Fluctuation Theorem and Chaos

The heat theorem (i.e. the second law of thermodynamics or the existence of entropy) is a manifestation of a general property of hamiltonian mechanics and of the ergodic Hypothesis. In nonequilibrium thermodynamics of stationary states the chaotic hypothesis plays a similar role: it allows a unique determination of the probability distribution (called SRB distribution on phase space providing the time averages of the observables. It also implies an expression for a few averages concrete enough to derive consequences of symmetry properties like the fluctuation theorem or to formulate a theory of coarse graining unifying the foundations of equilibrium and of nonequilibrium. 1 Boltzmann’s Heat Theorem In equilibrium statistical mechanics states are identificed with time invariant probability distributions μ on phase space. Thermodynamic functions, identified with time averages of mechanical observables, are expressed as integrals 〈F 〉 of suitable mechanical obeservables F . The averages depend on control parameters α, like volume, energy, kinetic energy. Under changes b dα of the control parameters the thermodynamic quantities change so that the variation of the average energy and the variation of the volume are dU and dV and are related to the time averages of the kinetic energy 〈K〉 def = 〈 ∑N i=1 mẋi 2 〉 def = 3 2NkBT and of p def = 〈−∂V U〉, with U being the total potential energy, so that, expressing p, T in terms of the control parameters: Heat Theorem: (HT) Changing α → α+dα induce changes dU , dV , with