Fluctuations near instability points

At thermodynamic equilibrium fluctuations constitute a negligible correction to the macroscopic description of matter except near instability points such as phase transitions or critical points. Similarly, in nonequilibrium situations as well one expects that fluctuations become important only near points of nonequilibrium instability (see Ref. 17 for a detailed discussion). For nonlinear systems in which such instabilities and transitions can occur, the role of fluctuations is even more dramatic than at equilibrium, due to the existence in general of many distinct "phases" compatible with the external conditions imposed on the system. This fact has been known for several years now, since the first (numerical) demonstration on simple chemical models of multiple solutions to the governing macroscopic differential equations (44j 4 The variety of possible solutions... 

We return to the bistable situation with clearly separated time scales and suppose that the system starts out at a site near the boundary of Da, i.e., close to the macroscopically unstable point (f)b. Then in the initial stage fluctuations across are not improbable. There is therefore a non-negligible probability that the system does not follow the macroscopic path towards (j>a but ends up in 4>c instead. Thus near a point of macroscopic instability fluctuations give rise to a macroscopic effect. It is therefore no longer possible to separate a macroscopic part from a fluctuation term as in (X.2.9) and treat the fluctuations as a small perturbation. The conclusion is that there exists no mesostate related to the stationary macrostate. Any probability distribution originally peaked near (j)b evolves in time and does not remain localized. The evolution occurs in three successive stages. 

The system operates near an instability point of a deterministic model. In this case small fluctuations may be amplified and produce observable, even macroscopic effects. It may also happen that the deterministic model of a system is structurally stable while the stochastic model is not, or vice versa. 

The analysis of the behaviour of the Lotka-Volterra model leads us to investigate fluctuations near an instability point, and we return to this point in Section 5.6. 

Fig. 5.10 Anomalous fluctuation near the instability point I t) denotes the most probable path or deterministic motion y t) and D [y t)] is the variance of the order parameter.

Suzuki, M. (1976b). Scaling theory of nonequilibrium systems near the instability point. II. Anomalous fluctuation theorems in the extensive region. Progr. Theor. Phys., 56, 477-... 

These model dissipative structures demonstrate that the capacity of biological systems to store information, transmit information, and regulate chemical processes finds its primitive counterparts in such models so long as they are based upon nonlinearity and removal from thermodynamic equilibrium. In such systems, fluctuations play a crucial role near the point of instability. Thus, the intuitive concept that life is tenuous may be related, on physical-chemical grounds, to the importance of the generation of instabilities far from equilibrium. 

The linear instability theory of the behavior of a system near the bifurcation point can be successfully applied to many self-organization problems, such as thermal convection in hydrodynamics4 and crystal growth in solution.5 In these theories, various initial fluctuations play important roles. Occasionally the fluctuations arise from the thermal motion of atoms or molecules. If a system reaches an unstable mode over... 

In 1977. Professor Ilya Prigogine of the Free University of Brussels. Belgium, was awarded Ihe Nobel Prize in chemistry for his central role in the advances made in irreversible thermodynamics over the last ihrec decades. Prigogine and his associates investigated Ihe properties of systems far from equilibrium where a variety of phenomena exist that are not possible near or al equilibrium. These include chemical systems with multiple stationary states, chemical hysteresis, nucleation processes which give rise to transitions between multiple stationary states, oscillatory systems, the formation of stable and oscillatory macroscopic spatial structures, chemical waves, and Lhe critical behavior of fluctuations. As pointed out by I. Procaccia and J. Ross (Science. 198, 716—717, 1977). the central question concerns Ihe conditions of instability of the thermodynamic branch. The theory of stability of ordinary differential equations is well established. The problem that confronted Prigogine and his collaborators was to develop a thermodynamic theory of stability that spans the whole range of equilibrium and nonequilibrium phenomena. 

Equations (8) and (18) could allow, at least in principle, the determination of the values of r and <]> near which the spherical interface of the globules is no longer stable to thermal perturbations. Near such a point, fluctuations of the interface will occur and, as mentioned in the Introduction, this can explain the ill defined Interfaces detected experimentally (ref. 13) by the Fourier transform NMR method. No expressions are as yet available for y and C, to calculate the values of r and at which the spherical interface becomes unstable. However, inverting the problem, Eqs. (8) and (18) could provide some information about the values of y and C which are needed for the instability of the spherical shape of the globules to arise. From the above equations one easily obtains r... 

For instability appearance analysis it is essential not only to develop the main equations of transfer but to formulate boundary conditions, because dissipative structures appear more frequently near phase boundaries and may be caused by fluctuations of boundary conditions. It is of great practical interest to investigate the role of capillary [5] and electrosurface [6] forces in inducing the instability of mobile phase boundaries. That is the reason for determining boundary conditions for the transfer equation system characterizing the interface of two mobile media (liquid-gas or liquid-liquid). Let be the radius-vector drawn from any fixed space point to another on the phase boundary and n a unit vector normal to the interface and directed towards the interior of phase 1 (the second phase would be written as phase 2). The tangential unit vector at the interfacial surface would be t. Then conjugation conditions for velocities, temperatures, concentrations and electric fields at the interface would be ... 

In (ll.l), the summation is restricted to wavenumbers close to q<3. In the case of the R.B instability, the quadratic term is due to non-Boussinesq effects (e.g. temperature dependent transport coefficients) a similar term always appears in the case of the Marangoni or the Turing instability. When these equations have a gradient structure and this property is sometimes satisfied near the bifurcation point, one can define a Lyapunov functional that decreases in any dynamics. In that case if one describes local fluctuations by a gaussian white noise term then the (T.D.G.L.) equation can be written as ... 

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