Nonequilibrium Stationary Distributions Research Articles

Fractional and scaled Brownian motion on the sphere: The effects of long-time correlations on navigation strategies.

We analyze fractional Brownian motion and scaled Brownian motion on the two-dimensional sphere S^{2}. We find that the intrinsic long-time correlations that characterize fractional Brownian motion collude with the specific dynamics (navigation strategies) carried out on the surface giving rise to rich transport properties. We focus our study on two classes of navigation strategies: one induced by a specific set of coordinates chosen for S^{2} (we have chosen the spherical ones in the present analysis), for which we find that contrary to what occurs in the absence of such long-time correlations, nonequilibrium stationary distributions are attained. These results resemble those reported in confined flat spaces in one and two dimensions [Guggenberger et al. New J. Phys. 21, 022002 (2019)1367-263010.1088/1367-2630/ab075f; Vojta et al. Phys. Rev. E 102, 032108 (2020)2470-004510.1103/PhysRevE.102.032108]; however, in the case analyzed here, there are no boundaries that affect the motion on the sphere. In contrast, when the navigation strategy chosen corresponds to a frame of reference moving with the particle (a Frenet-Serret reference system), then the equilibrium distribution on the sphere is recovered in the long-time limit. For both navigation strategies, the relaxation times toward the stationary distribution depend on the particular value of the Hurst parameter. We also show that on S^{2}, scaled Brownian motion, distinguished by a time-dependent diffusion coefficient with a power-scaling, is independent of the navigation strategy finding a good agreement between the analytical calculations obtained from the solution of a time-dependent diffusion equationon S^{2}, and the numerical results obtained from our numerical method to generate ensemble of trajectories.

Fingerprints of nonequilibrium stationary distributions in dispersion relations

Distributions different from those predicted by equilibrium statistical mechanics are commonplace in a number of physical situations, such as plasmas and self-gravitating systems. The best strategy for probing these distributions and unavailing their origins consists in combining theoretical knowledge with experiments, involving both direct and indirect measurements, as those associated with dispersion relations. This paper addresses, in a quite general context, the signature of nonequilibrium distributions in dispersion relations. We consider the very general scenario of distributions corresponding to a superposition of equilibrium distributions, that are well-suited for systems exhibiting only local equilibrium, and discuss the general context of systems obeying the combination of the Schrödinger and Poisson equations, while allowing the Planck’s constant to smoothly go to zero, yielding the classical kinetic regime. Examples of media where this approach is applicable are plasmas, gravitational systems, and optical molasses. We analyse in more depth the case of classical dispersion relations for a pair plasma. We also discuss a possible experimental setup, based on spectroscopic methods, to directly observe these classes of distributions.

Approach to equilibrium and nonequilibrium stationary distributions of interacting many-particle systems that are coupled to different heat baths.

A Hamiltonian-based model of many harmonically interacting massive particles that are subject to linear friction and coupled to heat baths at different temperatures is used to study the dynamic approach to equilibrium and nonequilibrium stationary states. An equilibrium system is here defined as a system whose stationary distribution equals the Boltzmann distribution, the relation of this definition to the conditions of detailed balance and vanishing probability current is discussed both for underdamped as well as for overdamped systems. Based on the exactly calculated dynamic approach to the stationary distribution, the functional that governs this approach, which is called the free entropy S_{free}(t), is constructed. For the stationary distribution S_{free}(t) becomes maximal and its time derivative, the free entropy production S[over ̇]_{free}(t), is minimal and vanishes. Thus, S_{free}(t) characterizes equilibrium as well as nonequilibrium stationary distributions by their extremal and stability properties. For an equilibrium system, i.e., if all heat baths have the same temperature, the free entropy equals the negative free energy divided by temperature and thus corresponds to the Massieu function which was previously introduced in an alternative formulation of statistical mechanics. Using a systematic perturbative scheme for calculating velocity and position correlations in the overdamped massless limit, explicit results for few particles are presented: For two particles localization in position and momentum space is demonstrated in the nonequilibrium stationary state, indicative of a tendency to phase separate. For three elastically interacting particles heat flows from a particle coupled to a cold reservoir to a particle coupled to a warm reservoir if the third reservoir is sufficiently hot. This does not constitute a violation of the second law of thermodynamics, but rather demonstrates that a particle in such a nonequilibrium system is not characterized by an effective temperature which equals the temperature of the heat bath it is coupled to. Active particle models can be described in the same general framework, which thereby allows us to characterize their entropy production not only in the stationary state but also in the approach to the stationary nonequilibrium state. Finally, the connection to nonequilibrium thermodynamics formulations that include the reservoir entropy production is discussed.

On diffusions with stochastic resettings: noisy restarts, optimal rates and interaction modelling

In this paper we address a model of particle diffusion, introduced by Evans and Majumdar (2011 J. Phys. A: Math. Theor. 44 435001) where, at random times, at rate , a particle resets to a position drawn from a probability distribution with density . We study an invariance law for the standard error of first-passage time observed by Reuveni (2016 Phys. Rev. Lett. 116 170601) and find that, when dealing with restarts, noise around the resetting position breaches such a law. We show that the mean first-passage time is finite and can be minimized with respect to , for a rather general class of functions . Moreover, we propose a model of a particle system evolution where the particles independently diffuse and reset but also at random times interact. We exhibit the non-equilibrium stationary distribution for the case of two interacting particles.

The interplay between population genetics and diffusion with stochastic resetting

In this work we analyze the connection between the theory of diffusion with stochastic resetting and the theory of the Ohta–Kimura continuous ladder model. A first hint of this connection is readily given by the equivalence between the models’ partial differential equations for their distribution probabilities. In addition, it turns out that Fleming–Viot processes play a central role in connecting these theories. As an unfolding thereof, it is expected that relevant results obtained for one model can be translated to the other model framework. This is illustrated in this article by obtaining the mean first-passage time for the Ohta–Kimura continuous ladder model and by establishing the correlation between particles in the stochastic resetting model. We also study the non-equilibrium stationary distribution for two interacting resetting particles. In addition we make a brief excursion into another particle interaction model that does not have stationary distribution.

INVESTIGATION OF THE NON-EQUILIBRIUM IMPACT ON THE PROPERTIES OF A SUPERCONDUCTOR USING A PLASMA SUPERCONDUCTIVITY MECHANISM

The results of theoretical studies of a semiconductor structure are presented which, upon irradiation with a particle stream or light, transforms into a superconducting phase state. The work is carried out: – analysis of ion-acoustic oscillations in non-degenerate and degenerate plasma; – analysis of the possibility of using the jelly model to describe the weak coupling and strong coupling; – determination of the value of the order parameter when the equilibrium distribution is replaced by a non-equilibrium distribution function; – analysis of the dispersion properties of the medium for a non-equilibrium stationary distribution of charged particle; – determination of the critical temperature of a superconducting transition when using the plasma mechanism for describing superconductivity. The aim of research: determination of the frequency of oscillations of the "ion sound" type with sufficiently large mobility of the sites of the absence of electrons (holes); determination of the method for calculating the damping of ion-acoustic oscillations, which are determined by the imaginary part of the dielectric constant (decrement damping); numerically solution of the equation for the width of the energy gap; determination of the method of estimating the critical temperature of a superconductor in the plasma description of the process of the onset of superconductivity.

Maximum entropy prediction of non-equilibrium stationary distributions for stochastic reaction networks with oscillatory dynamics

Many chemical reaction networks in biological systems present complex oscillatory dynamics. In systems such as regulatory gene networks, cell cycle, and enzymatic processes, the number of molecules involved is often far from the thermodynamic limit. Although stochastic models based on the probabilistic approach of the Chemical Master Equation (CME) have been proposed, studies in the literature have been limited by the challenges of solving the CME and the lack of computational power to perform large-scale stochastic simulations.In this paper, we show that the infinite set of stationary moment equations describing the stochastic Brusselator and Schnakenberg oscillatory reactions networks can be truncated and solved using maximization of the entropy of the distributions. The results from our numerical experiments compare with the distributions obtained from well-established kinetic Monte Carlo methods and suggest that the accuracy of the prediction increases exponentially with the closure order chosen for the system.We conclude that maximum entropy models can be used as an efficient closure scheme alternative for moment equations to predict the non-equilibrium stationary distributions of stochastic chemical reactions with oscillatory dynamics. This prediction is accomplished without any prior knowledge of the system dynamics and without imposing any biased assumptions on the mathematical relations among species involved.

Focusing in multiwell potentials: Applications to ion channels

We investigate nonequilibrium stationary distributions induced by stochastic dichotomous noise in double-well and multiwell models of ion channel gating kinetics. The channel kinetics is analyzed using both overdamped Langevin equations and master equations. With the Langevin equation approach we show a nontrivial focusing effect due to the external stochastic noise, namely, the concentration of the probability distribution in one of the two wells of a double-well system or in one or more of the wells of the multiwell model. In the multiwell system, focusing in the outer wells is shown to be achievable under physiological conditions, while focusing in the central wells has proved possible so far only at very low temperatures. We also discuss the strength of the focusing effect and obtain the conditions necessary for maximal focusing to appear. These conditions cannot be predicted by a simple master equation approach.

Reconstructing Free Energy Profiles from Nonequilibrium Relaxation Trajectories

Reconstructing free energy profiles is an important problem in bimolecular reactions, protein folding or allosteric conformational changes. Nonequilibrium trajectories are readily measured experimentally, but their statistical significance and relation to equilibrium system properties still call for rigorous methods of assessment and interpretation. Here we introduce methods to compute the equilibrium free energy profile of a given variable from a set of short nonequilibrium trajectories, obtained by externally driving a system out of equilibrium and subsequently observing its relaxation. This protocol is not suitable for the Jarzynski equality since the irreversible work on the system is instantaneous. Assuming that the variable of interest satisfies an overdamped Langevin equation, which is frequently used for modeling biomolecular processes, we show that the trajectories sample a nonequilibrium stationary distribution that can be calculated in closed form. This allows for the estimation of the free energy via an inversion procedure that is analogous to that used in equilibrium and bypasses more complicated path integral methods, which we derive for comparison. We generalize the inversion procedure to systems with a diffusion constant that depends on the reaction coordinate, as is the case in protein folding, as well as to protocols in which the trajectories are initiated at random points. Using only a statistical pool of tens of synthetic trajectories, we demonstrate the versatility of these methods by reconstructing double and multi-well potentials, as well as a proposed profile for the hydrophobic collapse of a protein.

Rigorous meaning of McLennan ensembles

We analyze the exact meaning of expressions for nonequilibrium stationary distributions in terms of entropy changes. They were originally introduced by McLennan [“Statistical mechanics of the steady state,” Phys. Rev. 115, 1405 (1959)] for mechanical systems close to equilibrium and more recent work by Komatsu and Nakagawa [“An expression for stationary distribution in nonequilibrium steady states,” Phys. Rev. Lett. 100, 030601 (2008)] has shown their intimate relation to the transient fluctuation symmetry. Here we derive these distributions for jump and diffusion Markov processes and we clarify the order of the limits that take the system both to its stationary regime and to the close-to-equilibrium regime. In particular, we prove that it is exactly the (finite) transient component of the irreversible part of the entropy flux that corrects the Boltzmann distribution to first order in the driving. We add further connections with the notion of local equilibrium, with the Green–Kubo relation, and with a generalized expression for the stationary distribution in terms of a reference equilibrium process.

Stochastic approach to molecular interactions and computational theory of metabolic and genetic regulations

The underlying molecular mechanisms of metabolic and genetic regulations are computationally identical and can be described by a finite state Markov process. We establish a common computational model for both regulations based on the stationary distribution of the Markov process with the aim of establishing a unified, quantitative model of general biological regulations. Various existing results regarding intracellular regulations are derived including the classical Michaelis–Menten equation and its generalization to more complex allosteric enzymes in a systematic way. The notion of probability flow is introduced to distinguish the equilibrium stationary distribution from the non-equilibrium one; it plays a crucial role in the analysis of stationary state equations. A graphical criterion to guarantee the existence of an equilibrium stationary distribution is derived, which turns out to be identical to the classical Wegscheider condition. Simple graphical methods to compute the equilibrium and non-equilibrium stationary distributions are derived based crucially on the probability flow, which dramatically simplifies the classical methods still used in enzymology.

Photo-Hall effect measurements in P, N and B-doped diamond at low temperatures

Resistivity and Hall effect measurements are applied to understand the electronic properties of homoepitaxially grown n- and p-type CVD diamond in the wide temperature range 10–900 K. To overcome the high dark resistivity at low temperature, carrier concentration is enhanced by photo-excitation. It is found that the lifetime of photo-excited carriers decreases significantly at low temperature whereas their mobility increases. It is suggested that the short lifetime and the continuous photo-excitation can lead to the non-equilibrium stationary distribution of the photo-excited carriers at low lattice temperature. The enhanced average kinetic energy of the photo-excited carriers can effectively reduce ionized impurity scattering hence increasing the carrier mobility.

Gas-dynamic shock waves in a relaxing gas

Shock waves in the rarefied gas of anharmonic oscillators are considered. This paper studies the nonequilibrium stationary distributions of molecules in vibrational levels which are realized behind the direct shock wave under various physical conditions.

Plasma Kinetic Theory of Transverse Diffusion

The kinetic theory of Prigogine and Balescu is applied to the diffusion of a plasma column across a uniform magnetic field. The problem of deriving the kinetic equation for the single-particle distribution function is reduced to that of solving a linear integral equation with a complicated kernel. No restriction is made on the amplitude or length scale of the inhomogeneity across the magnetic field. Although a multicomponent plasma is considered, the calculation ignores macroscopic electric fields. An unusual linearization of the Vlasov equation is also examined. It yields essentially the same linear integral equation, where the kernal represents a generalized electric susceptibility. The problem of finding the unstable modes of oscillation is the eigenvalue problem for the corresponding homogeneous integral equation. If the collective terms are replaced by appropriate cutoffs in the integrations, the kinetic theory becomes relatively simple. In particular, a single kinetic equation describes the collisional transverse diffusion. For a gyrotropic plasma with slab geometry, this kinetic equation reduces to one previously obtained by Eleonskiǐ, Zyryanov, and Silin by a different method; an H-theorem has been established, and some nonequilibrium stationary distributions have been found.