Equilibrium step fluctuations

STEP FLUCTUATIONS FROM EQUILIBRIUM ANALYSIS TO STEP UNBUNCHING AND CLUSTER DIFFUSION IN A UNIFIED PICTURE... 

The relaxation of isolated, pairs of and ensembles of steps on crystal surfaces towards equilibrium is reviewed, for systems both above and below the roughening transition temperature. Results of Monte Carlo simulations are discussed, together with analytic theories and experimental findings. Elementary dynaniical processes are, below roughening, step fluctuations, step-step repulsion and annihilation of steps. Evaporation kinetics arid surface diffusion are considered. 

As a general conclusion of this section, the dynamical complexity of the active site in an enzymatic reaction can be satisfactorily considered within the framework of TST, either including protein vibrational motions in the definition of the reaction coordinate or treating them as equilibrium thermal fluctuations and incorporating a recrossing transmission coefiicient into the formalism. None of the reactions analysed until now has shown conclusive evidences of a severe failure of the equilibrium approximation. Such a behaviour could be expected for chemical reactions with a very low firee energy barrier (few kcal moF ), an improbable observation as far as there is no any evolutive pressure to optimize the chemical step in enzymes below the rate of the diffusion process. 

The current frontiers for the subject of non-equilibrium thennodynamics are rich and active. Two areas dommate interest non-linear effects and molecular bioenergetics. The linearization step used in the near equilibrium regime is inappropriate far from equilibrium. Progress with a microscopic kinetic theory [38] for non-linear fluctuation phenomena has been made. Carefiil experiments [39] confinn this theory. Non-equilibrium long range correlations play an important role in some of the light scattering effects in fluids in far from equilibrium states [38, 39]. 

There are tliree steps in the calculation first, solve the frill nonlinear set of hydrodynamic equations in the steady state, where the time derivatives of all quantities are zero second, linearize about the steady-state solutions third, postulate a non-equilibrium ensemble through a generalized fluctuation dissipation relation. 

Typical runs consist of 100 000 up to 300 000 MC moves per lattice site. Far from the phase transition in the lamellar phase, the typical equilibration run takes 10 000 Monte Carlo steps per site (MCS). In the vicinity of the phase transitions the equilibration takes up to 200 000 MCS. For the rough estimate of the equihbration time one can monitor internal energy as well as the Euler characteristic. The equilibration time for the energy and Euler characteristic are roughly the same. For go = /o = 0 it takes 10 000 MCS to obtain the equilibrium configuration in which one finds the lamellar phase without passages and consequently the Euler characteristic is zero. For go = —3.15 and/o = 0 (close to the phase transition) it takes more than 50 000 MCS for the equihbration and here the Euler characteristic fluctuates around its mean value of —48. 

In parallel with the studies described above, which concern perfectly deterministic equations of evolution, it appeared necessary to complete the theory by studying the spontaneous fluctuations. Near equilibrium, any deviation is rapidly damped but near a bifurcation point, a fluctuation may may lead the system across the barrier. The fluctuation is then stabilized, or even amplified this is the origin of the phenomenon which Prigogine liked calling creation of order through fluctuations. More specifically, one witnesses in this way a step toward self-organization. 

For an individual molecule, fluctuations of the instantaneous electronic charge density away from its quantum mechanical average are characterized by the fluctuation-dissipation theorem (3, 4). The molecule is assumed to be in equilibrium with a radiation bath at temperature T then in the final step of the derivation, the limit is taken as T — 0. The fluctuation correlations, which are defined by... 

Within non-equilibrium thermodynamics, the driving force for relaxation is provided by deviations in the local chemical potential from it s equilibrium value. The rate at which such deviations relax is determined by the dominant kinetics in the physical system of interest. In addition, the thermal noise in the system randomly generates fluctuations. We thus describe the dynamics of a step edge by the equation. 

The contact rate P depends on the probability that the outside step will elongate, by a fluctuation, across the terrace to meet a step of opposite sign. To analyze this probability, we use the known result of the following equilibrium problem a single step made up of X and y links of size a (atomic distance) and energy Wl, in a semi-infinite space x>0, depicted in Fig. 3, is attracted to a hne at x=0 with abinding energy Wo per link. 

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