Fluctuation-dissipation theorem of linear nonequilibrium thermodynamics

2 Fluctuation-dissipation theorem of linear nonequilibrium thermodynamics 

Since chemical reaction is considered as a stochastic process, and furthermore as a thermodynamic process, it is a natural question to ask what are the counterparts of the statements of the fluctuation theory of nonequilibrium thermodynamics. In the theory of thermodynamics the fluctuation-dissipation theorem is associated with the observation that the dissipative process leading to equilibrium is connected with fluctuations around that equilibrium. This fact was pointed out in a particular case (related to Brownian movement) by Einstein. Different representations of the theorem exist for linear thermodynamic processes (Callen Welton, 1951 Greene Callen, 1951 Kubo 1957 Lax, 1960 van Vliet Fasset, 1965 van Kampen, 1965.) 

The time representation gives the relationship among the velocities of conditional moments. For linear thermodynamic processes (which are associated with diffusion processes, i.e. D x, t) is nonzero for w = 1,2 only) we get 

The time representation can be converted to a spectral representation by the Wiener-Khintchine relation. A fundamental property of stochastic processes a(/) that are stationary in the broad sense is that they can be characterised by the time correlation function  

The time correlation function expresses the effect of the value of the process at t for values at / H- x. Characteristic properties of C when [a] = 0  

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