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We introduce an approach based on the Chapman-Kolmogorov equation to model heterogeneous stochastic circuits, namely, the circuits combining binary or multi-state stochastic memristive devices and continuum reactive components (capacitors and/or inductors). Such circuits are described in terms of occupation probabilities of memristive states that are functions of reactive variables. As an illustrative example, the series circuit of a binary memristor and capacitor is considered in detail. Some analytical solutions are found. Our work offers a novel analytical/numerical tool for modeling complex stochastic networks, which may find a broad range of applications.

Efficient simulation of stochastic memristors and their networks requires novel modeling approaches. Utilizing a master equation to find occupation probabilities of network states is a recent major departure from typical memristor modeling [Chaos, solitons fractals 142, 110385 (2021)]. In the present article we show how to implement such master equations in SPICE – a general purpose circuit simulation program. In the case studies we simulate the dynamics of acdriven probabilistic binary and multi-state memristors, and dc-driven networks of probabilistic binary and multi-state memristors. Our SPICE results are in perfect agreement with known analytical solutions. Examples of LTspice code are included.

Window functions are widely employed in memristor models to restrict the changes of the internal state variables to specified intervals. Here, we show that the actual choice of window function is of significant importance for the predictive modelling of memristors. Using a recently formulated theory of memristor attractors, we demonstrate that whether stable fixed points exist depends on the type of window function used in the model. Our main findings are formulated in terms of two memristor attractor theorems, which apply to broad classes of memristor models. As an example of our findings, we predict the existence of stable fixed points in Biolek window function memristors and their absence in memristors described by the Joglekar window function, when such memristors are driven by periodic alternating polarity pulses. It is anticipated that the results of this brief will contribute toward the development of more sophisticated models of memristive devices and systems.

Some memristors are quite interesting from the point of view of dynamical systems. When driven by narrow pulses of alternating polarities, their dynamics has a stable fixed point, which may be useful for future applications. We study the transient dynamics of two types of memristors characterized by a stable fixed point using a time-averaged evolution equation. Time-averaged trajectories of the Biolek window function memristor and resistor-threshold type memristor circuit (an effective memristor) are determined analytically, and the times of relaxation to the stable fixed point are found. Our analytical results are in perfect agreement with the results of numerical simulations.

The possibility of using non-deterministic circuit components has been gaining significant attention in recent years. The modeling and simulation of their circuits require novel approaches, as now the state of a circuit at an arbitrary moment in time cannot be predicted deterministically. Generally, these circuits should be described in terms of probabilities, the circuit variables should be calculated on average, and correlation functions should be used to explore interrelations among the variables. In this paper, we use, for the first time, a master equation to analyze the networks composed of probabilistic binary memristors. Analytical solutions of the master equation for the case of identical memristors connected in-series and in-parallel are found. Our analytical results are supplemented by results of numerical simulations that extend our findings beyond the case of identical memristors. The approach proposed in this paper facilitates the development of probabilistic/stochastic electronic circuits and advance their real-world applications.

One-dimensional (1D) memristive networks are the simplest type of memristive networks one can imagine. Yet, despite their morphological simplicity, such networks represent an important class of memory networks characterized by the strongest interaction among the network components. This chapter reviews several important dynamical features of 1D memristive networks composed of realistic threshold-type memristive systems. First of all, the accelerated and decelerated switching regimes of memristive systems are introduced and exemplified. Secondly, the phenomenon of switching synchronization is presented. Finally, it is shown that metastable transmission lines composed of metastable memristive circuits can be used to transfer the information from one space location to another. Here, the information transfer occurs in the form of a switching front propagating along the line resembling a kink in, say, classical \(\phi ^4\) field theory model. Importantly, such memristive kinks can also be used for information processing purposes. This chapter thus reveals the triad of memristive systems functionalities in their 1D networks: information processing, storage and transfer.

Traditional studies of memristive devices have mainly focused on their applications in nonvolatile information storage and information processing. Here, we demonstrate that the third fundamental component of information technologies-the transfer of information-can also be employed with memristive devices. For this purpose, we introduce a metastable memristive circuit. Combining metastable memristive circuits into a line, one obtains an architecture capable of transferring a signal edge from one space location to another. We emphasize that the suggested metastable memristive lines employ only resistive circuit components. Moreover, their networks (for example, Y-connected lines) have an information processing capability.

Kinks and antikinks of the classical ${\ensuremath{\varphi}}^{4}$ field model are topological solutions connecting its two distinct ground states. Here we establish an analogy between the excitations of a long graphene nanoribbon buckled in the transverse direction and ${\ensuremath{\varphi}}^{4}$ model results. Using molecular dynamics simulations, we investigated the dynamics of a buckled graphene nanoribbon with a single kink and with a kink-antikink pair. Several features of the ${\ensuremath{\varphi}}^{4}$ model have been observed including the kink-antikink capture at low energies, kink-antikink reflection at high energies, and a bounce resonance. Our results pave the way towards the experimental observation of a rich variety of ${\ensuremath{\varphi}}^{4}$ model predictions based on graphene.

This paper presents a study of bifurcation in the time-averaged dynamics of TaO memristors driven by narrow pulses of alternating polarities. The analysis, based on a physics-inspired model, focuses on the stable fixed points and on how these are affected by the pulse parameters. Our main finding is the identification of a driving regime when two stable fixed points exist simultaneously. To the best of our knowledge, such bistability is identified in a single memristor for the first time. This result can be readily tested experimentally, and is expected to be useful in future memristor circuit designs.

It is shown that the time-averaged dynamics of memristors and their networks periodically driven by alternating-polarity pulses may converge to fixed-point attractors. Starting with a general memristive system model, we derive basic equations describing the fixed-point attractors and investigate attractors in the dynamics of ideal, threshold-type and second-order memristors, and memristive networks. A memristor potential function is introduced, and it is shown that in some cases the attractor identification problem can be mapped to the problem of potential function minimization. Importantly, the fixed-point attractors may only exist if the function describing the internal state dynamics depends on an internal state variable. Our findings may be used to tune the states of analog memristors, and also be relevant to memristive synapses subjected to forward- and back-propagating spikes.

The problem of the creation of high-energy phonons (h-phonons) by a pulse of low-energy phonons (l-phonons) moving from a heater to a detector in superfluid helium is solved. The rate of h-phonon creation is obtained, and it is shown that the h-phonons created occupy a much smaller solid angle in momentum space than the l-phonons. An analytical expression for the creation rate of h-phonons, along the symmetry axis of a pulse, is derived. It allows us to get useful approximate analytical expressions for the creation rate of h-phonons. The time dependences of the parameters describing the l-phonon pulse are obtained. This shows that half of the initial energy of l-phonon pulse can be transferred into h-phonons. The results of the calculations are compared with experimental data, and we show that this theory explains a number of experimental results. The value of the momentum that separates the l- and h-phonon subsystems is found.

We study transmission lines made of memory capacitive (memcapacitive) materials. The transmission properties of these lines can be adjusted on demand using an appropriate sequence of pulses. In particular, we demonstrate a pulse combination that creates a periodic modulation of dielectric properties along the line. Such a structure resembles a distributed Bragg reflector having important optical applications. We present simulation results demonstrating all major steps of such a reconfigurable device operation including reset, programming, and transmission of small amplitude signals. The proposed reconfigurable transmission lines employ only passive memory materials and can be realized using available memcapacitive devices.

We report on an astonishing switching synchronization phenomenon in one-dimensional memristive networks, which occurs when several memristive systems with different switching constants are switched from the high to low resistance state. Our numerical simulations show that such a collective behavior is especially pronounced when the applied voltage slightly exceeds the combined threshold voltage of memristive systems. Moreover, a finite increase in the network switching time is found compared to the average switching time of individual systems. An analytical model is presented to explain our observations. Using this model, we have derived asymptotic expressions for memory resistances at short and long times, which are in excellent agreement with our numerical calculations.

The rate of one phonon decaying into three is calculated in the momentum range where one phonon decaying into two is forbidden in liquid 4He. Exact and useful approximate expressions for the rate are obtained. Its dependence on all parameters, and the physical reasons for these dependences, are analysed. It is shown that processes where one phonon scatters into three, and three phonons scatter to one phonon, rapidly establishes equilibrium in anisotropic and isotropic phonon systems. This shows that the momentum range, where one to three phonon processes are allowed, should be included in the subsystem of low-energy phonons where equilibrium occurs quickly, and not in the subsystem of high-energy phonons where spontaneous decay processes are forbidden and equilibrium occurs slowly.

The thermodynamic functions of anisotropic phonon systems in superfluid helium are calculated for all levels of anisotropy. The results show that the thermodynamic functions of strongly anisotropic phonon systems are essentially different from isotropic ones. It is shown that for strongly anisotropic phonon systems, in thermodynamic equilibrium, the energy density of high-energy phonons, e/kB≥10 K, is more than ten times higher than in a cone with the same total energy density and with the Bose–Einstein distribution for an isotropic system. The stability curve for anisotropic phonon systems is derived and it is shown that strongly anisotropic phonon systems are thermodynamically stable over a wide temperature range.

The longitudinal evolution of a low energy phonon pulse in superfluid helium 4He is analysed theoretically. The phonons are considered to be strongly interacting and are described by the exact local equilibrium distribution function. The theory describes how the longitudinal pulse shape develops. We find that the temperature change in the pulse is determined by a simple running wave which moves with a velocity very close that of the sound velocity. This results in only a small variation of the pulse length as the pulse propagates. This is in contrast to the case of a non-interacting phonon pulse, which shows considerable dispersion. The details of the phonon pulse deformation depend on whether the pulse contains phonons with relatively high energies or not. The higher energy phonons cause the single wave to eventually break at both the front and back of the pulse. Without these high energy phonons, the pulse only breaks at the front. We find that, in the first approximation with respect to the small parameters of the problem, the anisotropic nature of the phonon pulse is conserved as it propagates.

The equations for the transverse evolution of the anisotropic phonon system are solved for the case of cylindrical symmetry. The angular distribution of the phonon-energy density is obtained at large distances from the heater.

The evolution of a pulse of noninteracting quasiparticles, caused by their different velocities and angular distribution of momenta, is studied theoretically. Equations are found that describe the shape of the pulse surface at any time. The times of the beginning and end and the duration of the quasiparticle energy flux density is determined at a general spatial point. The quasiparticle energy density is considered at all times and positions, and it is shown that in the region of high energy density, in the middle of the pulse, it is equal to the initial energy density under certain conditions. These theoretical results are discussed in relation to experimental data on the evolution of a pulse of noninteracting phonons in superfluid helium.

The system of equations that describes the temporal and spatial evolution of low-energy phonons (l-phonons) is obtained for the cases of plane and axial symmetry. From the solutions of these equations, we discuss the time-resolved measurements of the angular distribution of l-phonons. These show a rather unusual mesa shape with a flat top in the direction normal to the heater. We consider that the flat top is due to the creation of high-energy phonons (h-phonons).

Here we suggest a hybrid spin noise spectroscopy technique, which is sensitive to the spin Hall effect. Our main idea is to study correlations between transverse voltage fluctuations and spin polarization fluctuations, which appear as a result of spin-charge coupling that is fundamental for the spin Hall effect. It is shown that, while the standard spin-spin correlation function is not sensitive to the spin Hall effect, spin-transverse voltage and transverse voltage--transverse voltage correlation functions provide the missing sensitivity, which is linear and quadratic in the spin Hall coefficient, respectively. We anticipate that the proposed method could find applications in studies of spin-charge coupling in semiconductors and other materials.

Spin Noise Spectroscopy (SNS) is an experimental approach to obtain correlators of mesoscopic spin fluctuations in time by purely optical means. We explore the information that this technique can provide when it is applied to a weakly non-equilibrium regime when an electric current is driven through a sample by an electric field. We find that the noise power spectrum of conducting electrons experiences a shift, which is proportional to the strength of the spin-orbit coupling for electrons moving along the electric field direction. We propose applications of this effect to measurements of spin orbit coupling anisotropy and separation of spin noise of conducting and localized electrons., 5 pages, 2 figures

Can we change the average state of a resistor by simply applying white noise? We show that the answer to this question is positive if the resistor has memory of its past dynamics (a memristive system). We also prove that, if the memory arises only from the charge flowing through the resistor -- an ideal memristor -- then the current flowing through such memristor can not charge a capacitor connected in series, and therefore cannot produce useful work. Moreover, the memristive system may skew the charge probability density on the capacitor, an effect which can be measured experimentally.

We propose a method of two-beam spin noise spectroscopy to test the spin transport at equilibrium via analysis of correlations between time-shifted spin fluctuations at different space locations. This method allows one to determine the strength of spin-orbit interaction and spin relaxation time and separate spin noise of conducting electrons from the background noise of localized electrons. We formulate a theory of two-beam spin noise spectroscopy in semiconductor wires with Bychkov-Rashba spin-orbit interaction taking into account several possible spin relaxation channels and finite size of laser beams. Our theory predicts a peak shift with respect to the Larmor frequency to higher or lower frequencies depending on the strength of spin orbit interaction and distance between the beams. The two-beam spin noise spectroscopy could find applications in experimental studies of semiconductors, emergent materials, and many other systems.

We study electron spin relaxation in one-dimensional structures of finite length in the presence of Bychkov-Rashba spin-orbit coupling and boundary spin relaxation. Using a spin kinetic equation approach, we formulate boundary conditions for the case of a partial spin polarization loss at the boundaries. These boundary conditions are used to derive corresponding boundary conditions for a spin drift-diffusion equation. The latter is solved analytically for the case of relaxation of a homogeneous spin polarization in one-dimensional finite-length structures. It is found that the spin relaxation consists of three stages (in some cases, two)---an initial D'yakonov-Perel' relaxation is followed by spin helix formation and its subsequent decay. Analytical expressions for the decay time are found. We support our analytical results by results of Monte Carlo simulations.

Memristive systems, namely resistive systems with memory, are attracting considerable attention due to their ubiquity in several phenomena and technological applications. Here, we show that even the simplest one-dimensional network formed by the most common memristive elements with voltage threshold bears non-trivial physical properties. In particular, by taking into account the single element variability we find i) dynamical acceleration and slowing down of the total resistance in adiabatic processes, ii) dependence of the final state on the history of the input signal with same initial conditions, iii) existence of switching avalanches in memristive ladders, and iv) independence of the dynamics voltage threshold with respect to the number of memristive elements in the network (scale invariance). An important criterion for this scale invariance is the presence of memristive systems with very small threshold voltages in the ensemble. These results elucidate the role of memory in complex networks and are relevant to technological applications of these systems.

In this paper we use a spin kinetic equation to study spin-polarization dynamics in one-dimensional (1D) wires and 2D channels. The spin kinetic equation is valid in both diffusive and ballistic spin transport regimes and therefore is more general than the usual spin drift-diffusion equations. In particular, we demonstrate that in infinite 1D wires with Rashba spin-orbit interaction the exponential spin-relaxation decay can be modulated by an oscillating function. In the case of spin relaxation in finite length 1D wires, it is shown that an initially homogeneous spin polarization spontaneously transforms into a persistent spin helix. We find that a propagating spin-polarization profile reflects from a system boundary and returns back to its initial position similarly to the reflectance of sound waves from an obstacle. The Green's function of the spin kinetic equation is derived for both finite and infinite 1D systems. Moreover, we demonstrate explicitly that the spin relaxation in specifically oriented 2D channels with Rashba and Dresselhaus spin-orbit interactions of equal strength occurs similarly to that in 1D wires of finite length. Finally, a simple transformation mapping 1D spin kinetic equation into the Klein-Gordon equation with an imaginary mass is found thus establishing an interesting connection between semiconductor spintronics and relativistic quantum mechanics.

We find an exact solution for the problem of electron spin relaxation in a two-dimensional (2D) circle with Rashba spin-orbit interaction. Our analysis shows that the spin relaxation in finite-size regions involves three stages and is described by multiple spin relaxation times. It is important that the longest spin relaxation time increases with the decrease in system radius but always remains finite. Therefore, at long times, the spin polarization in small 2D systems decays exponentially with a size-dependent rate. This prediction is supported by results of Monte Carlo simulations.

We demonstrate that a homogeneous spin polarization in one-dimensional structures of finite length in the presence of Bychkov-Rashba spin-orbit coupling decays spontaneously toward a persistent spin helix. Such a strikingly different and simple method enables us to generate robust spin structures whose properties can be tuned by the strength of the spin-orbit interaction and/or the structure's length. We generalize our results for the two-dimensional case predicting the formation of a persistent spin helix in two-dimensional channels from homogeneous spin polarization. An analysis of the formation of a spin helical state is presented within an approach based on a mapping of spin drift-diffusion equations into a heat equation for a complex field.

Spin relaxation dynamics in rings with Rashba spin-orbit coupling is investigated using spin kinetic equation. We find that the spin relaxation in rings occurs toward a persistent spin configuration whose final shape depends on the initial spin polarization profile. As an example, it is shown that a homogeneous parallel to the ring axis spin polarization transforms into a persistent crown-like spin structure. It is demonstrated that the ring geometry introduces a geometrical contribution to the spin relaxation rate speeding up the transient dynamics. Moreover, we identify several persistent spin configurations as well as calculate the Green function of spin kinetic equation.

We present a theory for the formation of a mesa shaped phonon pulse in superfluid $^4$He. Starting from the hydrodynamic equations of superfluid helium, we obtain the system of equations which describe the evolution of strongly anisotropic phonon systems. Such systems can be created experimentally. The solution of the equations are simple waves, which correspond to second sound in the moving phonon pulse. Using these exact solutions, we describe the expansion of phonon pulses in superfluid helium at zero temperature. This theory gives an explanation for the mesa shape observed in the measured phonon angular distributions. Almost all dependencies of the mesa shape on the system parameters can be qualitatively understood., 29 pages, 13 figures

The thermodynamic properties of anisotropic quasiparticle systems of He II are considered for all degrees of anisotropy. It is shown that the thermodynamic functions of a strongly anisotropic phonon-roton system are mainly determined by rotons at all temperatures. Analytical expressions for the roton thermodynamic functions are obtained for all degrees of anisotropy. The maximum anisotropy is limited by the criterion for thermodynamic stability, which is here derived for the whole temperature range.

The phenomenon of a hot line forming in liquid helium was observed in experiments carried out in the University of Exeter (UK). It arises when two phonon pulses interact and this is theoretically investigated in this paper. To develop the theory we start from the exact quasiequilibrium distribution function that describes anisotropic phonon systems such as a phonon pulse in superfluid helium. This is related to the approximate distribution function, which is more physically intuitive and was used earlier. The local equilibrium distribution function for phonons in the region of a hot line is obtained from the distribution functions for the phonons in the two interacting pulses. In order to explain the results of experiments, we analyze the effect of different pressures when the angle between the two moving pulses in superfluid helium is constant and also the effect of different angles at the saturated vapor pressure. The conditions suitable for the creation of a hot line are found. The results of the calculations are compared with the experimental data.

Starting from the solution of the kinetic equation, we have calculated the distribution function for the high-energy phonons, which are created by a short pulse of low-energy phonons moving in superfluid helium. This enables an explicit expression for the energy density flux to be derived. Hence we find the amplitude of the high-energy phonon signal as a function of time on a bolometer. We divide this signal into two halves: the "head" and "tail" which arrive before and after the peak signal, respectively. We analyze which high-energy phonons form the head and tail of the signal. The half-widths of head and tail are calculated and approximate formulas which describe the shapes of them are obtained. The partial contribution of high-energy phonons, with different momenta, to the total signal at different times is determined. These results are compared with the experimental results given in the preceding paper [R. V. Vovk, C. D. H. Williams, and A. F. G. Wyatt, Phys. Rev. B 69, 144524 (2004)].

We derive the system of equations which describes the evolution of a cool phonon pulse during its propagation in superfluid helium. Solutions are found both for longitudinal deformation of the pulse along the pulse axis and for the transverse evolution perpendicular to the pulse axis. It is shown that the longitudinal motion of the pulse is described by a simple running wave, and the transverse evolution is similar to the expansion of a gas in a vacuum. From these solutions, the angular distribution of phonon-energy density is calculated at various distances from the source. We discuss the very unusual evolution of the phonon pulse and compare it to experimentally observed phenomena.

The system of equations that describes the sound propagation in superfluid helium at arbitrary values of the relative velocity of the normal and superfluid component is obtained. By using the general dispersion relation of this system, we analyze the influence of a finite relative velocity on the first sound velocity. We find that the change in the first sound velocity is determined by the small value of the thermal expansion coefficient.