Dynamical systems steady state fluctuations

The fluctuation theorem deals with fluctuations. Since the statistics of fluctuations will be different in different statistical ensembles, the fluctuation theorem is a set of closely related theorems. Also some theorems consider nonequilibrium steady-state fluctuations, while others consider transient fluctuations. One of the fluctuation theorems state that in a time-reversible dynamic system in contact with constant temperature heat bath, the fluctuations in the time-averaged irreversible entropy productions in a nonequilibrium steady state satisfy Eqn (15.49) (Evans and Searles, 2002). 

In Chapter 3 the steady-state hydrodynamic aspects of two-phase flow were discussed and reference was made to their potential for instabilities. The instability of a system may be either static or dynamic. A flow is subject to a static instability if, when the flow conditions change by a small step from the original steady-state ones, another steady state is not possible in the vicinity of the original state. The cause of the phenomenon lies in the steady-state laws hence, the threshold of the instability can be predicted only by using steady-state laws. A static instability can lead either to a different steady-state condition or to a periodic behavior (Boure et al., 1973). A flow is subject to a dynamic instability when the inertia and other feedback effects have an essential part in the process. The system behaves like a servomechanism, and knowledge of the steady-state laws is not sufficient even for the threshold prediction. The steady-state may be a solution of the equations of the system, but is not the only solution. The above-mentioned fluctuations in a steady flow may be sufficient to start the instability. Three conditions are required for a system to possess a potential for oscillating instabilities ... 

At this point, our notion and implications of the term stability must be clarified. At the most basic level, and as utilized in Section VILA, dynamic stability implies that the system returns to its steady state after a small perturbation. More quantitatively, increased stability can be associated with a decreased amount of time required to return to the steady state as for example, quantified by the largest real part within the spectrum of eigenvalues. However, obviously, stability does not imply the absence of variability in metabolite concentrations. In the face of constantperturbations, the concentration and flux values will fluctuate around their... 

It is most remarkable that the entropy production in a nonequilibrium steady state is directly related to the time asymmetry in the dynamical randomness of nonequilibrium fluctuations. The entropy production turns out to be the difference in the amounts of temporal disorder between the backward and forward paths or histories. In nonequilibrium steady states, the temporal disorder of the time reversals is larger than the temporal disorder h of the paths themselves. This is expressed by the principle of temporal ordering, according to which the typical paths are more ordered than their corresponding time reversals in nonequilibrium steady states. This principle is proved with nonequilibrium statistical mechanics and is a corollary of the second law of thermodynamics. Temporal ordering is possible out of equilibrium because of the increase of spatial disorder. There is thus no contradiction with Boltzmann s interpretation of the second law. Contrary to Boltzmann s interpretation, which deals with disorder in space at a fixed time, the principle of temporal ordering is concerned by order or disorder along the time axis, in the sequence of pictures of the nonequilibrium process filmed as a movie. The emphasis of the dynamical aspects is a recent trend that finds its roots in Shannon s information theory and modem dynamical systems theory. This can explain why we had to wait the last decade before these dynamical aspects of the second law were discovered. 

For most macroscopic dynamic systems, the neglect of correlations and fluctuations is a legitimate approximation [383]. For these cases the deterministic and stochastic approaches are essentially equivalent, and one is free to use whichever approach turns out to be more convenient or efficient. If an analytical solution is required, then the deterministic approach will always be much easier than the stochastic approach. For systems that are driven to conditions of instability, correlations and fluctuations will give rise to transitions between nonequihbrium steady states and the usual deterministic approach is incapable of accurately describing the time behavior. On the other hand, the stochastic simulation algorithm is directly applicable to these studies. 

Quite different behavior is exhibited by the topologically constrained model, as illustrated by Fig. 89, which shows the dynamic evolution of the topologically constrained model out to 12,000 Monte Carlo sweeps, for r = A, r=1.2 (at this temperature, the order parameter steady-state structure develops rather quickly in this version of the model, and large-amplitude density fluctuations such as those observed in the topologically unconstrained model are absent. 

In dynamic situations forces are not balanced at every instant one special case of dynamic situations is the steady state, in which forces are constant but they are not balanced by opposing forces. In addition, dynamic situations may include states at equilibrium. In dynamic equilibria forces fluctuate at every instant, but the forces are balanced when they are averaged over finite durations and finite parts of the system. The relevant time and length scales may or may not be sensible or important to an observer. Moreover, these scales can differ substantially for systems in different phases of aggregation for example, property fluctuations in solids are typically orders of magnitude smaller than those in fluids. 

Studies of the dynamics of a system generally pay little attention to transient behavior and focus instead on the ultimate fate of the system, the asymptotic state it attains as time goes to infinity. These can be stationary states, time-dependent periodic states, or time-dependent aperiodic or chaotic states. Our interest here is the former, the stationary or steady states and their stability properties. Small fluctuations or perturbation are inevitable in any real system. If the system is in a steady state and then experiences a small perturbation, will it return to the steady state In other words, is the state stable against small perturbations ... 

A third method for measuring spectral dynamics of individual molecules in glasses involves fluorescence intensity fluctuations during steady-state excitation at a fixed frequency. [20] This method has been applied to the systems of Tr in PE [18,20] and TBT in PIB [15, 16]. In these experiments the fluorescence intensity fluctuates as the chromophore moves in and out of resonance because of coupling to flipping TLSs. Thus this very clever technique can provide a direct probe of TLS dynamics on all time scales. 

In simulations in which the crystals were allowed to fluctuate without external pressure, the systems quickly reached a steady-state, enabling us to compare dynamic and structural behavior with experimental results. After about 4 ps the radial distribution functions were not dependent on time and became indistinguishable among different initial structures, which means that all three... 

A steady state into which all initial states of the system will evolve for t—>00 is called globally and asymptotically stable. Such a situation is a particularly simple form of the stability behaviour of a dynamical system. It includes cases where the steady state is not unique but there are multiple steady states among which, however, only one is globally and asymptotically stable. The other steady states are then instable any arbitrarily small fluctuation drives the system out of them into the stable state. 

The results shown in Fig. 4 indicate that the photopigment is able to induce increases in membrane conductance upon illumination, once the steady state is reached in the dark. The mechanism underlying this phenomenon seems to be related to the ability of the system to form dynamic channelsSuch a behavior offers ground for speculations on possible mechanisms of photoinduced electrical excitation in photoreceptors. The discrepancy arises from the comparison of the time lag between the onset of light and the appearance of the current fluctuations and the delay of the physiological response in photoreceptors. 

While there are only small thermal fluctuations in equilibrium, shear can enlarge them on small spatial scales (> the mesh size I in the following simulations) in an early stage t < yxo). In later times the fluctuations on various spatial scales appear and the system tends to a strongly fluctuating, dynamical steady state. Hence, random-source terms in the dynamic equations are indispensable at the onset. In our simulations we have therefore added the Gaussian random source terms,/y(r, t) in (3), V ffR(r, t) in (8) or (12), and 0(r, t) in (13) or (15), which are related to l/ j, rjQ, and C, respectively, to satisfy the fluctuation-dissipation relations. In our model, however, even after the dynamic equations are made dimensionless, these random source terms are still proportional to a common parameter e = which has not yet been specified, d be-... 

When a nonlinear system ewolwes under far-from-equilibrium conditions in the vicinity of a bifurcation point, a purely deterministic description often proved to be incomplete. The fluctuations of the dynamical variables can play an essential role and obstruct the observation of a transition expected by a deterministic analysis. In the framework of the deterministic approach, the stability of the different states according to the values of the control parameters is studied through a mathematical analysis of the velocity field. In particular, the theory of normal forms leads to the determination of the various kinds of attractors [l,2]. As far as we are concerned with the stochastic approach, the rrLa te.n. equation, has been widely used to analyze bifurcations of homogeneous or spatially ordered steady states or of limit cycles [3,4]. Our aim in the present contribution is to insist on the generality of the method to analyze various kinds of bifurcations in nonlinear nonequilibrium systems. The general procedure proposed to obtain a local description of the probability, which allows us to determine the system s attractors, turns out to display marked analogies with the theory of normal forms. 

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