Detailed fluctuation theorem

The probability of the second law violations diminishes in a large system and occurs exponentially rarely. This observation essentially reconciles the effective validity of thermodynamics on the macroscale and states that even for large systems, in principle, such violations must occur in even macroscales. Detailed fluctuation theorems corresponding to... 

At nonequilibrium steady state, a net flux in the species occurs if it is possible to adjust the concentrations. Hence, the stationary state violates the detailed balance condition PnJnm = PmJmn where J is the rate of transformation and p is the probability. For such nonequilibrium steady states, a detailed fluctuation theorem is... 

Equation (3.12) is an identity that does not depend on the details of the kinetic reaction mechanism that is operating in a particular system [19], We [19] have shown that Equation (3.12) is intimately related to the Crooks fluctuation theorem [41] - an important result in non-equilibrium statistical thermodynamics - as well as to theories developed by Hill [87, 90], Ussing [201], and Hodgkin and Huxley [95],... 

Equation (64) expresses the condition of detailed balance which is at the root of the fluctuation dissipation theorem. 

By referring to Eqs.(3) and (41, the imaginary part of the Fourier transform of the OHD-OKE response divided by the Fourier transform of the lAF of the laser pulse is directly comparable to the LS spectrum divided by the Bose factor in addition to Because the dynamical behavior and the fluctuation ought to be correlated with each other through the fluctuation-dissipation theorem, this comparison can verify that the information provided by the two experiments are identical. To our knowledge, however, the direct comparison has not been made yet. In this report, we will describe the details of our experiments and thier results on this problem. 

We have thus seen that the requirement that the friction y and the random force 7 (Z) together act to bring the system to thennal equilibrium at long time, naturally leads to a relation between them, expressed by Eq. (8.20). This is a relation between fluctuations and dissipation in the system, which constitutes an example of the fluctuation-dissipation theorem (see also Chapter 11). In effect, the requirement that Eq. (8.20) holds is equivalent to the condition of detailed balance, imposed on transition rates in models described by master equations, in order to satisfy the requirement that thermal equilibrium is reached at long time (see Section 8.3). 

In this chapter, we developed a stochastic theory of single molecule fluorescence spectroscopy. Fluctuations described by Q are evaluated in terms of a three-time correlation function C iXi, X2, T3) related to the response function in nonlinear spectroscopy. This function depends on the characteristics of the spectral diffusion process. Important time-ordering properties of the three-time correlation function were investigated here in detail. Since the fluctuations (i.e., Q) depend on the three-time correlation function, necessarily they contain more information than the line shape that depends on the one-time correlation function Ci(ti) via the Wiener-Khintchine theorem. 

As was shown in Ref [42], the single component FPM system yields the Gibbs distribution as the steady-state solution to the Fokker-Planck equation under the condition of detailed balance. Consequently, it obeys the fluctuation-dissipation theorem, which defines the relationship between the normalized weight functions, which are chosen such that... 

This method can be applied in all cases. The Fourier transform procedure is included commonly in mathematical software products and is easily available for everyone. The drawbacks of this procedure lie in its laborious course and abstract nature because the calculations are performed in Fourier space. Those who lack experience in numerical Fourier transforms are advised to study some pitfalls such as the break-off effect and the sampling theorem, both obtained by numerical treatment. Under specific conditions, this simulates periodicities and fluctuations that do not reflect any actual processes in the sample. Furthermore, the signal-to-noise ratio turns worse for further details, readers are referred to the literature (Bracewell, 2000 Davies, 2002). 

In liquids and dense gases where collisions, intramolecular molecular motions and energy relaxation occur on the picosecond timescales, spectroscopic lineshape studies in the frequency domain were for a long time the principle source of dynamical information on the equilibrium state of manybody systems. These interpretations were based on the scattering of incident radiation as a consequence of molecular motion such as vibration, rotation and translation. Spectroscopic lineshape analyses were intepreted through arguments based on the fluctuation-dissipation theorem and linear response theory (9,10). In generating details of the dynamics of molecules, this approach relies on FT techniques, but the statistical physics depends on the fact that the radiation probe is only weakly coupled to the system. If the pertubation does not disturb the system from its equilibrium properties, then linear response theory allows one to evaluate the response in terms of the time correlation functions (TCF) of the equilibrium state. Since each spectroscopic technique probes the expectation value... 

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