medium. Significant deviations from the law of mass action have been found. The deviations result not only from fluctuations in concentrations of reagents. For the Brownian systems the simulations have given a positive value of the excess in the rate coefficient even for very long times, which contradicts with general expectations. The reason for the positive excess values is repulsion between \(B\) particles (the so called excluded volume effect). The effect is strongly influenced by the shape of the probability density function for \(B\)–\(B\) pairs. As a result, for the liquids the effect is weaker: for short times the excess in rate coefficient is positive but for long times it becomes negative.

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A flux qubit based on a non-superconducting mesoscopic ring coupled to two split heat baths at different temperatures is studied. Heat currents flowing in such a nonequilibrium quantum thermodynamic system are analyzed. A method of control of heat transfer via the qubit is proposed.

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Excitable chemical system can process information coded in excitation pulses. Here we discuss the simplest realization of a chemical signal diode that transmits pulses in one direction only. The construction of such diode just requires a geometrical combination of three areas characterized by different excitability. The proposed construction of chemical signal diode has been tested in numerical simulations and verified in experiments with ruthenium (Ru) catalyzed Belousov–Zhabotinsky (BZ) reaction.

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We consider an \(\mathrm {A+B\rightarrow 0}\) reaction in a flat subdiffusive medium in contact with two well mixed reservoirs of particles of both types on the sides. We show that the behaviour of the stationary concentration and reaction intensity profiles in subdiffusion differs strikingly from that observed in simple diffusion. The most marked differences correspond to accumulation peaks and depletion zones in the concentration profile. The height of these peaks as well as the height of the reaction zones exhibit a non-monotonic behaviour with respect to the reactant’s concentrations at the boundaries. These characteristics are due to an effectively nonlinear transport under reaction which emerges from the non-Markovian property of the subdiffusion process.

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We use the hyperbolic subdiffusion equation with fractional time derivatives (the generalized Cattaneo equation) to study the transport process of electrolytes in subdiffusive media such as gels and porous media. In particular, we obtain the formula of electrochemical subdiffusive impedance of a spatially limited sample for large pulsation of electric field.

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Comparing the experimental results to theoretical functions, we estimate the subdiffusion coefficient of PEG2000 in agarose gel. The experiment was performed with the two-membrane system where thin membranes separated homogeneous solution of PEG2000 for pure solvent at an initial moment. The theoretical function was found by solving analytically the subdiffusion equation.

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A model system of a bright state coupled to a manifold of dark states is analyzed with regard to the width distributions of the dark manifold induced in the bright state. Independent box shaped distributions are assumed for the energy distributions, the coupling distributions and the dark level width-distributions. The width distributions induced via the couplings from the dark levels into the bright state can be expressed in terms of Lévy functions in the limit of a sparse level density which relates to anomalous long-time relaxations of the state selected survival probabilities.

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We study systems with Hamiltonian dynamics type coupled to reservoirs providing free energy which may be converted into acceleration. In the first part we introduce general concepts, like canonical dissipative systems and find exact solutions of associated Fokker–Planck equations that describe time evolutions of systems at hand. Next we analyze dynamics in ratchets with energy support which might be treated by perturbation theory around canonical dissipative systems. Finally we discuss possible applications of these ratchet systems to model the mechanism of biological energy conversion and molecular motors.

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Recent investigations call attention to the dynamics of anomalous diffusion and its connection with basic principles of statistical mechanics. We present here a short review of those ideas and their implications.

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The geometrical Brownian motion driven by Gaussian or dichotomous color noise is considered. The ordinary Malthusian evolution is observed for long times, however the initial values seem lowered and additionally, in the case of dichotomous noise, the rate of growth is decreased. In the latter case the possibility of arbitrage is shown explicitly.

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We study mixing in a large real social network consisting of over one million individuals, who form an Internet community and organise themselves in groups of different sizes. We consider mixing according to discrete characteristics such as gender and scalar characteristics such as age. On the basis of the users’ list of friends and other data registered in the database we investigate the structure and time development of the network. We found that in the network under investigation assortative mixing is observed, i.e. the tendency for vertices in network to be connected to other vertices that are like them in some way.

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We study features of the 27-day variations of the galactic cosmic ray (GCR) intensity and anisotropy using data of neutron monitors and solar wind (SW) velocity. We found that the larger amplitudes of the 27-day variations of the galactic cosmic ray anisotropy and intensity for the positive polarity period (\(A\gt 0\)) of solar magnetic cycle than for the negative polarity period (\(A\lt 0\)) in the minima epoch of solar activity are related with the heliolongitudinal asymmetry of the solar wind velocity. We reveal the long-lived (\(\sim 22\) years) active regions of the heliolongitudes being the sources of the 27-day variation of the solar wind velocity during the \(A\gt 0\) period. Also, we demonstrate an existence of the clear stable 27-day waves of the GCR intensity and anisotropy with the amplitudes larger in the \(A\gt 0\) than in the \(A\lt 0\) for the several individual Carrington rotations in the minima epoch of solar activity. We show that the solution of the Parker’s transport equation with the heliolongitudinal asymmetry of the solar wind velocity is in the reasonable agreement with the neutron monitors experimental data. We conclude that the larger amplitudes of the 27-day variations of the galactic cosmic ray anisotropy and intensity observed by the experimental data in the \(A\gt 0\) are related with the superposition of two effects: (1) the existence of the more regular heliolongitudinal asymmetry of the solar wind velocity, and (2) a coincidence of convective and drift streams of the GCR particles in comparison with the \(A\lt 0\).