The fluctuation theorem (FT), the first derived consequence of the Chaotic Hypothesis (CH) of [1], can be considered as an extension to arbitrary forcing fields of the fluctuation dissipation theorem (FD) and the corresponding Onsager reciprocity (OR), in a class of reversible nonequilibrium statistical mechanical systems. 47.52.+j, 05.45.+b, 05.70.Ln, A typical system studied here will be N point particles subject to (a) mutual and external conservative forces with potential V ( q 1 , . . . , q N ), (b) external (non conservative) forces, forcing agents, { F j }, j = 1, . . . , N , whose strength is measured by parameters {G j }, j = 1, . . . , s, and (c) also to forces { ϕ j }, j = 1, . . . N , generating constraints that provides a model for the thermostatting mechanism that keeps the energy of the system from growing indefinitely (because of the continuing action of the forcing agents).An observable O({ q,˙ q}) evolves in time under the time evolution S t solving the equations of motion:with m = particles mass, and ϕ j "thermostatting" forces assuring that the system reaches a (non equilibrium) stationary state. The time evolution of the observable O on the motion with initial data x = ( q,˙ q) is the function t → O(S t x) so that the motion statistics is the probability distribution µ + on the phase space C such that:for all data x ∈ C except a set of zero measure with respect to the volume µ 0 on C. The distribution µ + is assumed to exist: a property called zero-th law, [2,3]. The thermostatting mechanism will be described by force laws ϕ j which enforce the constraint that the kinetic energy (or the total energy) of the particles, or of subgroups of the particles, remains constant, [4]. It is convenient also to imagine that the constraints keep the total kinetic energy bounded (hence phase space is bounded). The constraint forces { ϕ j } will be supposed such that the system is reversible: this means that there will be a map i, defined on phase space, anticommuting with time evolution: i.e. S t i ≡ i S −t .Reversibility is a key assumption, [1,5].In [2] the divergence of the r.h.s. of Eq. (1) is a quantity −σ(x) defined on phase space that has been identified with the entropy production rate.The chaotic hypothesis, (CH), of [1] implies a fluctuation theorem or (FT) which, [2], is a property of the fluctuations of the entropy production rate. Namely if we denote σ + = C σ(y)µ + (dy) the time average, over an infinite time interval by Eq. (2), then the dimensionless finite time average p = p(x):has a statistical distribution π τ (p) with respect to the stationary state distribution µ + such that:provided (of course) σ + > 0. Following [1] a reversible system for which σ + > 0 will be called dissipative. Ruelle's H-theorem, [5], states that σ + > 0 unless the stationary distribution µ + has the form ρ(x)µ 0 (dx).Hence we shall suppose that the system is dissipative when the forcing G does not vanish and, without real loss of generality, that σ(ix) = −σ(x) and that σ(x) ≡ 0 when the external forcing vanishes...
