Fluctuations and stability of stationary non-equilibrium systems in detailed balance

A generalized thermodynamic potential for Markoffian systems with detailed balance and far from thermal equilibrium has been derived in a previous paper. It was shown that the principle of detailed balance is equivalent to a set of conditions fulfilled by this potential (“potential conditions”). The properties of this potential allow us to extend the validity of a number of thermodynamic concepts well known for systems in or near thermal equilibrium to stationary states far from thermal equilibrium. The concept of symmetry breaking phase transitions for these systems is introduced in analogy to thermal equilibrium systems by considering the dependence of the stationary probability density of the system on a set of externally controlled parameters {λ}. A functional of the time dependent probability density of the system is defined in close analogy to the Gibb's definition of entropy. This functional has the properties of a Ljapunov functional of the governing Fokker-Planck equation showing the stability of the stationary probability density. The Langevin equations connected with the Fokker-Planck equation are considered. It is shown that, by means of the potential conditions, generalized “thermodynamic” fluxes and forces may be defined in such a way that the smoothly varying part of the Langevin equations (kinetic equations) constitutes a linear relation between fluxes and forces. The matrix of coefficients is given by the diffusion matrix of the Fokker-Planck equation. The symmetry relations which hold for this matrix due to the potential conditions then lead to the Onsager-Casimir symmetry relations extended to systems with detailed balance near stationary states far from thermal equilibrium. Finally it is shown that under certain additional assumptions the generalized thermodynamic potential may be used as a Ljapunov function of the kinetic equations.     

Exact Stationary Solution of Fokker-Planck Equation and Generalized Potential for Non-Equilibrium Systems Without Detailed Balance

An exact analogy is approached between systems in thermal equilibrium and those far from equilibrium which can be the cases without detailed balance. The analogy is based on the requirement that a given drift in the Fokker-Planck equation can be decomposed into two parts, one of which is divergence-free and the other can be derived from a potential which is invariant along the direction of the first part. If the conditions are fulfilled the Fokker-Planck equation changes in to a standard Poisson equation. The relations of this requirement to other conditions are diecussed. As a concrete example, the stationary Fokker-Planck equation for optical bistability is solved by using"this method.     

Quantum fluctuations,master equation and Fokker-Planck equation

In order to describe quantum fluctuations a general method is developed, which also may be applied to nonstationary systems as well as to states far from thermodynamic equilibrium. After a concise derivation of the master equation quantum mechanically determined dissipation and fluctuation coefficients are introduced, for which several theorems and relations are given. By using these coefficients there is set up a general Fokker-Planck equation for the diffusion of the statistical operator due to quantum fluctuations.     

Fokker-Planck equation approach to fluctuations about nonequilibrium steady states

We examine the properties of steady states in systems which interact at the boundary with a nonequilibrium environment. The examination is based on a nonlinear Fokker-Planck equation, the structure of which is determined by the fact that it also governs the time evolution of the equilibrium fluctuations of the system. The nonlinearities in the Fokker-Planck equation may have two origins: thermodynamic nonlinearities which arise if the thermodynamic potential is not a bilinear function of the state variables, and nonlinear mode coupling which arises if the transport coefficients depend on the state. While these nonlinearities have only a small effect on the equilibrium fluctuations of a system away from critical points, they are shown to be important for the determination of fluctuations about nonequilibrium steady states. In particular the state dependence of the transport coefficients may lead to deviations from local equilibrium and to a breakdown of detail balance. An explicit formula for the time correlations of fluctuations about the nonequilibrium steady state is obtained. The formula leads to long-range correlations in fluids in the presence of a temperature gradient. The result is compared with earlier approaches to the same problem. Finally, we study the linear response to external forces and obtain a generalization of the fluctuation-dissipation formula relating the response functions with the nonequilibrium correlation functions.     

No-pumping theorem for non-Arrhenius rates

The no-pumping theorem refers to a Markov system that holds the detailed balance, but is subject to a time-periodic external field. It states that the time-averaged probability currents nullify in the steady periodic (Floquet) state, provided that the Markov system holds the Arrhenius transition rates. This makes an analogy between features of steady periodic and equilibrium states, because in the latter situation all probability currents vanish explicitly. However, the assumption on the Arrhenius rates is fairly specific, and it need not be met in applications. Here a new mechanism is identified for the no-pumping theorem, which holds for symmetric time-periodic external fields and the so called destination rates. These rates are the ones that lead to the locally equilibrium form of the master equation, where dissipative effects are proportional to the difference between the actual probability and the equilibrium (Gibbsian) one. The mechanism also leads to an approximate no-pumping theorem for the Fokker-Planck rates that relate to the discrete-space Fokker-Planck equation.     

Irreversibility and fluctuation theorem in stationary time series

The relative entropy between the joint probability distribution of backward and forward sequences is used to quantify time asymmetry (or irreversibility) for stationary time series. The parallel with the thermodynamic theory of nonequilibrium steady states allows us to link the degree of asymmetry in the time signal with the distance from equilibrium and the lack of detailed balance among its states. We study the statistics of time asymmetry in terms of the fluctuation theorem, showing that this type of relationship derives from simple general symmetries valid for any stationary time series.     

Statistical mechanics of nonlinear hydrodynamic fluctuations

The paper studies nonlinear hydrodynamic fluctuations by the methods of nonequilibrium statistical mechanics. The generalized Fokker-Planck equation for the distribution function of coarse-grained densities of conserved quantities is derived from the Liouville equation and then is investigated by using the gradient expansions in the flux correlation matrix. We have obtained the functional-differential Fokker-Planck equation describing the nonlinear hydrodynamic fluctuations in spatially nonuniform systems to second order in gradients of coarse-grained fluctuating fields. An outline of the derivation of Fokker-Planck equations containing the Burnett terms is also given. The explicit coordinate representation for the hydrodynamic Fokker-Planck equation is discussed in the case of one-component simple fluid. The general scheme of a change of coarse-grained functional variables is developed for hydrodynamic Fokker-Planck equations. The corresponding transformation rules are found for “drift” terms, “diffusion coefficients” and thermodynamic forces. The dynamical equations and stationary conditions for averages of functions (functionals) of hydrodynamic fields are discussed by using the Fokker-Planck operators acting on such functions. The explicit form of these operators are found for various sets of fluctuating fields. As an application of the formalism the calculation of the stationary correlation functions is presented for a simple nonequilibrium steady state.     

Some properties of stochastic equations for coupled chemical reactions far from equilibrium

Solutions of master equations for coupled chemical reactions far from equilibrium with one varying molecule species are studied and used for getting information about nonlinear Fokker-Planck equations and slow time-dependent processes such as extinction to an absorbing state and transition between several steady states. The Fokker-Planck equation solution is compared to that of the master equation in a relative sense and it is shown that they agree quite well in some important situations but that in general the cases can deviate considerably, when, e.g., accounting for the mutual importance of two probability maxima.     

On the theory of fluctuations around non-equilibrium steady states: A generalized time-convolutionless projector formalism

A new method of finding nonlinear Langevin type equations of motion for relevant macrovariables and the corresponding master equation for systems far from thermal equilibrium is presented by generalizing the time-convolutionless formalism proposed previously for equilibrium hamiltoian systems by Tokuyama and Mori. The Langevin type equation consists of a fluctuating force, and the nonlinear drift coefficients which are always identical to those of the master equation. A simple formula which relates the drift coefficients to the time correlation of the fluctuating forces is derived. This is a generalization of the fluctuation-dissipation theorem of the second kind in equilibrium systems and is valid not only for transport phenomena due to internal fluctuations but also for transport phenomena due to externally-driven fluctuations. A new cumulant expansion of the master equation is also obtained. The conditions under which a Langevin and a Fokker-Planck equation of a generalized type for non-equilibrium open systems can be derived are clarified.The theory is illustrated by studying hydrodynamic fluctuations near the Rayleigh-Bénard instability. The effects of two kinds of fluctuations, internal fluctuations of irrelevant macrovariables and external (thermal) noises, on the convective instability are investigated. A stochastic Ginzburg-Landau type equation for the order parameter and the corresponding nonlinear Fokker-Planck equation are derived.     

A comparison of master equation and Fokker-Planck equation in the thermodynamic limit

The master equation and the Fokker-Planck equation are compared thoroughly in both of the stationary and the time-dependent cases for a one-dimensional system. The comparison includes the master equation without detailed balance.     

Generalized Fokker-Planck equation: Derivation and exact solutions

We derive the generalized Fokker-Planck equation associated with the Langevin equation (in the Ito sense) for an overdamped particle in an external potential driven by multiplicative noise with an arbitrary distribution of the increments of the noise generating process. We explicitly consider this equation for various specific types of noises, including Poisson white noise and Lévy stable noise, and show that it reproduces all Fokker-Planck equations that are known for these noises. Exact analytical, time-dependent and stationary solutions of the generalized Fokker-Planck equation are derived and analyzed in detail for the cases of a linear, a quadratic, and a tailored potential.     

Hydrodynamic synchronization of light driven microrotors

Hydrodynamic synchronization is a fundamental physical phenomenon by which self-sustained oscillators communicate through perturbations in the surrounding fluid and converge to a stable synchronized state. This is an important factor for the emergence of regular and coordinated patterns in the motions of cilia and flagella. When dealing with biological systems, however, it is always hard to disentangle internal signaling mechanisms from external purely physical couplings. We have used the combination of two-photon polymerization and holographic optical trapping to build a mesoscale model composed of chiral propellers rotated by radiation pressure. The two microrotors can be synchronized by hydrodynamic interactions alone although the relative torques have to be finely tuned. Dealing with a micron sized system we treat synchronization as a stochastic phenomenon and show that the phase lag between the two microrotors is distributed according to a stationary Fokker-Planck equation for an overdamped particle over a tilted periodic potential. Synchronized states correspond to minima in this potential whose locations are shown to depend critically on the detailed geometry of the propellers.     

Transition state theory: A generalization to nonequilibrium systems with power-law distributions

Transition state theory (TST) is generalized to nonequilibrium systems with power-law distributions. The stochastic dynamics that gives rise to the power-law distributions for the reaction coordinate and momentum is modeled by Langevin equations and corresponding Fokker-Planck equations. It is considered that a system far away from equilibrium does not have to relax to a thermal equilibrium state with Boltzmann-Gibbs distribution, but asymptotically approaches a nonequilibrium stationary state with a power-law distribution. Thus, we obtain a possible generalization of TST rates to nonequilibrium systems with power-law distributions. Furthermore, we derive the generalized TST rate constants for one-dimensional and n-dimensional Hamiltonian systems away from equilibrium, and obtain a generalized Arrhenius rate for systems with power-law distributions.     

Stable nonequilibrium probability densities and phase transitions for meanfield models in the thermodynamic limit

A nonlinear Fokker-Planck equation is derived to describe the cooperative behavior of general stochastic systems interacting via mean-field couplings, in the limit of an infinite number of such systems. Disordered systems are also considered. In the weak-noise limit; a general result yields the possibility of having bifurcations from stationary solutions of the nonlinear Fokker-Planck equation into stable time-dependent solutions. The latter are interpreted as non-equilibrium probability distributions (states), and the bifurcations to them as nonequilibrium phase transitions. In the thermodynamic limit, results for three models are given for illustrative purposes. A model of self-synchronization of nonlinear oscillators presents a Hopf bifurcation to a time-periodic probability density, which can be analyzed for any value of the noise. The effects of disorder are illustrated by a simplified version of the Sompolinsky-Zippelius model of spin-glasses. Finally, results for the Fukuyama-Lee-Fisher model of charge-density waves are given. A singular perturbation analysis shows that the depinning transition is a bifurcation problem modified by the disorder noise due to impurities. Far from the bifurcation point, the CDW is either pinned or free, obeying (to leading order) the Grüner-Zawadowki-Chaikin equation. Near the bifurcation, the disorder noise drastically modifies the pattern, giving a quenched average of the CDW current which is constant. Critical exponents are found to depend on the noise, and they are larger than Fisher's values for the two probability distributions considered.     

A new access to path integrals and Fokker Planck equations via the maximum calibre principle

Under the assumption that the underlying process is continuous Markovian and using simple short time correlation functions as constraints in the maximum calibre principle of Jaynes we derive the explicit path integrals from which then the corresponding Fokker-Planck equation may be deduced. Our approach is valid for all systems irrespective of whether they are close to or far away from thermal equilibrium and it applies even to nonphysical systems.     

Finite Larmor radius magnetohydrodynamic analysis of the ballooning modes in tokamaks

In this paper,the effect of finite Larmor radius (FLR) on high n ballooning modes is studied on the basis of FLR magnetohydrodynamic (FLR-MHD) theory.A linear FLR ballooning mode equation is derived in an ’s α’ type equilibrium of circular-flux-surfaces,which is reduced to the ideal ballooning mode equation when the FLR effect is neglected.The present model reproduces some basic features of FLR effects on ballooning mode obtained previously by kinetic ballooning mode theories.That is,the FLR introduces a real frequency into ballooning mode and has a stabilising effect on ballooning modes (e.g.,in the case of high magnetic shear s ≥ 0.8).In particular,some new properties of FLR effects on ballooning mode are discovered in the present research.Here it is found that in a high magnetic shear region (s ≥ 0.8) the critical pressure gradient (α c,FLR) of ballooning mode is larger than the ideal one (α c,IMHD) and becomes larger and larger with the increase of FLR parameter b 0.However,in a low magnetic shear region,the FLR ballooning mode is more unstable than the ideal one,and the α c,FLR is much lower than the α c,IMHD.Moreover,the present results indicate that there exist some new weaker instabilities near the second stability boundary (obtained from ideal MHD theory),which means that the second stable region becomes narrow.     

Stochastic systems with cross-correlated Gaussian white noises

This paper theoretically investigates three stochastic systems with cross-correlation Gaussian white noises.Both steady state properties of the stochastic nonlinear systems and the nonequilibrium transitions induced by the cross-correlated noises are studied.The stationary solutions of the Fokker-Planck equation for three specific examples are analysed.It is shown explicitly that the cross-correlation of white noises can induce nonequilibrium transitions.     

From the Perron-Frobenius equation to the Fokker-Planck equation

We show that for certain classes of deterministic dynamical systems the Perron-Frobenius equation reduces to the Fokker-Planck equation in an appropriate scaling limit. By perturbative expansion in a small time scale parameter, we also derive the equations that are obeyed by the first- and second-order correction terms to the Fokker-Planck limit case. In general, these equations describe non-Gaussian corrections to a Langevin dynamics due to an underlying deterministic chaotic dynamics. For double-symmetric maps, the first-order correction term turns out to satisfy a kind of inhomogeneous Fokker-Planck equation with a source term. For a special example, we are able solve the first- and second-order equations explicitly.