Fluctuation Theorem - Dissipation Function

Dissipation Function

Strictly speaking the fluctuation theorem refers to a quantity known as the dissipation function. In thermostatted nonequilibrium states that are close to equilibrium, the long time average of the dissipation function is equal to the average entropy production. However the FT refers to fluctuations rather than averages. The dissipation function is defined as,

where k is Boltzmann's constant, is the initial (t = 0) distribution of molecular states, and is the molecular state arrived at after time t, under the exact time reversible equations of motion. is the INITIAL distribution of those time evolved states.

Note: in order for the FT to be valid we require that . This condition is known as the condition of ergodic consistency. It is widely satisfied in common statistical ensembles - e.g. the canonical ensemble.

The system may be in contact with a large heat reservoir in order to thermostat the system of interest. If this is the case is the heat lost to the reservoir over the time (0,t) and T is the absolute equilibrium temperature of the reservoir - see Williams et al., Phys Rev E70, 066113(2004). With this definition of the dissipation function the precise statement of the FT simply replaces entropy production with the dissipation function in each of the FT equations above.

Example: If one considers electrical conduction across an electrical resistor in contact with a large heat reservoir at temperature T, then the dissipation function is

the total electric current density J multiplied by the voltage drop across the circuit, and the system volume V, divided by the absolute temperature T, of the heat reservoir times Boltzmann's constant. Thus the dissipation function is easily recognised as the Ohmic work done on the system divided by the temperature of the reservoir. Close to equilibrium the long time average of this quantity is (to leading order in the voltage drop), equal to the average spontaneous entropy production per unit time - see de Groot and Mazur "Nonequilibrium Thermodynamics" (Dover), equation (61), page 348. However, the Fluctuation Theorem applies to systems arbitrariliy far from equilibrium where the definition of the spontaneous entropy production is problematic.